DEFINITION csubst0_drop_eq()
TYPE =
       ∀n:nat
         .∀c1:C
           .∀c2:C
             .∀v:T
               .csubst0 n v c1 c2
                 →∀e:C
                      .drop n O c1 e
                        →(or4
                             drop n O c2 e
                             ex3_4
                               F
                               C
                               T
                               T
                               λf:F.λe0:C.λu:T.λ:T.eq C e (CHead e0 (Flat f) u)
                               λf:F.λe0:C.λ:T.λw:T.drop n O c2 (CHead e0 (Flat f) w)
                               λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                             ex3_4
                               F
                               C
                               C
                               T
                               λf:F.λe1:C.λ:C.λu:T.eq C e (CHead e1 (Flat f) u)
                               λf:F.λ:C.λe2:C.λu:T.drop n O c2 (CHead e2 (Flat f) u)
                               λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                             ex4_5
                               F
                               C
                               C
                               T
                               T
                               λf:F.λe1:C.λ:C.λu:T.λ:T.eq C e (CHead e1 (Flat f) u)
                               λf:F.λ:C.λe2:C.λ:T.λw:T.drop n O c2 (CHead e2 (Flat f) w)
                               λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                               λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2)
BODY =
       assume n: nat
          we proceed by induction on n to prove 
             ∀c1:C
               .∀c2:C
                 .∀v:T
                   .csubst0 n v c1 c2
                     →∀e:C
                          .drop n O c1 e
                            →(or4
                                 drop n O c2 e
                                 ex3_4
                                   F
                                   C
                                   T
                                   T
                                   λf:F.λe0:C.λu:T.λ:T.eq C e (CHead e0 (Flat f) u)
                                   λf:F.λe0:C.λ:T.λw:T.drop n O c2 (CHead e0 (Flat f) w)
                                   λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                 ex3_4
                                   F
                                   C
                                   C
                                   T
                                   λf:F.λe1:C.λ:C.λu:T.eq C e (CHead e1 (Flat f) u)
                                   λf:F.λ:C.λe2:C.λu:T.drop n O c2 (CHead e2 (Flat f) u)
                                   λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                 ex4_5
                                   F
                                   C
                                   C
                                   T
                                   T
                                   λf:F.λe1:C.λ:C.λu:T.λ:T.eq C e (CHead e1 (Flat f) u)
                                   λf:F.λ:C.λe2:C.λ:T.λw:T.drop n O c2 (CHead e2 (Flat f) w)
                                   λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                   λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2)
             case O : ⇒
                the thesis becomes 
                ∀c1:C
                  .∀c2:C
                    .∀v:T
                      .csubst0 O v c1 c2
                        →∀e:C
                             .drop O O c1 e
                               →(or4
                                    drop O O c2 e
                                    ex3_4
                                      F
                                      C
                                      T
                                      T
                                      λf:F.λe0:C.λu:T.λ:T.eq C e (CHead e0 (Flat f) u)
                                      λf:F.λe0:C.λ:T.λw:T.drop O O c2 (CHead e0 (Flat f) w)
                                      λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                    ex3_4
                                      F
                                      C
                                      C
                                      T
                                      λf:F.λe1:C.λ:C.λu:T.eq C e (CHead e1 (Flat f) u)
                                      λf:F.λ:C.λe2:C.λu:T.drop O O c2 (CHead e2 (Flat f) u)
                                      λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                    ex4_5
                                      F
                                      C
                                      C
                                      T
                                      T
                                      λf:F.λe1:C.λ:C.λu:T.λ:T.eq C e (CHead e1 (Flat f) u)
                                      λf:F.λ:C.λe2:C.λ:T.λw:T.drop O O c2 (CHead e2 (Flat f) w)
                                      λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                      λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2)
                    assume c1: C
                    assume c2: C
                    assume v: T
                    suppose H: csubst0 O v c1 c2
                    assume e: C
                    suppose H0: drop O O c1 e
                      by (drop_gen_refl . . H0)
                      we proved eq C c1 e
                      we proceed by induction on the previous result to prove 
                         or4
                           drop O O c2 e
                           ex3_4
                             F
                             C
                             T
                             T
                             λf:F.λe0:C.λu:T.λ:T.eq C e (CHead e0 (Flat f) u)
                             λf:F.λe0:C.λ:T.λw:T.drop O O c2 (CHead e0 (Flat f) w)
                             λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                           ex3_4
                             F
                             C
                             C
                             T
                             λf:F.λe1:C.λ:C.λu:T.eq C e (CHead e1 (Flat f) u)
                             λf:F.λ:C.λe2:C.λu:T.drop O O c2 (CHead e2 (Flat f) u)
                             λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                           ex4_5
                             F
                             C
                             C
                             T
                             T
                             λf:F.λe1:C.λ:C.λu:T.λ:T.eq C e (CHead e1 (Flat f) u)
                             λf:F.λ:C.λe2:C.λ:T.λw:T.drop O O c2 (CHead e2 (Flat f) w)
                             λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                             λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2
                         case refl_equal : ⇒
                            the thesis becomes 
                            or4
                              drop O O c2 c1
                              ex3_4
                                F
                                C
                                T
                                T
                                λf:F.λe0:C.λu:T.λ:T.eq C c1 (CHead e0 (Flat f) u)
                                λf:F.λe0:C.λ:T.λw:T.drop O O c2 (CHead e0 (Flat f) w)
                                λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                              ex3_4
                                F
                                C
                                C
                                T
                                λf:F.λe1:C.λ:C.λu:T.eq C c1 (CHead e1 (Flat f) u)
                                λf:F.λ:C.λe2:C.λu:T.drop O O c2 (CHead e2 (Flat f) u)
                                λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                              ex4_5
                                F
                                C
                                C
                                T
                                T
                                λf:F.λe1:C.λ:C.λu:T.λ:T.eq C c1 (CHead e1 (Flat f) u)
                                λf:F.λ:C.λe2:C.λ:T.λw:T.drop O O c2 (CHead e2 (Flat f) w)
                                λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2
                                assume y: nat
                                suppose H1: csubst0 y v c1 c2
                                  we proceed by induction on H1 to prove 
                                     eq nat y O
                                       →(or4
                                            drop y y c2 c1
                                            ex3_4
                                              F
                                              C
                                              T
                                              T
                                              λf:F.λe0:C.λu:T.λ:T.eq C c1 (CHead e0 (Flat f) u)
                                              λf:F.λe0:C.λ:T.λw:T.drop y y c2 (CHead e0 (Flat f) w)
                                              λ:F.λ:C.λu:T.λw:T.subst0 y v u w
                                            ex3_4
                                              F
                                              C
                                              C
                                              T
                                              λf:F.λe1:C.λ:C.λu:T.eq C c1 (CHead e1 (Flat f) u)
                                              λf:F.λ:C.λe2:C.λu:T.drop y y c2 (CHead e2 (Flat f) u)
                                              λ:F.λe1:C.λe2:C.λ:T.csubst0 y v e1 e2
                                            ex4_5
                                              F
                                              C
                                              C
                                              T
                                              T
                                              λf:F.λe1:C.λ:C.λu:T.λ:T.eq C c1 (CHead e1 (Flat f) u)
                                              λf:F.λ:C.λe2:C.λ:T.λw:T.drop y y c2 (CHead e2 (Flat f) w)
                                              λ:F.λ:C.λ:C.λu:T.λw:T.subst0 y v u w
                                              λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 y v e1 e2)
                                     case csubst0_snd ⇒
                                        we need to prove 
                                        ∀k:K
                                          .∀i:nat
                                            .∀v0:T
                                              .∀u1:T
                                                .∀u2:T
                                                  .subst0 i v0 u1 u2
                                                    →∀c:C
                                                         .eq nat (s k i) O
                                                           →(or4
                                                                drop (s k i) (s k i) (CHead c k u2) (CHead c k u1)
                                                                ex3_4
                                                                  F
                                                                  C
                                                                  T
                                                                  T
                                                                  λf:F.λe0:C.λu:T.λ:T.eq C (CHead c k u1) (CHead e0 (Flat f) u)
                                                                  λf:F.λe0:C.λ:T.λw:T.drop (s k i) (s k i) (CHead c k u2) (CHead e0 (Flat f) w)
                                                                  λ:F.λ:C.λu:T.λw:T.subst0 (s k i) v0 u w
                                                                ex3_4
                                                                  F
                                                                  C
                                                                  C
                                                                  T
                                                                  λf:F.λe1:C.λ:C.λu:T.eq C (CHead c k u1) (CHead e1 (Flat f) u)
                                                                  λf:F.λ:C.λe2:C.λu:T.drop (s k i) (s k i) (CHead c k u2) (CHead e2 (Flat f) u)
                                                                  λ:F.λe1:C.λe2:C.λ:T.csubst0 (s k i) v0 e1 e2
                                                                ex4_5
                                                                  F
                                                                  C
                                                                  C
                                                                  T
                                                                  T
                                                                  λf:F.λe1:C.λ:C.λu:T.λ:T.eq C (CHead c k u1) (CHead e1 (Flat f) u)
                                                                  λf:F.λ:C.λe2:C.λ:T.λw:T.drop (s k i) (s k i) (CHead c k u2) (CHead e2 (Flat f) w)
                                                                  λ:F.λ:C.λ:C.λu:T.λw:T.subst0 (s k i) v0 u w
                                                                  λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 (s k i) v0 e1 e2)
                                           assume k: K
                                              we proceed by induction on k to prove 
                                                 ∀i:nat
                                                   .∀v0:T
                                                     .∀u1:T
                                                       .∀u2:T
                                                         .subst0 i v0 u1 u2
                                                           →∀c:C
                                                                .eq nat (s k i) O
                                                                  →(or4
                                                                       drop (s k i) (s k i) (CHead c k u2) (CHead c k u1)
                                                                       ex3_4
                                                                         F
                                                                         C
                                                                         T
                                                                         T
                                                                         λf:F.λe0:C.λu:T.λ:T.eq C (CHead c k u1) (CHead e0 (Flat f) u)
                                                                         λf:F.λe0:C.λ:T.λw:T.drop (s k i) (s k i) (CHead c k u2) (CHead e0 (Flat f) w)
                                                                         λ:F.λ:C.λu:T.λw:T.subst0 (s k i) v0 u w
                                                                       ex3_4
                                                                         F
                                                                         C
                                                                         C
                                                                         T
                                                                         λf:F.λe1:C.λ:C.λu:T.eq C (CHead c k u1) (CHead e1 (Flat f) u)
                                                                         λf:F.λ:C.λe2:C.λu:T.drop (s k i) (s k i) (CHead c k u2) (CHead e2 (Flat f) u)
                                                                         λ:F.λe1:C.λe2:C.λ:T.csubst0 (s k i) v0 e1 e2
                                                                       ex4_5
                                                                         F
                                                                         C
                                                                         C
                                                                         T
                                                                         T
                                                                         λf:F.λe1:C.λ:C.λu:T.λ:T.eq C (CHead c k u1) (CHead e1 (Flat f) u)
                                                                         λf:F.λ:C.λe2:C.λ:T.λw:T.drop (s k i) (s k i) (CHead c k u2) (CHead e2 (Flat f) w)
                                                                         λ:F.λ:C.λ:C.λu:T.λw:T.subst0 (s k i) v0 u w
                                                                         λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 (s k i) v0 e1 e2)
                                                 case Bind : b:B ⇒
                                                    the thesis becomes 
                                                    ∀i:nat
                                                      .∀v0:T
                                                        .∀u1:T
                                                          .∀u2:T
                                                            .subst0 i v0 u1 u2
                                                              →∀c:C
                                                                   .eq nat (s (Bind b) i) O
                                                                     →(or4
                                                                          drop
                                                                            s (Bind b) i
                                                                            s (Bind b) i
                                                                            CHead c (Bind b) u2
                                                                            CHead c (Bind b) u1
                                                                          ex3_4
                                                                            F
                                                                            C
                                                                            T
                                                                            T
                                                                            λf:F.λe0:C.λu:T.λ:T.eq C (CHead c (Bind b) u1) (CHead e0 (Flat f) u)
                                                                            λf:F
                                                                              .λe0:C
                                                                                .λ:T
                                                                                  .λw:T
                                                                                    .drop
                                                                                      s (Bind b) i
                                                                                      s (Bind b) i
                                                                                      CHead c (Bind b) u2
                                                                                      CHead e0 (Flat f) w
                                                                            λ:F.λ:C.λu:T.λw:T.subst0 (s (Bind b) i) v0 u w
                                                                          ex3_4
                                                                            F
                                                                            C
                                                                            C
                                                                            T
                                                                            λf:F.λe1:C.λ:C.λu:T.eq C (CHead c (Bind b) u1) (CHead e1 (Flat f) u)
                                                                            λf:F
                                                                              .λ:C
                                                                                .λe2:C
                                                                                  .λu:T
                                                                                    .drop
                                                                                      s (Bind b) i
                                                                                      s (Bind b) i
                                                                                      CHead c (Bind b) u2
                                                                                      CHead e2 (Flat f) u
                                                                            λ:F.λe1:C.λe2:C.λ:T.csubst0 (s (Bind b) i) v0 e1 e2
                                                                          ex4_5
                                                                            F
                                                                            C
                                                                            C
                                                                            T
                                                                            T
                                                                            λf:F.λe1:C.λ:C.λu:T.λ:T.eq C (CHead c (Bind b) u1) (CHead e1 (Flat f) u)
                                                                            λf:F
                                                                              .λ:C
                                                                                .λe2:C
                                                                                  .λ:T
                                                                                    .λw:T
                                                                                      .drop
                                                                                        s (Bind b) i
                                                                                        s (Bind b) i
                                                                                        CHead c (Bind b) u2
                                                                                        CHead e2 (Flat f) w
                                                                            λ:F.λ:C.λ:C.λu:T.λw:T.subst0 (s (Bind b) i) v0 u w
                                                                            λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 (s (Bind b) i) v0 e1 e2)
                                                        assume i: nat
                                                        assume v0: T
                                                        assume u1: T
                                                        assume u2: T
                                                        suppose : subst0 i v0 u1 u2
                                                        assume c: C
                                                          we must prove 
                                                                eq nat (s (Bind b) i) O
                                                                  →(or4
                                                                       drop
                                                                         s (Bind b) i
                                                                         s (Bind b) i
                                                                         CHead c (Bind b) u2
                                                                         CHead c (Bind b) u1
                                                                       ex3_4
                                                                         F
                                                                         C
                                                                         T
                                                                         T
                                                                         λf:F.λe0:C.λu:T.λ:T.eq C (CHead c (Bind b) u1) (CHead e0 (Flat f) u)
                                                                         λf:F
                                                                           .λe0:C
                                                                             .λ:T
                                                                               .λw:T
                                                                                 .drop
                                                                                   s (Bind b) i
                                                                                   s (Bind b) i
                                                                                   CHead c (Bind b) u2
                                                                                   CHead e0 (Flat f) w
                                                                         λ:F.λ:C.λu:T.λw:T.subst0 (s (Bind b) i) v0 u w
                                                                       ex3_4
                                                                         F
                                                                         C
                                                                         C
                                                                         T
                                                                         λf:F.λe1:C.λ:C.λu:T.eq C (CHead c (Bind b) u1) (CHead e1 (Flat f) u)
                                                                         λf:F
                                                                           .λ:C
                                                                             .λe2:C
                                                                               .λu:T
                                                                                 .drop
                                                                                   s (Bind b) i
                                                                                   s (Bind b) i
                                                                                   CHead c (Bind b) u2
                                                                                   CHead e2 (Flat f) u
                                                                         λ:F.λe1:C.λe2:C.λ:T.csubst0 (s (Bind b) i) v0 e1 e2
                                                                       ex4_5
                                                                         F
                                                                         C
                                                                         C
                                                                         T
                                                                         T
                                                                         λf:F.λe1:C.λ:C.λu:T.λ:T.eq C (CHead c (Bind b) u1) (CHead e1 (Flat f) u)
                                                                         λf:F
                                                                           .λ:C
                                                                             .λe2:C
                                                                               .λ:T
                                                                                 .λw:T
                                                                                   .drop
                                                                                     s (Bind b) i
                                                                                     s (Bind b) i
                                                                                     CHead c (Bind b) u2
                                                                                     CHead e2 (Flat f) w
                                                                         λ:F.λ:C.λ:C.λu:T.λw:T.subst0 (s (Bind b) i) v0 u w
                                                                         λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 (s (Bind b) i) v0 e1 e2)
                                                          or equivalently 
                                                                eq nat (S i) O
                                                                  →(or4
                                                                       drop
                                                                         s (Bind b) i
                                                                         s (Bind b) i
                                                                         CHead c (Bind b) u2
                                                                         CHead c (Bind b) u1
                                                                       ex3_4
                                                                         F
                                                                         C
                                                                         T
                                                                         T
                                                                         λf:F.λe0:C.λu:T.λ:T.eq C (CHead c (Bind b) u1) (CHead e0 (Flat f) u)
                                                                         λf:F
                                                                           .λe0:C
                                                                             .λ:T
                                                                               .λw:T
                                                                                 .drop
                                                                                   s (Bind b) i
                                                                                   s (Bind b) i
                                                                                   CHead c (Bind b) u2
                                                                                   CHead e0 (Flat f) w
                                                                         λ:F.λ:C.λu:T.λw:T.subst0 (s (Bind b) i) v0 u w
                                                                       ex3_4
                                                                         F
                                                                         C
                                                                         C
                                                                         T
                                                                         λf:F.λe1:C.λ:C.λu:T.eq C (CHead c (Bind b) u1) (CHead e1 (Flat f) u)
                                                                         λf:F
                                                                           .λ:C
                                                                             .λe2:C
                                                                               .λu:T
                                                                                 .drop
                                                                                   s (Bind b) i
                                                                                   s (Bind b) i
                                                                                   CHead c (Bind b) u2
                                                                                   CHead e2 (Flat f) u
                                                                         λ:F.λe1:C.λe2:C.λ:T.csubst0 (s (Bind b) i) v0 e1 e2
                                                                       ex4_5
                                                                         F
                                                                         C
                                                                         C
                                                                         T
                                                                         T
                                                                         λf:F.λe1:C.λ:C.λu:T.λ:T.eq C (CHead c (Bind b) u1) (CHead e1 (Flat f) u)
                                                                         λf:F
                                                                           .λ:C
                                                                             .λe2:C
                                                                               .λ:T
                                                                                 .λw:T
                                                                                   .drop
                                                                                     s (Bind b) i
                                                                                     s (Bind b) i
                                                                                     CHead c (Bind b) u2
                                                                                     CHead e2 (Flat f) w
                                                                         λ:F.λ:C.λ:C.λu:T.λw:T.subst0 (s (Bind b) i) v0 u w
                                                                         λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 (s (Bind b) i) v0 e1 e2)
                                                          suppose H3: eq nat (S i) O
                                                             (H4) 
                                                                we proceed by induction on H3 to prove <λ:nat.Prop> CASE O OF O⇒False | S ⇒True
                                                                   case refl_equal : ⇒
                                                                      the thesis becomes <λ:nat.Prop> CASE S i OF O⇒False | S ⇒True
                                                                         consider I
                                                                         we proved True
<λ:nat.Prop> CASE S i OF O⇒False | S ⇒True
<λ:nat.Prop> CASE O OF O⇒False | S ⇒True
                                                             end of H4
                                                             consider H4
                                                             we proved <λ:nat.Prop> CASE O OF O⇒False | S ⇒True
                                                             that is equivalent to False
                                                             we proceed by induction on the previous result to prove 
                                                                or4
                                                                  drop (S i) (S i) (CHead c (Bind b) u2) (CHead c (Bind b) u1)
                                                                  ex3_4
                                                                    F
                                                                    C
                                                                    T
                                                                    T
                                                                    λf:F.λe0:C.λu:T.λ:T.eq C (CHead c (Bind b) u1) (CHead e0 (Flat f) u)
                                                                    λf:F
                                                                      .λe0:C.λ:T.λw:T.drop (S i) (S i) (CHead c (Bind b) u2) (CHead e0 (Flat f) w)
                                                                    λ:F.λ:C.λu:T.λw:T.subst0 (S i) v0 u w
                                                                  ex3_4
                                                                    F
                                                                    C
                                                                    C
                                                                    T
                                                                    λf:F.λe1:C.λ:C.λu:T.eq C (CHead c (Bind b) u1) (CHead e1 (Flat f) u)
                                                                    λf:F
                                                                      .λ:C.λe2:C.λu:T.drop (S i) (S i) (CHead c (Bind b) u2) (CHead e2 (Flat f) u)
                                                                    λ:F.λe1:C.λe2:C.λ:T.csubst0 (S i) v0 e1 e2
                                                                  ex4_5
                                                                    F
                                                                    C
                                                                    C
                                                                    T
                                                                    T
                                                                    λf:F.λe1:C.λ:C.λu:T.λ:T.eq C (CHead c (Bind b) u1) (CHead e1 (Flat f) u)
                                                                    λf:F
                                                                      .λ:C
                                                                        .λe2:C.λ:T.λw:T.drop (S i) (S i) (CHead c (Bind b) u2) (CHead e2 (Flat f) w)
                                                                    λ:F.λ:C.λ:C.λu:T.λw:T.subst0 (S i) v0 u w
                                                                    λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 (S i) v0 e1 e2
                                                             we proved 
                                                                or4
                                                                  drop (S i) (S i) (CHead c (Bind b) u2) (CHead c (Bind b) u1)
                                                                  ex3_4
                                                                    F
                                                                    C
                                                                    T
                                                                    T
                                                                    λf:F.λe0:C.λu:T.λ:T.eq C (CHead c (Bind b) u1) (CHead e0 (Flat f) u)
                                                                    λf:F
                                                                      .λe0:C.λ:T.λw:T.drop (S i) (S i) (CHead c (Bind b) u2) (CHead e0 (Flat f) w)
                                                                    λ:F.λ:C.λu:T.λw:T.subst0 (S i) v0 u w
                                                                  ex3_4
                                                                    F
                                                                    C
                                                                    C
                                                                    T
                                                                    λf:F.λe1:C.λ:C.λu:T.eq C (CHead c (Bind b) u1) (CHead e1 (Flat f) u)
                                                                    λf:F
                                                                      .λ:C.λe2:C.λu:T.drop (S i) (S i) (CHead c (Bind b) u2) (CHead e2 (Flat f) u)
                                                                    λ:F.λe1:C.λe2:C.λ:T.csubst0 (S i) v0 e1 e2
                                                                  ex4_5
                                                                    F
                                                                    C
                                                                    C
                                                                    T
                                                                    T
                                                                    λf:F.λe1:C.λ:C.λu:T.λ:T.eq C (CHead c (Bind b) u1) (CHead e1 (Flat f) u)
                                                                    λf:F
                                                                      .λ:C
                                                                        .λe2:C.λ:T.λw:T.drop (S i) (S i) (CHead c (Bind b) u2) (CHead e2 (Flat f) w)
                                                                    λ:F.λ:C.λ:C.λu:T.λw:T.subst0 (S i) v0 u w
                                                                    λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 (S i) v0 e1 e2
                                                             that is equivalent to 
                                                                or4
                                                                  drop
                                                                    s (Bind b) i
                                                                    s (Bind b) i
                                                                    CHead c (Bind b) u2
                                                                    CHead c (Bind b) u1
                                                                  ex3_4
                                                                    F
                                                                    C
                                                                    T
                                                                    T
                                                                    λf:F.λe0:C.λu:T.λ:T.eq C (CHead c (Bind b) u1) (CHead e0 (Flat f) u)
                                                                    λf:F
                                                                      .λe0:C
                                                                        .λ:T
                                                                          .λw:T
                                                                            .drop
                                                                              s (Bind b) i
                                                                              s (Bind b) i
                                                                              CHead c (Bind b) u2
                                                                              CHead e0 (Flat f) w
                                                                    λ:F.λ:C.λu:T.λw:T.subst0 (s (Bind b) i) v0 u w
                                                                  ex3_4
                                                                    F
                                                                    C
                                                                    C
                                                                    T
                                                                    λf:F.λe1:C.λ:C.λu:T.eq C (CHead c (Bind b) u1) (CHead e1 (Flat f) u)
                                                                    λf:F
                                                                      .λ:C
                                                                        .λe2:C
                                                                          .λu:T
                                                                            .drop
                                                                              s (Bind b) i
                                                                              s (Bind b) i
                                                                              CHead c (Bind b) u2
                                                                              CHead e2 (Flat f) u
                                                                    λ:F.λe1:C.λe2:C.λ:T.csubst0 (s (Bind b) i) v0 e1 e2
                                                                  ex4_5
                                                                    F
                                                                    C
                                                                    C
                                                                    T
                                                                    T
                                                                    λf:F.λe1:C.λ:C.λu:T.λ:T.eq C (CHead c (Bind b) u1) (CHead e1 (Flat f) u)
                                                                    λf:F
                                                                      .λ:C
                                                                        .λe2:C
                                                                          .λ:T
                                                                            .λw:T
                                                                              .drop
                                                                                s (Bind b) i
                                                                                s (Bind b) i
                                                                                CHead c (Bind b) u2
                                                                                CHead e2 (Flat f) w
                                                                    λ:F.λ:C.λ:C.λu:T.λw:T.subst0 (s (Bind b) i) v0 u w
                                                                    λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 (s (Bind b) i) v0 e1 e2
                                                          we proved 
                                                             eq nat (S i) O
                                                               →(or4
                                                                    drop
                                                                      s (Bind b) i
                                                                      s (Bind b) i
                                                                      CHead c (Bind b) u2
                                                                      CHead c (Bind b) u1
                                                                    ex3_4
                                                                      F
                                                                      C
                                                                      T
                                                                      T
                                                                      λf:F.λe0:C.λu:T.λ:T.eq C (CHead c (Bind b) u1) (CHead e0 (Flat f) u)
                                                                      λf:F
                                                                        .λe0:C
                                                                          .λ:T
                                                                            .λw:T
                                                                              .drop
                                                                                s (Bind b) i
                                                                                s (Bind b) i
                                                                                CHead c (Bind b) u2
                                                                                CHead e0 (Flat f) w
                                                                      λ:F.λ:C.λu:T.λw:T.subst0 (s (Bind b) i) v0 u w
                                                                    ex3_4
                                                                      F
                                                                      C
                                                                      C
                                                                      T
                                                                      λf:F.λe1:C.λ:C.λu:T.eq C (CHead c (Bind b) u1) (CHead e1 (Flat f) u)
                                                                      λf:F
                                                                        .λ:C
                                                                          .λe2:C
                                                                            .λu:T
                                                                              .drop
                                                                                s (Bind b) i
                                                                                s (Bind b) i
                                                                                CHead c (Bind b) u2
                                                                                CHead e2 (Flat f) u
                                                                      λ:F.λe1:C.λe2:C.λ:T.csubst0 (s (Bind b) i) v0 e1 e2
                                                                    ex4_5
                                                                      F
                                                                      C
                                                                      C
                                                                      T
                                                                      T
                                                                      λf:F.λe1:C.λ:C.λu:T.λ:T.eq C (CHead c (Bind b) u1) (CHead e1 (Flat f) u)
                                                                      λf:F
                                                                        .λ:C
                                                                          .λe2:C
                                                                            .λ:T
                                                                              .λw:T
                                                                                .drop
                                                                                  s (Bind b) i
                                                                                  s (Bind b) i
                                                                                  CHead c (Bind b) u2
                                                                                  CHead e2 (Flat f) w
                                                                      λ:F.λ:C.λ:C.λu:T.λw:T.subst0 (s (Bind b) i) v0 u w
                                                                      λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 (s (Bind b) i) v0 e1 e2)
                                                          that is equivalent to 
                                                             eq nat (s (Bind b) i) O
                                                               →(or4
                                                                    drop
                                                                      s (Bind b) i
                                                                      s (Bind b) i
                                                                      CHead c (Bind b) u2
                                                                      CHead c (Bind b) u1
                                                                    ex3_4
                                                                      F
                                                                      C
                                                                      T
                                                                      T
                                                                      λf:F.λe0:C.λu:T.λ:T.eq C (CHead c (Bind b) u1) (CHead e0 (Flat f) u)
                                                                      λf:F
                                                                        .λe0:C
                                                                          .λ:T
                                                                            .λw:T
                                                                              .drop
                                                                                s (Bind b) i
                                                                                s (Bind b) i
                                                                                CHead c (Bind b) u2
                                                                                CHead e0 (Flat f) w
                                                                      λ:F.λ:C.λu:T.λw:T.subst0 (s (Bind b) i) v0 u w
                                                                    ex3_4
                                                                      F
                                                                      C
                                                                      C
                                                                      T
                                                                      λf:F.λe1:C.λ:C.λu:T.eq C (CHead c (Bind b) u1) (CHead e1 (Flat f) u)
                                                                      λf:F
                                                                        .λ:C
                                                                          .λe2:C
                                                                            .λu:T
                                                                              .drop
                                                                                s (Bind b) i
                                                                                s (Bind b) i
                                                                                CHead c (Bind b) u2
                                                                                CHead e2 (Flat f) u
                                                                      λ:F.λe1:C.λe2:C.λ:T.csubst0 (s (Bind b) i) v0 e1 e2
                                                                    ex4_5
                                                                      F
                                                                      C
                                                                      C
                                                                      T
                                                                      T
                                                                      λf:F.λe1:C.λ:C.λu:T.λ:T.eq C (CHead c (Bind b) u1) (CHead e1 (Flat f) u)
                                                                      λf:F
                                                                        .λ:C
                                                                          .λe2:C
                                                                            .λ:T
                                                                              .λw:T
                                                                                .drop
                                                                                  s (Bind b) i
                                                                                  s (Bind b) i
                                                                                  CHead c (Bind b) u2
                                                                                  CHead e2 (Flat f) w
                                                                      λ:F.λ:C.λ:C.λu:T.λw:T.subst0 (s (Bind b) i) v0 u w
                                                                      λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 (s (Bind b) i) v0 e1 e2)

                                                          ∀i:nat
                                                            .∀v0:T
                                                              .∀u1:T
                                                                .∀u2:T
                                                                  .subst0 i v0 u1 u2
                                                                    →∀c:C
                                                                         .eq nat (s (Bind b) i) O
                                                                           →(or4
                                                                                drop
                                                                                  s (Bind b) i
                                                                                  s (Bind b) i
                                                                                  CHead c (Bind b) u2
                                                                                  CHead c (Bind b) u1
                                                                                ex3_4
                                                                                  F
                                                                                  C
                                                                                  T
                                                                                  T
                                                                                  λf:F.λe0:C.λu:T.λ:T.eq C (CHead c (Bind b) u1) (CHead e0 (Flat f) u)
                                                                                  λf:F
                                                                                    .λe0:C
                                                                                      .λ:T
                                                                                        .λw:T
                                                                                          .drop
                                                                                            s (Bind b) i
                                                                                            s (Bind b) i
                                                                                            CHead c (Bind b) u2
                                                                                            CHead e0 (Flat f) w
                                                                                  λ:F.λ:C.λu:T.λw:T.subst0 (s (Bind b) i) v0 u w
                                                                                ex3_4
                                                                                  F
                                                                                  C
                                                                                  C
                                                                                  T
                                                                                  λf:F.λe1:C.λ:C.λu:T.eq C (CHead c (Bind b) u1) (CHead e1 (Flat f) u)
                                                                                  λf:F
                                                                                    .λ:C
                                                                                      .λe2:C
                                                                                        .λu:T
                                                                                          .drop
                                                                                            s (Bind b) i
                                                                                            s (Bind b) i
                                                                                            CHead c (Bind b) u2
                                                                                            CHead e2 (Flat f) u
                                                                                  λ:F.λe1:C.λe2:C.λ:T.csubst0 (s (Bind b) i) v0 e1 e2
                                                                                ex4_5
                                                                                  F
                                                                                  C
                                                                                  C
                                                                                  T
                                                                                  T
                                                                                  λf:F.λe1:C.λ:C.λu:T.λ:T.eq C (CHead c (Bind b) u1) (CHead e1 (Flat f) u)
                                                                                  λf:F
                                                                                    .λ:C
                                                                                      .λe2:C
                                                                                        .λ:T
                                                                                          .λw:T
                                                                                            .drop
                                                                                              s (Bind b) i
                                                                                              s (Bind b) i
                                                                                              CHead c (Bind b) u2
                                                                                              CHead e2 (Flat f) w
                                                                                  λ:F.λ:C.λ:C.λu:T.λw:T.subst0 (s (Bind b) i) v0 u w
                                                                                  λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 (s (Bind b) i) v0 e1 e2)
                                                 case Flat : f:F ⇒
                                                    the thesis becomes 
                                                    ∀i:nat
                                                      .∀v0:T
                                                        .∀u1:T
                                                          .∀u2:T
                                                            .∀H2:subst0 i v0 u1 u2
                                                              .∀c:C
                                                                .eq nat (s (Flat f) i) O
                                                                  →(or4
                                                                       drop
                                                                         s (Flat f) i
                                                                         s (Flat f) i
                                                                         CHead c (Flat f) u2
                                                                         CHead c (Flat f) u1
                                                                       ex3_4
                                                                         F
                                                                         C
                                                                         T
                                                                         T
                                                                         λf:F.λe0:C.λu:T.λ:T.eq C (CHead c (Flat f) u1) (CHead e0 (Flat f) u)
                                                                         λf:F
                                                                           .λe0:C
                                                                             .λ:T
                                                                               .λw:T
                                                                                 .drop
                                                                                   s (Flat f) i
                                                                                   s (Flat f) i
                                                                                   CHead c (Flat f) u2
                                                                                   CHead e0 (Flat f) w
                                                                         λ:F.λ:C.λu:T.λw:T.subst0 (s (Flat f) i) v0 u w
                                                                       ex3_4
                                                                         F
                                                                         C
                                                                         C
                                                                         T
                                                                         λf:F.λe1:C.λ:C.λu:T.eq C (CHead c (Flat f) u1) (CHead e1 (Flat f) u)
                                                                         λf:F
                                                                           .λ:C
                                                                             .λe2:C
                                                                               .λu:T
                                                                                 .drop
                                                                                   s (Flat f) i
                                                                                   s (Flat f) i
                                                                                   CHead c (Flat f) u2
                                                                                   CHead e2 (Flat f) u
                                                                         λ:F.λe1:C.λe2:C.λ:T.csubst0 (s (Flat f) i) v0 e1 e2
                                                                       ex4_5
                                                                         F
                                                                         C
                                                                         C
                                                                         T
                                                                         T
                                                                         λf:F.λe1:C.λ:C.λu:T.λ:T.eq C (CHead c (Flat f) u1) (CHead e1 (Flat f) u)
                                                                         λf:F
                                                                           .λ:C
                                                                             .λe2:C
                                                                               .λ:T
                                                                                 .λw:T
                                                                                   .drop
                                                                                     s (Flat f) i
                                                                                     s (Flat f) i
                                                                                     CHead c (Flat f) u2
                                                                                     CHead e2 (Flat f) w
                                                                         λ:F.λ:C.λ:C.λu:T.λw:T.subst0 (s (Flat f) i) v0 u w
                                                                         λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 (s (Flat f) i) v0 e1 e2)
                                                        assume i: nat
                                                        assume v0: T
                                                        assume u1: T
                                                        assume u2: T
                                                        suppose H2: subst0 i v0 u1 u2
                                                        assume c: C
                                                          we must prove 
                                                                eq nat (s (Flat f) i) O
                                                                  →(or4
                                                                       drop
                                                                         s (Flat f) i
                                                                         s (Flat f) i
                                                                         CHead c (Flat f) u2
                                                                         CHead c (Flat f) u1
                                                                       ex3_4
                                                                         F
                                                                         C
                                                                         T
                                                                         T
                                                                         λf:F.λe0:C.λu:T.λ:T.eq C (CHead c (Flat f) u1) (CHead e0 (Flat f) u)
                                                                         λf:F
                                                                           .λe0:C
                                                                             .λ:T
                                                                               .λw:T
                                                                                 .drop
                                                                                   s (Flat f) i
                                                                                   s (Flat f) i
                                                                                   CHead c (Flat f) u2
                                                                                   CHead e0 (Flat f) w
                                                                         λ:F.λ:C.λu:T.λw:T.subst0 (s (Flat f) i) v0 u w
                                                                       ex3_4
                                                                         F
                                                                         C
                                                                         C
                                                                         T
                                                                         λf:F.λe1:C.λ:C.λu:T.eq C (CHead c (Flat f) u1) (CHead e1 (Flat f) u)
                                                                         λf:F
                                                                           .λ:C
                                                                             .λe2:C
                                                                               .λu:T
                                                                                 .drop
                                                                                   s (Flat f) i
                                                                                   s (Flat f) i
                                                                                   CHead c (Flat f) u2
                                                                                   CHead e2 (Flat f) u
                                                                         λ:F.λe1:C.λe2:C.λ:T.csubst0 (s (Flat f) i) v0 e1 e2
                                                                       ex4_5
                                                                         F
                                                                         C
                                                                         C
                                                                         T
                                                                         T
                                                                         λf:F.λe1:C.λ:C.λu:T.λ:T.eq C (CHead c (Flat f) u1) (CHead e1 (Flat f) u)
                                                                         λf:F
                                                                           .λ:C
                                                                             .λe2:C
                                                                               .λ:T
                                                                                 .λw:T
                                                                                   .drop
                                                                                     s (Flat f) i
                                                                                     s (Flat f) i
                                                                                     CHead c (Flat f) u2
                                                                                     CHead e2 (Flat f) w
                                                                         λ:F.λ:C.λ:C.λu:T.λw:T.subst0 (s (Flat f) i) v0 u w
                                                                         λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 (s (Flat f) i) v0 e1 e2)
                                                          or equivalently 
                                                                eq nat i O
                                                                  →(or4
                                                                       drop
                                                                         s (Flat f) i
                                                                         s (Flat f) i
                                                                         CHead c (Flat f) u2
                                                                         CHead c (Flat f) u1
                                                                       ex3_4
                                                                         F
                                                                         C
                                                                         T
                                                                         T
                                                                         λf:F.λe0:C.λu:T.λ:T.eq C (CHead c (Flat f) u1) (CHead e0 (Flat f) u)
                                                                         λf:F
                                                                           .λe0:C
                                                                             .λ:T
                                                                               .λw:T
                                                                                 .drop
                                                                                   s (Flat f) i
                                                                                   s (Flat f) i
                                                                                   CHead c (Flat f) u2
                                                                                   CHead e0 (Flat f) w
                                                                         λ:F.λ:C.λu:T.λw:T.subst0 (s (Flat f) i) v0 u w
                                                                       ex3_4
                                                                         F
                                                                         C
                                                                         C
                                                                         T
                                                                         λf:F.λe1:C.λ:C.λu:T.eq C (CHead c (Flat f) u1) (CHead e1 (Flat f) u)
                                                                         λf:F
                                                                           .λ:C
                                                                             .λe2:C
                                                                               .λu:T
                                                                                 .drop
                                                                                   s (Flat f) i
                                                                                   s (Flat f) i
                                                                                   CHead c (Flat f) u2
                                                                                   CHead e2 (Flat f) u
                                                                         λ:F.λe1:C.λe2:C.λ:T.csubst0 (s (Flat f) i) v0 e1 e2
                                                                       ex4_5
                                                                         F
                                                                         C
                                                                         C
                                                                         T
                                                                         T
                                                                         λf:F.λe1:C.λ:C.λu:T.λ:T.eq C (CHead c (Flat f) u1) (CHead e1 (Flat f) u)
                                                                         λf:F
                                                                           .λ:C
                                                                             .λe2:C
                                                                               .λ:T
                                                                                 .λw:T
                                                                                   .drop
                                                                                     s (Flat f) i
                                                                                     s (Flat f) i
                                                                                     CHead c (Flat f) u2
                                                                                     CHead e2 (Flat f) w
                                                                         λ:F.λ:C.λ:C.λu:T.λw:T.subst0 (s (Flat f) i) v0 u w
                                                                         λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 (s (Flat f) i) v0 e1 e2)
                                                          suppose H3: eq nat i O
                                                             (H4) 
                                                                we proceed by induction on H3 to prove subst0 O v0 u1 u2
                                                                   case refl_equal : ⇒
                                                                      the thesis becomes the hypothesis H2
subst0 O v0 u1 u2
                                                             end of H4
                                                             (h1) 
                                                                by (refl_equal . .)
eq C (CHead c (Flat f) u1) (CHead c (Flat f) u1)
                                                             end of h1
                                                             (h2) 
                                                                by (drop_refl .)
drop O O (CHead c (Flat f) u2) (CHead c (Flat f) u2)
                                                             end of h2
                                                             by (ex3_4_intro . . . . . . . . . . . h1 h2 H4)
                                                             we proved 
                                                                ex3_4
                                                                  F
                                                                  C
                                                                  T
                                                                  T
                                                                  λf0:F.λe0:C.λu:T.λ:T.eq C (CHead c (Flat f) u1) (CHead e0 (Flat f0) u)
                                                                  λf0:F.λe0:C.λ:T.λw:T.drop O O (CHead c (Flat f) u2) (CHead e0 (Flat f0) w)
                                                                  λ:F.λ:C.λu:T.λw:T.subst0 O v0 u w
                                                             by (or4_intro1 . . . . previous)
                                                             we proved 
                                                                or4
                                                                  drop O O (CHead c (Flat f) u2) (CHead c (Flat f) u1)
                                                                  ex3_4
                                                                    F
                                                                    C
                                                                    T
                                                                    T
                                                                    λf0:F.λe0:C.λu:T.λ:T.eq C (CHead c (Flat f) u1) (CHead e0 (Flat f0) u)
                                                                    λf0:F.λe0:C.λ:T.λw:T.drop O O (CHead c (Flat f) u2) (CHead e0 (Flat f0) w)
                                                                    λ:F.λ:C.λu:T.λw:T.subst0 O v0 u w
                                                                  ex3_4
                                                                    F
                                                                    C
                                                                    C
                                                                    T
                                                                    λf0:F.λe1:C.λ:C.λu:T.eq C (CHead c (Flat f) u1) (CHead e1 (Flat f0) u)
                                                                    λf0:F.λ:C.λe2:C.λu:T.drop O O (CHead c (Flat f) u2) (CHead e2 (Flat f0) u)
                                                                    λ:F.λe1:C.λe2:C.λ:T.csubst0 O v0 e1 e2
                                                                  ex4_5
                                                                    F
                                                                    C
                                                                    C
                                                                    T
                                                                    T
                                                                    λf0:F.λe1:C.λ:C.λu:T.λ:T.eq C (CHead c (Flat f) u1) (CHead e1 (Flat f0) u)
                                                                    λf0:F.λ:C.λe2:C.λ:T.λw:T.drop O O (CHead c (Flat f) u2) (CHead e2 (Flat f0) w)
                                                                    λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v0 u w
                                                                    λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v0 e1 e2
                                                             by (eq_ind_r . . . previous . H3)
                                                             we proved 
                                                                or4
                                                                  drop i i (CHead c (Flat f) u2) (CHead c (Flat f) u1)
                                                                  ex3_4
                                                                    F
                                                                    C
                                                                    T
                                                                    T
                                                                    λf0:F.λe0:C.λu:T.λ:T.eq C (CHead c (Flat f) u1) (CHead e0 (Flat f0) u)
                                                                    λf0:F.λe0:C.λ:T.λw:T.drop i i (CHead c (Flat f) u2) (CHead e0 (Flat f0) w)
                                                                    λ:F.λ:C.λu:T.λw:T.subst0 i v0 u w
                                                                  ex3_4
                                                                    F
                                                                    C
                                                                    C
                                                                    T
                                                                    λf0:F.λe1:C.λ:C.λu:T.eq C (CHead c (Flat f) u1) (CHead e1 (Flat f0) u)
                                                                    λf0:F.λ:C.λe2:C.λu:T.drop i i (CHead c (Flat f) u2) (CHead e2 (Flat f0) u)
                                                                    λ:F.λe1:C.λe2:C.λ:T.csubst0 i v0 e1 e2
                                                                  ex4_5
                                                                    F
                                                                    C
                                                                    C
                                                                    T
                                                                    T
                                                                    λf0:F.λe1:C.λ:C.λu:T.λ:T.eq C (CHead c (Flat f) u1) (CHead e1 (Flat f0) u)
                                                                    λf0:F.λ:C.λe2:C.λ:T.λw:T.drop i i (CHead c (Flat f) u2) (CHead e2 (Flat f0) w)
                                                                    λ:F.λ:C.λ:C.λu:T.λw:T.subst0 i v0 u w
                                                                    λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 i v0 e1 e2
                                                             that is equivalent to 
                                                                or4
                                                                  drop
                                                                    s (Flat f) i
                                                                    s (Flat f) i
                                                                    CHead c (Flat f) u2
                                                                    CHead c (Flat f) u1
                                                                  ex3_4
                                                                    F
                                                                    C
                                                                    T
                                                                    T
                                                                    λf:F.λe0:C.λu:T.λ:T.eq C (CHead c (Flat f) u1) (CHead e0 (Flat f) u)
                                                                    λf:F
                                                                      .λe0:C
                                                                        .λ:T
                                                                          .λw:T
                                                                            .drop
                                                                              s (Flat f) i
                                                                              s (Flat f) i
                                                                              CHead c (Flat f) u2
                                                                              CHead e0 (Flat f) w
                                                                    λ:F.λ:C.λu:T.λw:T.subst0 (s (Flat f) i) v0 u w
                                                                  ex3_4
                                                                    F
                                                                    C
                                                                    C
                                                                    T
                                                                    λf:F.λe1:C.λ:C.λu:T.eq C (CHead c (Flat f) u1) (CHead e1 (Flat f) u)
                                                                    λf:F
                                                                      .λ:C
                                                                        .λe2:C
                                                                          .λu:T
                                                                            .drop
                                                                              s (Flat f) i
                                                                              s (Flat f) i
                                                                              CHead c (Flat f) u2
                                                                              CHead e2 (Flat f) u
                                                                    λ:F.λe1:C.λe2:C.λ:T.csubst0 (s (Flat f) i) v0 e1 e2
                                                                  ex4_5
                                                                    F
                                                                    C
                                                                    C
                                                                    T
                                                                    T
                                                                    λf:F.λe1:C.λ:C.λu:T.λ:T.eq C (CHead c (Flat f) u1) (CHead e1 (Flat f) u)
                                                                    λf:F
                                                                      .λ:C
                                                                        .λe2:C
                                                                          .λ:T
                                                                            .λw:T
                                                                              .drop
                                                                                s (Flat f) i
                                                                                s (Flat f) i
                                                                                CHead c (Flat f) u2
                                                                                CHead e2 (Flat f) w
                                                                    λ:F.λ:C.λ:C.λu:T.λw:T.subst0 (s (Flat f) i) v0 u w
                                                                    λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 (s (Flat f) i) v0 e1 e2
                                                          we proved 
                                                             eq nat i O
                                                               →(or4
                                                                    drop
                                                                      s (Flat f) i
                                                                      s (Flat f) i
                                                                      CHead c (Flat f) u2
                                                                      CHead c (Flat f) u1
                                                                    ex3_4
                                                                      F
                                                                      C
                                                                      T
                                                                      T
                                                                      λf:F.λe0:C.λu:T.λ:T.eq C (CHead c (Flat f) u1) (CHead e0 (Flat f) u)
                                                                      λf:F
                                                                        .λe0:C
                                                                          .λ:T
                                                                            .λw:T
                                                                              .drop
                                                                                s (Flat f) i
                                                                                s (Flat f) i
                                                                                CHead c (Flat f) u2
                                                                                CHead e0 (Flat f) w
                                                                      λ:F.λ:C.λu:T.λw:T.subst0 (s (Flat f) i) v0 u w
                                                                    ex3_4
                                                                      F
                                                                      C
                                                                      C
                                                                      T
                                                                      λf:F.λe1:C.λ:C.λu:T.eq C (CHead c (Flat f) u1) (CHead e1 (Flat f) u)
                                                                      λf:F
                                                                        .λ:C
                                                                          .λe2:C
                                                                            .λu:T
                                                                              .drop
                                                                                s (Flat f) i
                                                                                s (Flat f) i
                                                                                CHead c (Flat f) u2
                                                                                CHead e2 (Flat f) u
                                                                      λ:F.λe1:C.λe2:C.λ:T.csubst0 (s (Flat f) i) v0 e1 e2
                                                                    ex4_5
                                                                      F
                                                                      C
                                                                      C
                                                                      T
                                                                      T
                                                                      λf:F.λe1:C.λ:C.λu:T.λ:T.eq C (CHead c (Flat f) u1) (CHead e1 (Flat f) u)
                                                                      λf:F
                                                                        .λ:C
                                                                          .λe2:C
                                                                            .λ:T
                                                                              .λw:T
                                                                                .drop
                                                                                  s (Flat f) i
                                                                                  s (Flat f) i
                                                                                  CHead c (Flat f) u2
                                                                                  CHead e2 (Flat f) w
                                                                      λ:F.λ:C.λ:C.λu:T.λw:T.subst0 (s (Flat f) i) v0 u w
                                                                      λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 (s (Flat f) i) v0 e1 e2)
                                                          that is equivalent to 
                                                             eq nat (s (Flat f) i) O
                                                               →(or4
                                                                    drop
                                                                      s (Flat f) i
                                                                      s (Flat f) i
                                                                      CHead c (Flat f) u2
                                                                      CHead c (Flat f) u1
                                                                    ex3_4
                                                                      F
                                                                      C
                                                                      T
                                                                      T
                                                                      λf:F.λe0:C.λu:T.λ:T.eq C (CHead c (Flat f) u1) (CHead e0 (Flat f) u)
                                                                      λf:F
                                                                        .λe0:C
                                                                          .λ:T
                                                                            .λw:T
                                                                              .drop
                                                                                s (Flat f) i
                                                                                s (Flat f) i
                                                                                CHead c (Flat f) u2
                                                                                CHead e0 (Flat f) w
                                                                      λ:F.λ:C.λu:T.λw:T.subst0 (s (Flat f) i) v0 u w
                                                                    ex3_4
                                                                      F
                                                                      C
                                                                      C
                                                                      T
                                                                      λf:F.λe1:C.λ:C.λu:T.eq C (CHead c (Flat f) u1) (CHead e1 (Flat f) u)
                                                                      λf:F
                                                                        .λ:C
                                                                          .λe2:C
                                                                            .λu:T
                                                                              .drop
                                                                                s (Flat f) i
                                                                                s (Flat f) i
                                                                                CHead c (Flat f) u2
                                                                                CHead e2 (Flat f) u
                                                                      λ:F.λe1:C.λe2:C.λ:T.csubst0 (s (Flat f) i) v0 e1 e2
                                                                    ex4_5
                                                                      F
                                                                      C
                                                                      C
                                                                      T
                                                                      T
                                                                      λf:F.λe1:C.λ:C.λu:T.λ:T.eq C (CHead c (Flat f) u1) (CHead e1 (Flat f) u)
                                                                      λf:F
                                                                        .λ:C
                                                                          .λe2:C
                                                                            .λ:T
                                                                              .λw:T
                                                                                .drop
                                                                                  s (Flat f) i
                                                                                  s (Flat f) i
                                                                                  CHead c (Flat f) u2
                                                                                  CHead e2 (Flat f) w
                                                                      λ:F.λ:C.λ:C.λu:T.λw:T.subst0 (s (Flat f) i) v0 u w
                                                                      λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 (s (Flat f) i) v0 e1 e2)

                                                          ∀i:nat
                                                            .∀v0:T
                                                              .∀u1:T
                                                                .∀u2:T
                                                                  .∀H2:subst0 i v0 u1 u2
                                                                    .∀c:C
                                                                      .eq nat (s (Flat f) i) O
                                                                        →(or4
                                                                             drop
                                                                               s (Flat f) i
                                                                               s (Flat f) i
                                                                               CHead c (Flat f) u2
                                                                               CHead c (Flat f) u1
                                                                             ex3_4
                                                                               F
                                                                               C
                                                                               T
                                                                               T
                                                                               λf:F.λe0:C.λu:T.λ:T.eq C (CHead c (Flat f) u1) (CHead e0 (Flat f) u)
                                                                               λf:F
                                                                                 .λe0:C
                                                                                   .λ:T
                                                                                     .λw:T
                                                                                       .drop
                                                                                         s (Flat f) i
                                                                                         s (Flat f) i
                                                                                         CHead c (Flat f) u2
                                                                                         CHead e0 (Flat f) w
                                                                               λ:F.λ:C.λu:T.λw:T.subst0 (s (Flat f) i) v0 u w
                                                                             ex3_4
                                                                               F
                                                                               C
                                                                               C
                                                                               T
                                                                               λf:F.λe1:C.λ:C.λu:T.eq C (CHead c (Flat f) u1) (CHead e1 (Flat f) u)
                                                                               λf:F
                                                                                 .λ:C
                                                                                   .λe2:C
                                                                                     .λu:T
                                                                                       .drop
                                                                                         s (Flat f) i
                                                                                         s (Flat f) i
                                                                                         CHead c (Flat f) u2
                                                                                         CHead e2 (Flat f) u
                                                                               λ:F.λe1:C.λe2:C.λ:T.csubst0 (s (Flat f) i) v0 e1 e2
                                                                             ex4_5
                                                                               F
                                                                               C
                                                                               C
                                                                               T
                                                                               T
                                                                               λf:F.λe1:C.λ:C.λu:T.λ:T.eq C (CHead c (Flat f) u1) (CHead e1 (Flat f) u)
                                                                               λf:F
                                                                                 .λ:C
                                                                                   .λe2:C
                                                                                     .λ:T
                                                                                       .λw:T
                                                                                         .drop
                                                                                           s (Flat f) i
                                                                                           s (Flat f) i
                                                                                           CHead c (Flat f) u2
                                                                                           CHead e2 (Flat f) w
                                                                               λ:F.λ:C.λ:C.λu:T.λw:T.subst0 (s (Flat f) i) v0 u w
                                                                               λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 (s (Flat f) i) v0 e1 e2)
                                              we proved 
                                                 ∀i:nat
                                                   .∀v0:T
                                                     .∀u1:T
                                                       .∀u2:T
                                                         .subst0 i v0 u1 u2
                                                           →∀c:C
                                                                .eq nat (s k i) O
                                                                  →(or4
                                                                       drop (s k i) (s k i) (CHead c k u2) (CHead c k u1)
                                                                       ex3_4
                                                                         F
                                                                         C
                                                                         T
                                                                         T
                                                                         λf:F.λe0:C.λu:T.λ:T.eq C (CHead c k u1) (CHead e0 (Flat f) u)
                                                                         λf:F.λe0:C.λ:T.λw:T.drop (s k i) (s k i) (CHead c k u2) (CHead e0 (Flat f) w)
                                                                         λ:F.λ:C.λu:T.λw:T.subst0 (s k i) v0 u w
                                                                       ex3_4
                                                                         F
                                                                         C
                                                                         C
                                                                         T
                                                                         λf:F.λe1:C.λ:C.λu:T.eq C (CHead c k u1) (CHead e1 (Flat f) u)
                                                                         λf:F.λ:C.λe2:C.λu:T.drop (s k i) (s k i) (CHead c k u2) (CHead e2 (Flat f) u)
                                                                         λ:F.λe1:C.λe2:C.λ:T.csubst0 (s k i) v0 e1 e2
                                                                       ex4_5
                                                                         F
                                                                         C
                                                                         C
                                                                         T
                                                                         T
                                                                         λf:F.λe1:C.λ:C.λu:T.λ:T.eq C (CHead c k u1) (CHead e1 (Flat f) u)
                                                                         λf:F.λ:C.λe2:C.λ:T.λw:T.drop (s k i) (s k i) (CHead c k u2) (CHead e2 (Flat f) w)
                                                                         λ:F.λ:C.λ:C.λu:T.λw:T.subst0 (s k i) v0 u w
                                                                         λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 (s k i) v0 e1 e2)

                                              ∀k:K
                                                .∀i:nat
                                                  .∀v0:T
                                                    .∀u1:T
                                                      .∀u2:T
                                                        .subst0 i v0 u1 u2
                                                          →∀c:C
                                                               .eq nat (s k i) O
                                                                 →(or4
                                                                      drop (s k i) (s k i) (CHead c k u2) (CHead c k u1)
                                                                      ex3_4
                                                                        F
                                                                        C
                                                                        T
                                                                        T
                                                                        λf:F.λe0:C.λu:T.λ:T.eq C (CHead c k u1) (CHead e0 (Flat f) u)
                                                                        λf:F.λe0:C.λ:T.λw:T.drop (s k i) (s k i) (CHead c k u2) (CHead e0 (Flat f) w)
                                                                        λ:F.λ:C.λu:T.λw:T.subst0 (s k i) v0 u w
                                                                      ex3_4
                                                                        F
                                                                        C
                                                                        C
                                                                        T
                                                                        λf:F.λe1:C.λ:C.λu:T.eq C (CHead c k u1) (CHead e1 (Flat f) u)
                                                                        λf:F.λ:C.λe2:C.λu:T.drop (s k i) (s k i) (CHead c k u2) (CHead e2 (Flat f) u)
                                                                        λ:F.λe1:C.λe2:C.λ:T.csubst0 (s k i) v0 e1 e2
                                                                      ex4_5
                                                                        F
                                                                        C
                                                                        C
                                                                        T
                                                                        T
                                                                        λf:F.λe1:C.λ:C.λu:T.λ:T.eq C (CHead c k u1) (CHead e1 (Flat f) u)
                                                                        λf:F.λ:C.λe2:C.λ:T.λw:T.drop (s k i) (s k i) (CHead c k u2) (CHead e2 (Flat f) w)
                                                                        λ:F.λ:C.λ:C.λu:T.λw:T.subst0 (s k i) v0 u w
                                                                        λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 (s k i) v0 e1 e2)
                                     case csubst0_fst ⇒
                                        we need to prove 
                                        ∀k:K
                                          .∀i:nat
                                            .∀c3:C
                                              .∀c4:C
                                                .∀v0:T
                                                  .csubst0 i v0 c3 c4
                                                    →(eq nat i O
                                                           →(or4
                                                                drop i i c4 c3
                                                                ex3_4
                                                                  F
                                                                  C
                                                                  T
                                                                  T
                                                                  λf:F.λe0:C.λu:T.λ:T.eq C c3 (CHead e0 (Flat f) u)
                                                                  λf:F.λe0:C.λ:T.λw:T.drop i i c4 (CHead e0 (Flat f) w)
                                                                  λ:F.λ:C.λu:T.λw:T.subst0 i v0 u w
                                                                ex3_4
                                                                  F
                                                                  C
                                                                  C
                                                                  T
                                                                  λf:F.λe1:C.λ:C.λu:T.eq C c3 (CHead e1 (Flat f) u)
                                                                  λf:F.λ:C.λe2:C.λu:T.drop i i c4 (CHead e2 (Flat f) u)
                                                                  λ:F.λe1:C.λe2:C.λ:T.csubst0 i v0 e1 e2
                                                                ex4_5
                                                                  F
                                                                  C
                                                                  C
                                                                  T
                                                                  T
                                                                  λf:F.λe1:C.λ:C.λu:T.λ:T.eq C c3 (CHead e1 (Flat f) u)
                                                                  λf:F.λ:C.λe2:C.λ:T.λw:T.drop i i c4 (CHead e2 (Flat f) w)
                                                                  λ:F.λ:C.λ:C.λu:T.λw:T.subst0 i v0 u w
                                                                  λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 i v0 e1 e2)
                                                         →∀u:T
                                                              .eq nat (s k i) O
                                                                →(or4
                                                                     drop (s k i) (s k i) (CHead c4 k u) (CHead c3 k u)
                                                                     ex3_4
                                                                       F
                                                                       C
                                                                       T
                                                                       T
                                                                       λf:F.λe0:C.λu0:T.λ:T.eq C (CHead c3 k u) (CHead e0 (Flat f) u0)
                                                                       λf:F.λe0:C.λ:T.λw:T.drop (s k i) (s k i) (CHead c4 k u) (CHead e0 (Flat f) w)
                                                                       λ:F.λ:C.λu0:T.λw:T.subst0 (s k i) v0 u0 w
                                                                     ex3_4
                                                                       F
                                                                       C
                                                                       C
                                                                       T
                                                                       λf:F.λe1:C.λ:C.λu0:T.eq C (CHead c3 k u) (CHead e1 (Flat f) u0)
                                                                       λf:F.λ:C.λe2:C.λu0:T.drop (s k i) (s k i) (CHead c4 k u) (CHead e2 (Flat f) u0)
                                                                       λ:F.λe1:C.λe2:C.λ:T.csubst0 (s k i) v0 e1 e2
                                                                     ex4_5
                                                                       F
                                                                       C
                                                                       C
                                                                       T
                                                                       T
                                                                       λf:F.λe1:C.λ:C.λu0:T.λ:T.eq C (CHead c3 k u) (CHead e1 (Flat f) u0)
                                                                       λf:F.λ:C.λe2:C.λ:T.λw:T.drop (s k i) (s k i) (CHead c4 k u) (CHead e2 (Flat f) w)
                                                                       λ:F.λ:C.λ:C.λu0:T.λw:T.subst0 (s k i) v0 u0 w
                                                                       λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 (s k i) v0 e1 e2))
                                           assume k: K
                                              we proceed by induction on k to prove 
                                                 ∀i:nat
                                                   .∀c3:C
                                                     .∀c4:C
                                                       .∀v0:T
                                                         .csubst0 i v0 c3 c4
                                                           →(eq nat i O
                                                                  →(or4
                                                                       drop i i c4 c3
                                                                       ex3_4
                                                                         F
                                                                         C
                                                                         T
                                                                         T
                                                                         λf:F.λe0:C.λu:T.λ:T.eq C c3 (CHead e0 (Flat f) u)
                                                                         λf:F.λe0:C.λ:T.λw:T.drop i i c4 (CHead e0 (Flat f) w)
                                                                         λ:F.λ:C.λu:T.λw:T.subst0 i v0 u w
                                                                       ex3_4
                                                                         F
                                                                         C
                                                                         C
                                                                         T
                                                                         λf:F.λe1:C.λ:C.λu:T.eq C c3 (CHead e1 (Flat f) u)
                                                                         λf:F.λ:C.λe2:C.λu:T.drop i i c4 (CHead e2 (Flat f) u)
                                                                         λ:F.λe1:C.λe2:C.λ:T.csubst0 i v0 e1 e2
                                                                       ex4_5
                                                                         F
                                                                         C
                                                                         C
                                                                         T
                                                                         T
                                                                         λf:F.λe1:C.λ:C.λu:T.λ:T.eq C c3 (CHead e1 (Flat f) u)
                                                                         λf:F.λ:C.λe2:C.λ:T.λw:T.drop i i c4 (CHead e2 (Flat f) w)
                                                                         λ:F.λ:C.λ:C.λu:T.λw:T.subst0 i v0 u w
                                                                         λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 i v0 e1 e2)
                                                                →∀u:T
                                                                     .eq nat (s k i) O
                                                                       →(or4
                                                                            drop (s k i) (s k i) (CHead c4 k u) (CHead c3 k u)
                                                                            ex3_4
                                                                              F
                                                                              C
                                                                              T
                                                                              T
                                                                              λf:F.λe0:C.λu0:T.λ:T.eq C (CHead c3 k u) (CHead e0 (Flat f) u0)
                                                                              λf:F.λe0:C.λ:T.λw:T.drop (s k i) (s k i) (CHead c4 k u) (CHead e0 (Flat f) w)
                                                                              λ:F.λ:C.λu0:T.λw:T.subst0 (s k i) v0 u0 w
                                                                            ex3_4
                                                                              F
                                                                              C
                                                                              C
                                                                              T
                                                                              λf:F.λe1:C.λ:C.λu0:T.eq C (CHead c3 k u) (CHead e1 (Flat f) u0)
                                                                              λf:F.λ:C.λe2:C.λu0:T.drop (s k i) (s k i) (CHead c4 k u) (CHead e2 (Flat f) u0)
                                                                              λ:F.λe1:C.λe2:C.λ:T.csubst0 (s k i) v0 e1 e2
                                                                            ex4_5
                                                                              F
                                                                              C
                                                                              C
                                                                              T
                                                                              T
                                                                              λf:F.λe1:C.λ:C.λu0:T.λ:T.eq C (CHead c3 k u) (CHead e1 (Flat f) u0)
                                                                              λf:F.λ:C.λe2:C.λ:T.λw:T.drop (s k i) (s k i) (CHead c4 k u) (CHead e2 (Flat f) w)
                                                                              λ:F.λ:C.λ:C.λu0:T.λw:T.subst0 (s k i) v0 u0 w
                                                                              λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 (s k i) v0 e1 e2))
                                                 case Bind : b:B ⇒
                                                    the thesis becomes 
                                                    ∀i:nat
                                                      .∀c3:C
                                                        .∀c4:C
                                                          .∀v0:T
                                                            .csubst0 i v0 c3 c4
                                                              →(eq nat i O
                                                                     →(or4
                                                                          drop i i c4 c3
                                                                          ex3_4
                                                                            F
                                                                            C
                                                                            T
                                                                            T
                                                                            λf:F.λe0:C.λu:T.λ:T.eq C c3 (CHead e0 (Flat f) u)
                                                                            λf:F.λe0:C.λ:T.λw:T.drop i i c4 (CHead e0 (Flat f) w)
                                                                            λ:F.λ:C.λu:T.λw:T.subst0 i v0 u w
                                                                          ex3_4
                                                                            F
                                                                            C
                                                                            C
                                                                            T
                                                                            λf:F.λe1:C.λ:C.λu:T.eq C c3 (CHead e1 (Flat f) u)
                                                                            λf:F.λ:C.λe2:C.λu:T.drop i i c4 (CHead e2 (Flat f) u)
                                                                            λ:F.λe1:C.λe2:C.λ:T.csubst0 i v0 e1 e2
                                                                          ex4_5
                                                                            F
                                                                            C
                                                                            C
                                                                            T
                                                                            T
                                                                            λf:F.λe1:C.λ:C.λu:T.λ:T.eq C c3 (CHead e1 (Flat f) u)
                                                                            λf:F.λ:C.λe2:C.λ:T.λw:T.drop i i c4 (CHead e2 (Flat f) w)
                                                                            λ:F.λ:C.λ:C.λu:T.λw:T.subst0 i v0 u w
                                                                            λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 i v0 e1 e2)
                                                                   →∀u:T
                                                                        .eq nat (s (Bind b) i) O
                                                                          →(or4
                                                                               drop
                                                                                 s (Bind b) i
                                                                                 s (Bind b) i
                                                                                 CHead c4 (Bind b) u
                                                                                 CHead c3 (Bind b) u
                                                                               ex3_4
                                                                                 F
                                                                                 C
                                                                                 T
                                                                                 T
                                                                                 λf:F.λe0:C.λu0:T.λ:T.eq C (CHead c3 (Bind b) u) (CHead e0 (Flat f) u0)
                                                                                 λf:F
                                                                                   .λe0:C
                                                                                     .λ:T
                                                                                       .λw:T
                                                                                         .drop
                                                                                           s (Bind b) i
                                                                                           s (Bind b) i
                                                                                           CHead c4 (Bind b) u
                                                                                           CHead e0 (Flat f) w
                                                                                 λ:F.λ:C.λu0:T.λw:T.subst0 (s (Bind b) i) v0 u0 w
                                                                               ex3_4
                                                                                 F
                                                                                 C
                                                                                 C
                                                                                 T
                                                                                 λf:F.λe1:C.λ:C.λu0:T.eq C (CHead c3 (Bind b) u) (CHead e1 (Flat f) u0)
                                                                                 λf:F
                                                                                   .λ:C
                                                                                     .λe2:C
                                                                                       .λu0:T
                                                                                         .drop
                                                                                           s (Bind b) i
                                                                                           s (Bind b) i
                                                                                           CHead c4 (Bind b) u
                                                                                           CHead e2 (Flat f) u0
                                                                                 λ:F.λe1:C.λe2:C.λ:T.csubst0 (s (Bind b) i) v0 e1 e2
                                                                               ex4_5
                                                                                 F
                                                                                 C
                                                                                 C
                                                                                 T
                                                                                 T
                                                                                 λf:F.λe1:C.λ:C.λu0:T.λ:T.eq C (CHead c3 (Bind b) u) (CHead e1 (Flat f) u0)
                                                                                 λf:F
                                                                                   .λ:C
                                                                                     .λe2:C
                                                                                       .λ:T
                                                                                         .λw:T
                                                                                           .drop
                                                                                             s (Bind b) i
                                                                                             s (Bind b) i
                                                                                             CHead c4 (Bind b) u
                                                                                             CHead e2 (Flat f) w
                                                                                 λ:F.λ:C.λ:C.λu0:T.λw:T.subst0 (s (Bind b) i) v0 u0 w
                                                                                 λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 (s (Bind b) i) v0 e1 e2))
                                                        assume i: nat
                                                        assume c3: C
                                                        assume c4: C
                                                        assume v0: T
                                                        suppose : csubst0 i v0 c3 c4
                                                        suppose : 
                                                           eq nat i O
                                                             →(or4
                                                                  drop i i c4 c3
                                                                  ex3_4
                                                                    F
                                                                    C
                                                                    T
                                                                    T
                                                                    λf:F.λe0:C.λu:T.λ:T.eq C c3 (CHead e0 (Flat f) u)
                                                                    λf:F.λe0:C.λ:T.λw:T.drop i i c4 (CHead e0 (Flat f) w)
                                                                    λ:F.λ:C.λu:T.λw:T.subst0 i v0 u w
                                                                  ex3_4
                                                                    F
                                                                    C
                                                                    C
                                                                    T
                                                                    λf:F.λe1:C.λ:C.λu:T.eq C c3 (CHead e1 (Flat f) u)
                                                                    λf:F.λ:C.λe2:C.λu:T.drop i i c4 (CHead e2 (Flat f) u)
                                                                    λ:F.λe1:C.λe2:C.λ:T.csubst0 i v0 e1 e2
                                                                  ex4_5
                                                                    F
                                                                    C
                                                                    C
                                                                    T
                                                                    T
                                                                    λf:F.λe1:C.λ:C.λu:T.λ:T.eq C c3 (CHead e1 (Flat f) u)
                                                                    λf:F.λ:C.λe2:C.λ:T.λw:T.drop i i c4 (CHead e2 (Flat f) w)
                                                                    λ:F.λ:C.λ:C.λu:T.λw:T.subst0 i v0 u w
                                                                    λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 i v0 e1 e2)
                                                        assume u: T
                                                          we must prove 
                                                                eq nat (s (Bind b) i) O
                                                                  →(or4
                                                                       drop
                                                                         s (Bind b) i
                                                                         s (Bind b) i
                                                                         CHead c4 (Bind b) u
                                                                         CHead c3 (Bind b) u
                                                                       ex3_4
                                                                         F
                                                                         C
                                                                         T
                                                                         T
                                                                         λf:F.λe0:C.λu0:T.λ:T.eq C (CHead c3 (Bind b) u) (CHead e0 (Flat f) u0)
                                                                         λf:F
                                                                           .λe0:C
                                                                             .λ:T
                                                                               .λw:T
                                                                                 .drop
                                                                                   s (Bind b) i
                                                                                   s (Bind b) i
                                                                                   CHead c4 (Bind b) u
                                                                                   CHead e0 (Flat f) w
                                                                         λ:F.λ:C.λu0:T.λw:T.subst0 (s (Bind b) i) v0 u0 w
                                                                       ex3_4
                                                                         F
                                                                         C
                                                                         C
                                                                         T
                                                                         λf:F.λe1:C.λ:C.λu0:T.eq C (CHead c3 (Bind b) u) (CHead e1 (Flat f) u0)
                                                                         λf:F
                                                                           .λ:C
                                                                             .λe2:C
                                                                               .λu0:T
                                                                                 .drop
                                                                                   s (Bind b) i
                                                                                   s (Bind b) i
                                                                                   CHead c4 (Bind b) u
                                                                                   CHead e2 (Flat f) u0
                                                                         λ:F.λe1:C.λe2:C.λ:T.csubst0 (s (Bind b) i) v0 e1 e2
                                                                       ex4_5
                                                                         F
                                                                         C
                                                                         C
                                                                         T
                                                                         T
                                                                         λf:F.λe1:C.λ:C.λu0:T.λ:T.eq C (CHead c3 (Bind b) u) (CHead e1 (Flat f) u0)
                                                                         λf:F
                                                                           .λ:C
                                                                             .λe2:C
                                                                               .λ:T
                                                                                 .λw:T
                                                                                   .drop
                                                                                     s (Bind b) i
                                                                                     s (Bind b) i
                                                                                     CHead c4 (Bind b) u
                                                                                     CHead e2 (Flat f) w
                                                                         λ:F.λ:C.λ:C.λu0:T.λw:T.subst0 (s (Bind b) i) v0 u0 w
                                                                         λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 (s (Bind b) i) v0 e1 e2)
                                                          or equivalently 
                                                                eq nat (S i) O
                                                                  →(or4
                                                                       drop
                                                                         s (Bind b) i
                                                                         s (Bind b) i
                                                                         CHead c4 (Bind b) u
                                                                         CHead c3 (Bind b) u
                                                                       ex3_4
                                                                         F
                                                                         C
                                                                         T
                                                                         T
                                                                         λf:F.λe0:C.λu0:T.λ:T.eq C (CHead c3 (Bind b) u) (CHead e0 (Flat f) u0)
                                                                         λf:F
                                                                           .λe0:C
                                                                             .λ:T
                                                                               .λw:T
                                                                                 .drop
                                                                                   s (Bind b) i
                                                                                   s (Bind b) i
                                                                                   CHead c4 (Bind b) u
                                                                                   CHead e0 (Flat f) w
                                                                         λ:F.λ:C.λu0:T.λw:T.subst0 (s (Bind b) i) v0 u0 w
                                                                       ex3_4
                                                                         F
                                                                         C
                                                                         C
                                                                         T
                                                                         λf:F.λe1:C.λ:C.λu0:T.eq C (CHead c3 (Bind b) u) (CHead e1 (Flat f) u0)
                                                                         λf:F
                                                                           .λ:C
                                                                             .λe2:C
                                                                               .λu0:T
                                                                                 .drop
                                                                                   s (Bind b) i
                                                                                   s (Bind b) i
                                                                                   CHead c4 (Bind b) u
                                                                                   CHead e2 (Flat f) u0
                                                                         λ:F.λe1:C.λe2:C.λ:T.csubst0 (s (Bind b) i) v0 e1 e2
                                                                       ex4_5
                                                                         F
                                                                         C
                                                                         C
                                                                         T
                                                                         T
                                                                         λf:F.λe1:C.λ:C.λu0:T.λ:T.eq C (CHead c3 (Bind b) u) (CHead e1 (Flat f) u0)
                                                                         λf:F
                                                                           .λ:C
                                                                             .λe2:C
                                                                               .λ:T
                                                                                 .λw:T
                                                                                   .drop
                                                                                     s (Bind b) i
                                                                                     s (Bind b) i
                                                                                     CHead c4 (Bind b) u
                                                                                     CHead e2 (Flat f) w
                                                                         λ:F.λ:C.λ:C.λu0:T.λw:T.subst0 (s (Bind b) i) v0 u0 w
                                                                         λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 (s (Bind b) i) v0 e1 e2)
                                                          suppose H4: eq nat (S i) O
                                                             (H5) 
                                                                we proceed by induction on H4 to prove <λ:nat.Prop> CASE O OF O⇒False | S ⇒True
                                                                   case refl_equal : ⇒
                                                                      the thesis becomes <λ:nat.Prop> CASE S i OF O⇒False | S ⇒True
                                                                         consider I
                                                                         we proved True
<λ:nat.Prop> CASE S i OF O⇒False | S ⇒True
<λ:nat.Prop> CASE O OF O⇒False | S ⇒True
                                                             end of H5
                                                             consider H5
                                                             we proved <λ:nat.Prop> CASE O OF O⇒False | S ⇒True
                                                             that is equivalent to False
                                                             we proceed by induction on the previous result to prove 
                                                                or4
                                                                  drop (S i) (S i) (CHead c4 (Bind b) u) (CHead c3 (Bind b) u)
                                                                  ex3_4
                                                                    F
                                                                    C
                                                                    T
                                                                    T
                                                                    λf:F.λe0:C.λu0:T.λ:T.eq C (CHead c3 (Bind b) u) (CHead e0 (Flat f) u0)
                                                                    λf:F
                                                                      .λe0:C.λ:T.λw:T.drop (S i) (S i) (CHead c4 (Bind b) u) (CHead e0 (Flat f) w)
                                                                    λ:F.λ:C.λu0:T.λw:T.subst0 (S i) v0 u0 w
                                                                  ex3_4
                                                                    F
                                                                    C
                                                                    C
                                                                    T
                                                                    λf:F.λe1:C.λ:C.λu0:T.eq C (CHead c3 (Bind b) u) (CHead e1 (Flat f) u0)
                                                                    λf:F.λ:C.λe2:C.λu0:T.drop (S i) (S i) (CHead c4 (Bind b) u) (CHead e2 (Flat f) u0)
                                                                    λ:F.λe1:C.λe2:C.λ:T.csubst0 (S i) v0 e1 e2
                                                                  ex4_5
                                                                    F
                                                                    C
                                                                    C
                                                                    T
                                                                    T
                                                                    λf:F.λe1:C.λ:C.λu0:T.λ:T.eq C (CHead c3 (Bind b) u) (CHead e1 (Flat f) u0)
                                                                    λf:F
                                                                      .λ:C
                                                                        .λe2:C.λ:T.λw:T.drop (S i) (S i) (CHead c4 (Bind b) u) (CHead e2 (Flat f) w)
                                                                    λ:F.λ:C.λ:C.λu0:T.λw:T.subst0 (S i) v0 u0 w
                                                                    λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 (S i) v0 e1 e2
                                                             we proved 
                                                                or4
                                                                  drop (S i) (S i) (CHead c4 (Bind b) u) (CHead c3 (Bind b) u)
                                                                  ex3_4
                                                                    F
                                                                    C
                                                                    T
                                                                    T
                                                                    λf:F.λe0:C.λu0:T.λ:T.eq C (CHead c3 (Bind b) u) (CHead e0 (Flat f) u0)
                                                                    λf:F
                                                                      .λe0:C.λ:T.λw:T.drop (S i) (S i) (CHead c4 (Bind b) u) (CHead e0 (Flat f) w)
                                                                    λ:F.λ:C.λu0:T.λw:T.subst0 (S i) v0 u0 w
                                                                  ex3_4
                                                                    F
                                                                    C
                                                                    C
                                                                    T
                                                                    λf:F.λe1:C.λ:C.λu0:T.eq C (CHead c3 (Bind b) u) (CHead e1 (Flat f) u0)
                                                                    λf:F.λ:C.λe2:C.λu0:T.drop (S i) (S i) (CHead c4 (Bind b) u) (CHead e2 (Flat f) u0)
                                                                    λ:F.λe1:C.λe2:C.λ:T.csubst0 (S i) v0 e1 e2
                                                                  ex4_5
                                                                    F
                                                                    C
                                                                    C
                                                                    T
                                                                    T
                                                                    λf:F.λe1:C.λ:C.λu0:T.λ:T.eq C (CHead c3 (Bind b) u) (CHead e1 (Flat f) u0)
                                                                    λf:F
                                                                      .λ:C
                                                                        .λe2:C.λ:T.λw:T.drop (S i) (S i) (CHead c4 (Bind b) u) (CHead e2 (Flat f) w)
                                                                    λ:F.λ:C.λ:C.λu0:T.λw:T.subst0 (S i) v0 u0 w
                                                                    λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 (S i) v0 e1 e2
                                                             that is equivalent to 
                                                                or4
                                                                  drop
                                                                    s (Bind b) i
                                                                    s (Bind b) i
                                                                    CHead c4 (Bind b) u
                                                                    CHead c3 (Bind b) u
                                                                  ex3_4
                                                                    F
                                                                    C
                                                                    T
                                                                    T
                                                                    λf:F.λe0:C.λu0:T.λ:T.eq C (CHead c3 (Bind b) u) (CHead e0 (Flat f) u0)
                                                                    λf:F
                                                                      .λe0:C
                                                                        .λ:T
                                                                          .λw:T
                                                                            .drop
                                                                              s (Bind b) i
                                                                              s (Bind b) i
                                                                              CHead c4 (Bind b) u
                                                                              CHead e0 (Flat f) w
                                                                    λ:F.λ:C.λu0:T.λw:T.subst0 (s (Bind b) i) v0 u0 w
                                                                  ex3_4
                                                                    F
                                                                    C
                                                                    C
                                                                    T
                                                                    λf:F.λe1:C.λ:C.λu0:T.eq C (CHead c3 (Bind b) u) (CHead e1 (Flat f) u0)
                                                                    λf:F
                                                                      .λ:C
                                                                        .λe2:C
                                                                          .λu0:T
                                                                            .drop
                                                                              s (Bind b) i
                                                                              s (Bind b) i
                                                                              CHead c4 (Bind b) u
                                                                              CHead e2 (Flat f) u0
                                                                    λ:F.λe1:C.λe2:C.λ:T.csubst0 (s (Bind b) i) v0 e1 e2
                                                                  ex4_5
                                                                    F
                                                                    C
                                                                    C
                                                                    T
                                                                    T
                                                                    λf:F.λe1:C.λ:C.λu0:T.λ:T.eq C (CHead c3 (Bind b) u) (CHead e1 (Flat f) u0)
                                                                    λf:F
                                                                      .λ:C
                                                                        .λe2:C
                                                                          .λ:T
                                                                            .λw:T
                                                                              .drop
                                                                                s (Bind b) i
                                                                                s (Bind b) i
                                                                                CHead c4 (Bind b) u
                                                                                CHead e2 (Flat f) w
                                                                    λ:F.λ:C.λ:C.λu0:T.λw:T.subst0 (s (Bind b) i) v0 u0 w
                                                                    λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 (s (Bind b) i) v0 e1 e2
                                                          we proved 
                                                             eq nat (S i) O
                                                               →(or4
                                                                    drop
                                                                      s (Bind b) i
                                                                      s (Bind b) i
                                                                      CHead c4 (Bind b) u
                                                                      CHead c3 (Bind b) u
                                                                    ex3_4
                                                                      F
                                                                      C
                                                                      T
                                                                      T
                                                                      λf:F.λe0:C.λu0:T.λ:T.eq C (CHead c3 (Bind b) u) (CHead e0 (Flat f) u0)
                                                                      λf:F
                                                                        .λe0:C
                                                                          .λ:T
                                                                            .λw:T
                                                                              .drop
                                                                                s (Bind b) i
                                                                                s (Bind b) i
                                                                                CHead c4 (Bind b) u
                                                                                CHead e0 (Flat f) w
                                                                      λ:F.λ:C.λu0:T.λw:T.subst0 (s (Bind b) i) v0 u0 w
                                                                    ex3_4
                                                                      F
                                                                      C
                                                                      C
                                                                      T
                                                                      λf:F.λe1:C.λ:C.λu0:T.eq C (CHead c3 (Bind b) u) (CHead e1 (Flat f) u0)
                                                                      λf:F
                                                                        .λ:C
                                                                          .λe2:C
                                                                            .λu0:T
                                                                              .drop
                                                                                s (Bind b) i
                                                                                s (Bind b) i
                                                                                CHead c4 (Bind b) u
                                                                                CHead e2 (Flat f) u0
                                                                      λ:F.λe1:C.λe2:C.λ:T.csubst0 (s (Bind b) i) v0 e1 e2
                                                                    ex4_5
                                                                      F
                                                                      C
                                                                      C
                                                                      T
                                                                      T
                                                                      λf:F.λe1:C.λ:C.λu0:T.λ:T.eq C (CHead c3 (Bind b) u) (CHead e1 (Flat f) u0)
                                                                      λf:F
                                                                        .λ:C
                                                                          .λe2:C
                                                                            .λ:T
                                                                              .λw:T
                                                                                .drop
                                                                                  s (Bind b) i
                                                                                  s (Bind b) i
                                                                                  CHead c4 (Bind b) u
                                                                                  CHead e2 (Flat f) w
                                                                      λ:F.λ:C.λ:C.λu0:T.λw:T.subst0 (s (Bind b) i) v0 u0 w
                                                                      λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 (s (Bind b) i) v0 e1 e2)
                                                          that is equivalent to 
                                                             eq nat (s (Bind b) i) O
                                                               →(or4
                                                                    drop
                                                                      s (Bind b) i
                                                                      s (Bind b) i
                                                                      CHead c4 (Bind b) u
                                                                      CHead c3 (Bind b) u
                                                                    ex3_4
                                                                      F
                                                                      C
                                                                      T
                                                                      T
                                                                      λf:F.λe0:C.λu0:T.λ:T.eq C (CHead c3 (Bind b) u) (CHead e0 (Flat f) u0)
                                                                      λf:F
                                                                        .λe0:C
                                                                          .λ:T
                                                                            .λw:T
                                                                              .drop
                                                                                s (Bind b) i
                                                                                s (Bind b) i
                                                                                CHead c4 (Bind b) u
                                                                                CHead e0 (Flat f) w
                                                                      λ:F.λ:C.λu0:T.λw:T.subst0 (s (Bind b) i) v0 u0 w
                                                                    ex3_4
                                                                      F
                                                                      C
                                                                      C
                                                                      T
                                                                      λf:F.λe1:C.λ:C.λu0:T.eq C (CHead c3 (Bind b) u) (CHead e1 (Flat f) u0)
                                                                      λf:F
                                                                        .λ:C
                                                                          .λe2:C
                                                                            .λu0:T
                                                                              .drop
                                                                                s (Bind b) i
                                                                                s (Bind b) i
                                                                                CHead c4 (Bind b) u
                                                                                CHead e2 (Flat f) u0
                                                                      λ:F.λe1:C.λe2:C.λ:T.csubst0 (s (Bind b) i) v0 e1 e2
                                                                    ex4_5
                                                                      F
                                                                      C
                                                                      C
                                                                      T
                                                                      T
                                                                      λf:F.λe1:C.λ:C.λu0:T.λ:T.eq C (CHead c3 (Bind b) u) (CHead e1 (Flat f) u0)
                                                                      λf:F
                                                                        .λ:C
                                                                          .λe2:C
                                                                            .λ:T
                                                                              .λw:T
                                                                                .drop
                                                                                  s (Bind b) i
                                                                                  s (Bind b) i
                                                                                  CHead c4 (Bind b) u
                                                                                  CHead e2 (Flat f) w
                                                                      λ:F.λ:C.λ:C.λu0:T.λw:T.subst0 (s (Bind b) i) v0 u0 w
                                                                      λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 (s (Bind b) i) v0 e1 e2)

                                                          ∀i:nat
                                                            .∀c3:C
                                                              .∀c4:C
                                                                .∀v0:T
                                                                  .csubst0 i v0 c3 c4
                                                                    →(eq nat i O
                                                                           →(or4
                                                                                drop i i c4 c3
                                                                                ex3_4
                                                                                  F
                                                                                  C
                                                                                  T
                                                                                  T
                                                                                  λf:F.λe0:C.λu:T.λ:T.eq C c3 (CHead e0 (Flat f) u)
                                                                                  λf:F.λe0:C.λ:T.λw:T.drop i i c4 (CHead e0 (Flat f) w)
                                                                                  λ:F.λ:C.λu:T.λw:T.subst0 i v0 u w
                                                                                ex3_4
                                                                                  F
                                                                                  C
                                                                                  C
                                                                                  T
                                                                                  λf:F.λe1:C.λ:C.λu:T.eq C c3 (CHead e1 (Flat f) u)
                                                                                  λf:F.λ:C.λe2:C.λu:T.drop i i c4 (CHead e2 (Flat f) u)
                                                                                  λ:F.λe1:C.λe2:C.λ:T.csubst0 i v0 e1 e2
                                                                                ex4_5
                                                                                  F
                                                                                  C
                                                                                  C
                                                                                  T
                                                                                  T
                                                                                  λf:F.λe1:C.λ:C.λu:T.λ:T.eq C c3 (CHead e1 (Flat f) u)
                                                                                  λf:F.λ:C.λe2:C.λ:T.λw:T.drop i i c4 (CHead e2 (Flat f) w)
                                                                                  λ:F.λ:C.λ:C.λu:T.λw:T.subst0 i v0 u w
                                                                                  λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 i v0 e1 e2)
                                                                         →∀u:T
                                                                              .eq nat (s (Bind b) i) O
                                                                                →(or4
                                                                                     drop
                                                                                       s (Bind b) i
                                                                                       s (Bind b) i
                                                                                       CHead c4 (Bind b) u
                                                                                       CHead c3 (Bind b) u
                                                                                     ex3_4
                                                                                       F
                                                                                       C
                                                                                       T
                                                                                       T
                                                                                       λf:F.λe0:C.λu0:T.λ:T.eq C (CHead c3 (Bind b) u) (CHead e0 (Flat f) u0)
                                                                                       λf:F
                                                                                         .λe0:C
                                                                                           .λ:T
                                                                                             .λw:T
                                                                                               .drop
                                                                                                 s (Bind b) i
                                                                                                 s (Bind b) i
                                                                                                 CHead c4 (Bind b) u
                                                                                                 CHead e0 (Flat f) w
                                                                                       λ:F.λ:C.λu0:T.λw:T.subst0 (s (Bind b) i) v0 u0 w
                                                                                     ex3_4
                                                                                       F
                                                                                       C
                                                                                       C
                                                                                       T
                                                                                       λf:F.λe1:C.λ:C.λu0:T.eq C (CHead c3 (Bind b) u) (CHead e1 (Flat f) u0)
                                                                                       λf:F
                                                                                         .λ:C
                                                                                           .λe2:C
                                                                                             .λu0:T
                                                                                               .drop
                                                                                                 s (Bind b) i
                                                                                                 s (Bind b) i
                                                                                                 CHead c4 (Bind b) u
                                                                                                 CHead e2 (Flat f) u0
                                                                                       λ:F.λe1:C.λe2:C.λ:T.csubst0 (s (Bind b) i) v0 e1 e2
                                                                                     ex4_5
                                                                                       F
                                                                                       C
                                                                                       C
                                                                                       T
                                                                                       T
                                                                                       λf:F.λe1:C.λ:C.λu0:T.λ:T.eq C (CHead c3 (Bind b) u) (CHead e1 (Flat f) u0)
                                                                                       λf:F
                                                                                         .λ:C
                                                                                           .λe2:C
                                                                                             .λ:T
                                                                                               .λw:T
                                                                                                 .drop
                                                                                                   s (Bind b) i
                                                                                                   s (Bind b) i
                                                                                                   CHead c4 (Bind b) u
                                                                                                   CHead e2 (Flat f) w
                                                                                       λ:F.λ:C.λ:C.λu0:T.λw:T.subst0 (s (Bind b) i) v0 u0 w
                                                                                       λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 (s (Bind b) i) v0 e1 e2))
                                                 case Flat : f:F ⇒
                                                    the thesis becomes 
                                                    ∀i:nat
                                                      .∀c3:C
                                                        .∀c4:C
                                                          .∀v0:T
                                                            .∀H2:csubst0 i v0 c3 c4
                                                              .∀H3:eq nat i O
                                                                         →(or4
                                                                              drop i i c4 c3
                                                                              ex3_4
                                                                                F
                                                                                C
                                                                                T
                                                                                T
                                                                                λf0:F.λe0:C.λu:T.λ:T.eq C c3 (CHead e0 (Flat f0) u)
                                                                                λf0:F.λe0:C.λ:T.λw:T.drop i i c4 (CHead e0 (Flat f0) w)
                                                                                λ:F.λ:C.λu:T.λw:T.subst0 i v0 u w
                                                                              ex3_4
                                                                                F
                                                                                C
                                                                                C
                                                                                T
                                                                                λf0:F.λe1:C.λ:C.λu:T.eq C c3 (CHead e1 (Flat f0) u)
                                                                                λf0:F.λ:C.λe2:C.λu:T.drop i i c4 (CHead e2 (Flat f0) u)
                                                                                λ:F.λe1:C.λe2:C.λ:T.csubst0 i v0 e1 e2
                                                                              ex4_5
                                                                                F
                                                                                C
                                                                                C
                                                                                T
                                                                                T
                                                                                λf0:F.λe1:C.λ:C.λu:T.λ:T.eq C c3 (CHead e1 (Flat f0) u)
                                                                                λf0:F.λ:C.λe2:C.λ:T.λw:T.drop i i c4 (CHead e2 (Flat f0) w)
                                                                                λ:F.λ:C.λ:C.λu:T.λw:T.subst0 i v0 u w
                                                                                λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 i v0 e1 e2)
                                                                .∀u:T
                                                                  .eq nat (s (Flat f) i) O
                                                                    →(or4
                                                                         drop
                                                                           s (Flat f) i
                                                                           s (Flat f) i
                                                                           CHead c4 (Flat f) u
                                                                           CHead c3 (Flat f) u
                                                                         ex3_4
                                                                           F
                                                                           C
                                                                           T
                                                                           T
                                                                           λf:F.λe0:C.λu0:T.λ:T.eq C (CHead c3 (Flat f) u) (CHead e0 (Flat f) u0)
                                                                           λf:F
                                                                             .λe0:C
                                                                               .λ:T
                                                                                 .λw:T
                                                                                   .drop
                                                                                     s (Flat f) i
                                                                                     s (Flat f) i
                                                                                     CHead c4 (Flat f) u
                                                                                     CHead e0 (Flat f) w
                                                                           λ:F.λ:C.λu0:T.λw:T.subst0 (s (Flat f) i) v0 u0 w
                                                                         ex3_4
                                                                           F
                                                                           C
                                                                           C
                                                                           T
                                                                           λf:F.λe1:C.λ:C.λu0:T.eq C (CHead c3 (Flat f) u) (CHead e1 (Flat f) u0)
                                                                           λf:F
                                                                             .λ:C
                                                                               .λe2:C
                                                                                 .λu0:T
                                                                                   .drop
                                                                                     s (Flat f) i
                                                                                     s (Flat f) i
                                                                                     CHead c4 (Flat f) u
                                                                                     CHead e2 (Flat f) u0
                                                                           λ:F.λe1:C.λe2:C.λ:T.csubst0 (s (Flat f) i) v0 e1 e2
                                                                         ex4_5
                                                                           F
                                                                           C
                                                                           C
                                                                           T
                                                                           T
                                                                           λf:F.λe1:C.λ:C.λu0:T.λ:T.eq C (CHead c3 (Flat f) u) (CHead e1 (Flat f) u0)
                                                                           λf:F
                                                                             .λ:C
                                                                               .λe2:C
                                                                                 .λ:T
                                                                                   .λw:T
                                                                                     .drop
                                                                                       s (Flat f) i
                                                                                       s (Flat f) i
                                                                                       CHead c4 (Flat f) u
                                                                                       CHead e2 (Flat f) w
                                                                           λ:F.λ:C.λ:C.λu0:T.λw:T.subst0 (s (Flat f) i) v0 u0 w
                                                                           λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 (s (Flat f) i) v0 e1 e2)
                                                        assume i: nat
                                                        assume c3: C
                                                        assume c4: C
                                                        assume v0: T
                                                        suppose H2: csubst0 i v0 c3 c4
                                                        suppose H3: 
                                                           eq nat i O
                                                             →(or4
                                                                  drop i i c4 c3
                                                                  ex3_4
                                                                    F
                                                                    C
                                                                    T
                                                                    T
                                                                    λf0:F.λe0:C.λu:T.λ:T.eq C c3 (CHead e0 (Flat f0) u)
                                                                    λf0:F.λe0:C.λ:T.λw:T.drop i i c4 (CHead e0 (Flat f0) w)
                                                                    λ:F.λ:C.λu:T.λw:T.subst0 i v0 u w
                                                                  ex3_4
                                                                    F
                                                                    C
                                                                    C
                                                                    T
                                                                    λf0:F.λe1:C.λ:C.λu:T.eq C c3 (CHead e1 (Flat f0) u)
                                                                    λf0:F.λ:C.λe2:C.λu:T.drop i i c4 (CHead e2 (Flat f0) u)
                                                                    λ:F.λe1:C.λe2:C.λ:T.csubst0 i v0 e1 e2
                                                                  ex4_5
                                                                    F
                                                                    C
                                                                    C
                                                                    T
                                                                    T
                                                                    λf0:F.λe1:C.λ:C.λu:T.λ:T.eq C c3 (CHead e1 (Flat f0) u)
                                                                    λf0:F.λ:C.λe2:C.λ:T.λw:T.drop i i c4 (CHead e2 (Flat f0) w)
                                                                    λ:F.λ:C.λ:C.λu:T.λw:T.subst0 i v0 u w
                                                                    λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 i v0 e1 e2)
                                                        assume u: T
                                                          we must prove 
                                                                eq nat (s (Flat f) i) O
                                                                  →(or4
                                                                       drop
                                                                         s (Flat f) i
                                                                         s (Flat f) i
                                                                         CHead c4 (Flat f) u
                                                                         CHead c3 (Flat f) u
                                                                       ex3_4
                                                                         F
                                                                         C
                                                                         T
                                                                         T
                                                                         λf:F.λe0:C.λu0:T.λ:T.eq C (CHead c3 (Flat f) u) (CHead e0 (Flat f) u0)
                                                                         λf:F
                                                                           .λe0:C
                                                                             .λ:T
                                                                               .λw:T
                                                                                 .drop
                                                                                   s (Flat f) i
                                                                                   s (Flat f) i
                                                                                   CHead c4 (Flat f) u
                                                                                   CHead e0 (Flat f) w
                                                                         λ:F.λ:C.λu0:T.λw:T.subst0 (s (Flat f) i) v0 u0 w
                                                                       ex3_4
                                                                         F
                                                                         C
                                                                         C
                                                                         T
                                                                         λf:F.λe1:C.λ:C.λu0:T.eq C (CHead c3 (Flat f) u) (CHead e1 (Flat f) u0)
                                                                         λf:F
                                                                           .λ:C
                                                                             .λe2:C
                                                                               .λu0:T
                                                                                 .drop
                                                                                   s (Flat f) i
                                                                                   s (Flat f) i
                                                                                   CHead c4 (Flat f) u
                                                                                   CHead e2 (Flat f) u0
                                                                         λ:F.λe1:C.λe2:C.λ:T.csubst0 (s (Flat f) i) v0 e1 e2
                                                                       ex4_5
                                                                         F
                                                                         C
                                                                         C
                                                                         T
                                                                         T
                                                                         λf:F.λe1:C.λ:C.λu0:T.λ:T.eq C (CHead c3 (Flat f) u) (CHead e1 (Flat f) u0)
                                                                         λf:F
                                                                           .λ:C
                                                                             .λe2:C
                                                                               .λ:T
                                                                                 .λw:T
                                                                                   .drop
                                                                                     s (Flat f) i
                                                                                     s (Flat f) i
                                                                                     CHead c4 (Flat f) u
                                                                                     CHead e2 (Flat f) w
                                                                         λ:F.λ:C.λ:C.λu0:T.λw:T.subst0 (s (Flat f) i) v0 u0 w
                                                                         λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 (s (Flat f) i) v0 e1 e2)
                                                          or equivalently 
                                                                eq nat i O
                                                                  →(or4
                                                                       drop
                                                                         s (Flat f) i
                                                                         s (Flat f) i
                                                                         CHead c4 (Flat f) u
                                                                         CHead c3 (Flat f) u
                                                                       ex3_4
                                                                         F
                                                                         C
                                                                         T
                                                                         T
                                                                         λf:F.λe0:C.λu0:T.λ:T.eq C (CHead c3 (Flat f) u) (CHead e0 (Flat f) u0)
                                                                         λf:F
                                                                           .λe0:C
                                                                             .λ:T
                                                                               .λw:T
                                                                                 .drop
                                                                                   s (Flat f) i
                                                                                   s (Flat f) i
                                                                                   CHead c4 (Flat f) u
                                                                                   CHead e0 (Flat f) w
                                                                         λ:F.λ:C.λu0:T.λw:T.subst0 (s (Flat f) i) v0 u0 w
                                                                       ex3_4
                                                                         F
                                                                         C
                                                                         C
                                                                         T
                                                                         λf:F.λe1:C.λ:C.λu0:T.eq C (CHead c3 (Flat f) u) (CHead e1 (Flat f) u0)
                                                                         λf:F
                                                                           .λ:C
                                                                             .λe2:C
                                                                               .λu0:T
                                                                                 .drop
                                                                                   s (Flat f) i
                                                                                   s (Flat f) i
                                                                                   CHead c4 (Flat f) u
                                                                                   CHead e2 (Flat f) u0
                                                                         λ:F.λe1:C.λe2:C.λ:T.csubst0 (s (Flat f) i) v0 e1 e2
                                                                       ex4_5
                                                                         F
                                                                         C
                                                                         C
                                                                         T
                                                                         T
                                                                         λf:F.λe1:C.λ:C.λu0:T.λ:T.eq C (CHead c3 (Flat f) u) (CHead e1 (Flat f) u0)
                                                                         λf:F
                                                                           .λ:C
                                                                             .λe2:C
                                                                               .λ:T
                                                                                 .λw:T
                                                                                   .drop
                                                                                     s (Flat f) i
                                                                                     s (Flat f) i
                                                                                     CHead c4 (Flat f) u
                                                                                     CHead e2 (Flat f) w
                                                                         λ:F.λ:C.λ:C.λu0:T.λw:T.subst0 (s (Flat f) i) v0 u0 w
                                                                         λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 (s (Flat f) i) v0 e1 e2)
                                                          suppose H4: eq nat i O
                                                             (H6) 
                                                                we proceed by induction on H4 to prove csubst0 O v0 c3 c4
                                                                   case refl_equal : ⇒
                                                                      the thesis becomes the hypothesis H2
csubst0 O v0 c3 c4
                                                             end of H6
                                                             (h1) 
                                                                by (refl_equal . .)
eq C (CHead c3 (Flat f) u) (CHead c3 (Flat f) u)
                                                             end of h1
                                                             (h2) 
                                                                by (drop_refl .)
drop O O (CHead c4 (Flat f) u) (CHead c4 (Flat f) u)
                                                             end of h2
                                                             by (ex3_4_intro . . . . . . . . . . . h1 h2 H6)
                                                             we proved 
                                                                ex3_4
                                                                  F
                                                                  C
                                                                  C
                                                                  T
                                                                  λf0:F.λe1:C.λ:C.λu0:T.eq C (CHead c3 (Flat f) u) (CHead e1 (Flat f0) u0)
                                                                  λf0:F.λ:C.λe2:C.λu0:T.drop O O (CHead c4 (Flat f) u) (CHead e2 (Flat f0) u0)
                                                                  λ:F.λe1:C.λe2:C.λ:T.csubst0 O v0 e1 e2
                                                             by (or4_intro2 . . . . previous)
                                                             we proved 
                                                                or4
                                                                  drop O O (CHead c4 (Flat f) u) (CHead c3 (Flat f) u)
                                                                  ex3_4
                                                                    F
                                                                    C
                                                                    T
                                                                    T
                                                                    λf0:F.λe0:C.λu0:T.λ:T.eq C (CHead c3 (Flat f) u) (CHead e0 (Flat f0) u0)
                                                                    λf0:F.λe0:C.λ:T.λw:T.drop O O (CHead c4 (Flat f) u) (CHead e0 (Flat f0) w)
                                                                    λ:F.λ:C.λu0:T.λw:T.subst0 O v0 u0 w
                                                                  ex3_4
                                                                    F
                                                                    C
                                                                    C
                                                                    T
                                                                    λf0:F.λe1:C.λ:C.λu0:T.eq C (CHead c3 (Flat f) u) (CHead e1 (Flat f0) u0)
                                                                    λf0:F.λ:C.λe2:C.λu0:T.drop O O (CHead c4 (Flat f) u) (CHead e2 (Flat f0) u0)
                                                                    λ:F.λe1:C.λe2:C.λ:T.csubst0 O v0 e1 e2
                                                                  ex4_5
                                                                    F
                                                                    C
                                                                    C
                                                                    T
                                                                    T
                                                                    λf0:F.λe1:C.λ:C.λu0:T.λ:T.eq C (CHead c3 (Flat f) u) (CHead e1 (Flat f0) u0)
                                                                    λf0:F.λ:C.λe2:C.λ:T.λw:T.drop O O (CHead c4 (Flat f) u) (CHead e2 (Flat f0) w)
                                                                    λ:F.λ:C.λ:C.λu0:T.λw:T.subst0 O v0 u0 w
                                                                    λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v0 e1 e2
                                                             by (eq_ind_r . . . previous . H4)
                                                             we proved 
                                                                or4
                                                                  drop i i (CHead c4 (Flat f) u) (CHead c3 (Flat f) u)
                                                                  ex3_4
                                                                    F
                                                                    C
                                                                    T
                                                                    T
                                                                    λf0:F.λe0:C.λu0:T.λ:T.eq C (CHead c3 (Flat f) u) (CHead e0 (Flat f0) u0)
                                                                    λf0:F.λe0:C.λ:T.λw:T.drop i i (CHead c4 (Flat f) u) (CHead e0 (Flat f0) w)
                                                                    λ:F.λ:C.λu0:T.λw:T.subst0 i v0 u0 w
                                                                  ex3_4
                                                                    F
                                                                    C
                                                                    C
                                                                    T
                                                                    λf0:F.λe1:C.λ:C.λu0:T.eq C (CHead c3 (Flat f) u) (CHead e1 (Flat f0) u0)
                                                                    λf0:F.λ:C.λe2:C.λu0:T.drop i i (CHead c4 (Flat f) u) (CHead e2 (Flat f0) u0)
                                                                    λ:F.λe1:C.λe2:C.λ:T.csubst0 i v0 e1 e2
                                                                  ex4_5
                                                                    F
                                                                    C
                                                                    C
                                                                    T
                                                                    T
                                                                    λf0:F.λe1:C.λ:C.λu0:T.λ:T.eq C (CHead c3 (Flat f) u) (CHead e1 (Flat f0) u0)
                                                                    λf0:F.λ:C.λe2:C.λ:T.λw:T.drop i i (CHead c4 (Flat f) u) (CHead e2 (Flat f0) w)
                                                                    λ:F.λ:C.λ:C.λu0:T.λw:T.subst0 i v0 u0 w
                                                                    λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 i v0 e1 e2
                                                             that is equivalent to 
                                                                or4
                                                                  drop
                                                                    s (Flat f) i
                                                                    s (Flat f) i
                                                                    CHead c4 (Flat f) u
                                                                    CHead c3 (Flat f) u
                                                                  ex3_4
                                                                    F
                                                                    C
                                                                    T
                                                                    T
                                                                    λf:F.λe0:C.λu0:T.λ:T.eq C (CHead c3 (Flat f) u) (CHead e0 (Flat f) u0)
                                                                    λf:F
                                                                      .λe0:C
                                                                        .λ:T
                                                                          .λw:T
                                                                            .drop
                                                                              s (Flat f) i
                                                                              s (Flat f) i
                                                                              CHead c4 (Flat f) u
                                                                              CHead e0 (Flat f) w
                                                                    λ:F.λ:C.λu0:T.λw:T.subst0 (s (Flat f) i) v0 u0 w
                                                                  ex3_4
                                                                    F
                                                                    C
                                                                    C
                                                                    T
                                                                    λf:F.λe1:C.λ:C.λu0:T.eq C (CHead c3 (Flat f) u) (CHead e1 (Flat f) u0)
                                                                    λf:F
                                                                      .λ:C
                                                                        .λe2:C
                                                                          .λu0:T
                                                                            .drop
                                                                              s (Flat f) i
                                                                              s (Flat f) i
                                                                              CHead c4 (Flat f) u
                                                                              CHead e2 (Flat f) u0
                                                                    λ:F.λe1:C.λe2:C.λ:T.csubst0 (s (Flat f) i) v0 e1 e2
                                                                  ex4_5
                                                                    F
                                                                    C
                                                                    C
                                                                    T
                                                                    T
                                                                    λf:F.λe1:C.λ:C.λu0:T.λ:T.eq C (CHead c3 (Flat f) u) (CHead e1 (Flat f) u0)
                                                                    λf:F
                                                                      .λ:C
                                                                        .λe2:C
                                                                          .λ:T
                                                                            .λw:T
                                                                              .drop
                                                                                s (Flat f) i
                                                                                s (Flat f) i
                                                                                CHead c4 (Flat f) u
                                                                                CHead e2 (Flat f) w
                                                                    λ:F.λ:C.λ:C.λu0:T.λw:T.subst0 (s (Flat f) i) v0 u0 w
                                                                    λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 (s (Flat f) i) v0 e1 e2
                                                          we proved 
                                                             eq nat i O
                                                               →(or4
                                                                    drop
                                                                      s (Flat f) i
                                                                      s (Flat f) i
                                                                      CHead c4 (Flat f) u
                                                                      CHead c3 (Flat f) u
                                                                    ex3_4
                                                                      F
                                                                      C
                                                                      T
                                                                      T
                                                                      λf:F.λe0:C.λu0:T.λ:T.eq C (CHead c3 (Flat f) u) (CHead e0 (Flat f) u0)
                                                                      λf:F
                                                                        .λe0:C
                                                                          .λ:T
                                                                            .λw:T
                                                                              .drop
                                                                                s (Flat f) i
                                                                                s (Flat f) i
                                                                                CHead c4 (Flat f) u
                                                                                CHead e0 (Flat f) w
                                                                      λ:F.λ:C.λu0:T.λw:T.subst0 (s (Flat f) i) v0 u0 w
                                                                    ex3_4
                                                                      F
                                                                      C
                                                                      C
                                                                      T
                                                                      λf:F.λe1:C.λ:C.λu0:T.eq C (CHead c3 (Flat f) u) (CHead e1 (Flat f) u0)
                                                                      λf:F
                                                                        .λ:C
                                                                          .λe2:C
                                                                            .λu0:T
                                                                              .drop
                                                                                s (Flat f) i
                                                                                s (Flat f) i
                                                                                CHead c4 (Flat f) u
                                                                                CHead e2 (Flat f) u0
                                                                      λ:F.λe1:C.λe2:C.λ:T.csubst0 (s (Flat f) i) v0 e1 e2
                                                                    ex4_5
                                                                      F
                                                                      C
                                                                      C
                                                                      T
                                                                      T
                                                                      λf:F.λe1:C.λ:C.λu0:T.λ:T.eq C (CHead c3 (Flat f) u) (CHead e1 (Flat f) u0)
                                                                      λf:F
                                                                        .λ:C
                                                                          .λe2:C
                                                                            .λ:T
                                                                              .λw:T
                                                                                .drop
                                                                                  s (Flat f) i
                                                                                  s (Flat f) i
                                                                                  CHead c4 (Flat f) u
                                                                                  CHead e2 (Flat f) w
                                                                      λ:F.λ:C.λ:C.λu0:T.λw:T.subst0 (s (Flat f) i) v0 u0 w
                                                                      λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 (s (Flat f) i) v0 e1 e2)
                                                          that is equivalent to 
                                                             eq nat (s (Flat f) i) O
                                                               →(or4
                                                                    drop
                                                                      s (Flat f) i
                                                                      s (Flat f) i
                                                                      CHead c4 (Flat f) u
                                                                      CHead c3 (Flat f) u
                                                                    ex3_4
                                                                      F
                                                                      C
                                                                      T
                                                                      T
                                                                      λf:F.λe0:C.λu0:T.λ:T.eq C (CHead c3 (Flat f) u) (CHead e0 (Flat f) u0)
                                                                      λf:F
                                                                        .λe0:C
                                                                          .λ:T
                                                                            .λw:T
                                                                              .drop
                                                                                s (Flat f) i
                                                                                s (Flat f) i
                                                                                CHead c4 (Flat f) u
                                                                                CHead e0 (Flat f) w
                                                                      λ:F.λ:C.λu0:T.λw:T.subst0 (s (Flat f) i) v0 u0 w
                                                                    ex3_4
                                                                      F
                                                                      C
                                                                      C
                                                                      T
                                                                      λf:F.λe1:C.λ:C.λu0:T.eq C (CHead c3 (Flat f) u) (CHead e1 (Flat f) u0)
                                                                      λf:F
                                                                        .λ:C
                                                                          .λe2:C
                                                                            .λu0:T
                                                                              .drop
                                                                                s (Flat f) i
                                                                                s (Flat f) i
                                                                                CHead c4 (Flat f) u
                                                                                CHead e2 (Flat f) u0
                                                                      λ:F.λe1:C.λe2:C.λ:T.csubst0 (s (Flat f) i) v0 e1 e2
                                                                    ex4_5
                                                                      F
                                                                      C
                                                                      C
                                                                      T
                                                                      T
                                                                      λf:F.λe1:C.λ:C.λu0:T.λ:T.eq C (CHead c3 (Flat f) u) (CHead e1 (Flat f) u0)
                                                                      λf:F
                                                                        .λ:C
                                                                          .λe2:C
                                                                            .λ:T
                                                                              .λw:T
                                                                                .drop
                                                                                  s (Flat f) i
                                                                                  s (Flat f) i
                                                                                  CHead c4 (Flat f) u
                                                                                  CHead e2 (Flat f) w
                                                                      λ:F.λ:C.λ:C.λu0:T.λw:T.subst0 (s (Flat f) i) v0 u0 w
                                                                      λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 (s (Flat f) i) v0 e1 e2)

                                                          ∀i:nat
                                                            .∀c3:C
                                                              .∀c4:C
                                                                .∀v0:T
                                                                  .∀H2:csubst0 i v0 c3 c4
                                                                    .∀H3:eq nat i O
                                                                               →(or4
                                                                                    drop i i c4 c3
                                                                                    ex3_4
                                                                                      F
                                                                                      C
                                                                                      T
                                                                                      T
                                                                                      λf0:F.λe0:C.λu:T.λ:T.eq C c3 (CHead e0 (Flat f0) u)
                                                                                      λf0:F.λe0:C.λ:T.λw:T.drop i i c4 (CHead e0 (Flat f0) w)
                                                                                      λ:F.λ:C.λu:T.λw:T.subst0 i v0 u w
                                                                                    ex3_4
                                                                                      F
                                                                                      C
                                                                                      C
                                                                                      T
                                                                                      λf0:F.λe1:C.λ:C.λu:T.eq C c3 (CHead e1 (Flat f0) u)
                                                                                      λf0:F.λ:C.λe2:C.λu:T.drop i i c4 (CHead e2 (Flat f0) u)
                                                                                      λ:F.λe1:C.λe2:C.λ:T.csubst0 i v0 e1 e2
                                                                                    ex4_5
                                                                                      F
                                                                                      C
                                                                                      C
                                                                                      T
                                                                                      T
                                                                                      λf0:F.λe1:C.λ:C.λu:T.λ:T.eq C c3 (CHead e1 (Flat f0) u)
                                                                                      λf0:F.λ:C.λe2:C.λ:T.λw:T.drop i i c4 (CHead e2 (Flat f0) w)
                                                                                      λ:F.λ:C.λ:C.λu:T.λw:T.subst0 i v0 u w
                                                                                      λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 i v0 e1 e2)
                                                                      .∀u:T
                                                                        .eq nat (s (Flat f) i) O
                                                                          →(or4
                                                                               drop
                                                                                 s (Flat f) i
                                                                                 s (Flat f) i
                                                                                 CHead c4 (Flat f) u
                                                                                 CHead c3 (Flat f) u
                                                                               ex3_4
                                                                                 F
                                                                                 C
                                                                                 T
                                                                                 T
                                                                                 λf:F.λe0:C.λu0:T.λ:T.eq C (CHead c3 (Flat f) u) (CHead e0 (Flat f) u0)
                                                                                 λf:F
                                                                                   .λe0:C
                                                                                     .λ:T
                                                                                       .λw:T
                                                                                         .drop
                                                                                           s (Flat f) i
                                                                                           s (Flat f) i
                                                                                           CHead c4 (Flat f) u
                                                                                           CHead e0 (Flat f) w
                                                                                 λ:F.λ:C.λu0:T.λw:T.subst0 (s (Flat f) i) v0 u0 w
                                                                               ex3_4
                                                                                 F
                                                                                 C
                                                                                 C
                                                                                 T
                                                                                 λf:F.λe1:C.λ:C.λu0:T.eq C (CHead c3 (Flat f) u) (CHead e1 (Flat f) u0)
                                                                                 λf:F
                                                                                   .λ:C
                                                                                     .λe2:C
                                                                                       .λu0:T
                                                                                         .drop
                                                                                           s (Flat f) i
                                                                                           s (Flat f) i
                                                                                           CHead c4 (Flat f) u
                                                                                           CHead e2 (Flat f) u0
                                                                                 λ:F.λe1:C.λe2:C.λ:T.csubst0 (s (Flat f) i) v0 e1 e2
                                                                               ex4_5
                                                                                 F
                                                                                 C
                                                                                 C
                                                                                 T
                                                                                 T
                                                                                 λf:F.λe1:C.λ:C.λu0:T.λ:T.eq C (CHead c3 (Flat f) u) (CHead e1 (Flat f) u0)
                                                                                 λf:F
                                                                                   .λ:C
                                                                                     .λe2:C
                                                                                       .λ:T
                                                                                         .λw:T
                                                                                           .drop
                                                                                             s (Flat f) i
                                                                                             s (Flat f) i
                                                                                             CHead c4 (Flat f) u
                                                                                             CHead e2 (Flat f) w
                                                                                 λ:F.λ:C.λ:C.λu0:T.λw:T.subst0 (s (Flat f) i) v0 u0 w
                                                                                 λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 (s (Flat f) i) v0 e1 e2)
                                              we proved 
                                                 ∀i:nat
                                                   .∀c3:C
                                                     .∀c4:C
                                                       .∀v0:T
                                                         .csubst0 i v0 c3 c4
                                                           →(eq nat i O
                                                                  →(or4
                                                                       drop i i c4 c3
                                                                       ex3_4
                                                                         F
                                                                         C
                                                                         T
                                                                         T
                                                                         λf:F.λe0:C.λu:T.λ:T.eq C c3 (CHead e0 (Flat f) u)
                                                                         λf:F.λe0:C.λ:T.λw:T.drop i i c4 (CHead e0 (Flat f) w)
                                                                         λ:F.λ:C.λu:T.λw:T.subst0 i v0 u w
                                                                       ex3_4
                                                                         F
                                                                         C
                                                                         C
                                                                         T
                                                                         λf:F.λe1:C.λ:C.λu:T.eq C c3 (CHead e1 (Flat f) u)
                                                                         λf:F.λ:C.λe2:C.λu:T.drop i i c4 (CHead e2 (Flat f) u)
                                                                         λ:F.λe1:C.λe2:C.λ:T.csubst0 i v0 e1 e2
                                                                       ex4_5
                                                                         F
                                                                         C
                                                                         C
                                                                         T
                                                                         T
                                                                         λf:F.λe1:C.λ:C.λu:T.λ:T.eq C c3 (CHead e1 (Flat f) u)
                                                                         λf:F.λ:C.λe2:C.λ:T.λw:T.drop i i c4 (CHead e2 (Flat f) w)
                                                                         λ:F.λ:C.λ:C.λu:T.λw:T.subst0 i v0 u w
                                                                         λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 i v0 e1 e2)
                                                                →∀u:T
                                                                     .eq nat (s k i) O
                                                                       →(or4
                                                                            drop (s k i) (s k i) (CHead c4 k u) (CHead c3 k u)
                                                                            ex3_4
                                                                              F
                                                                              C
                                                                              T
                                                                              T
                                                                              λf:F.λe0:C.λu0:T.λ:T.eq C (CHead c3 k u) (CHead e0 (Flat f) u0)
                                                                              λf:F.λe0:C.λ:T.λw:T.drop (s k i) (s k i) (CHead c4 k u) (CHead e0 (Flat f) w)
                                                                              λ:F.λ:C.λu0:T.λw:T.subst0 (s k i) v0 u0 w
                                                                            ex3_4
                                                                              F
                                                                              C
                                                                              C
                                                                              T
                                                                              λf:F.λe1:C.λ:C.λu0:T.eq C (CHead c3 k u) (CHead e1 (Flat f) u0)
                                                                              λf:F.λ:C.λe2:C.λu0:T.drop (s k i) (s k i) (CHead c4 k u) (CHead e2 (Flat f) u0)
                                                                              λ:F.λe1:C.λe2:C.λ:T.csubst0 (s k i) v0 e1 e2
                                                                            ex4_5
                                                                              F
                                                                              C
                                                                              C
                                                                              T
                                                                              T
                                                                              λf:F.λe1:C.λ:C.λu0:T.λ:T.eq C (CHead c3 k u) (CHead e1 (Flat f) u0)
                                                                              λf:F.λ:C.λe2:C.λ:T.λw:T.drop (s k i) (s k i) (CHead c4 k u) (CHead e2 (Flat f) w)
                                                                              λ:F.λ:C.λ:C.λu0:T.λw:T.subst0 (s k i) v0 u0 w
                                                                              λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 (s k i) v0 e1 e2))

                                              ∀k:K
                                                .∀i:nat
                                                  .∀c3:C
                                                    .∀c4:C
                                                      .∀v0:T
                                                        .csubst0 i v0 c3 c4
                                                          →(eq nat i O
                                                                 →(or4
                                                                      drop i i c4 c3
                                                                      ex3_4
                                                                        F
                                                                        C
                                                                        T
                                                                        T
                                                                        λf:F.λe0:C.λu:T.λ:T.eq C c3 (CHead e0 (Flat f) u)
                                                                        λf:F.λe0:C.λ:T.λw:T.drop i i c4 (CHead e0 (Flat f) w)
                                                                        λ:F.λ:C.λu:T.λw:T.subst0 i v0 u w
                                                                      ex3_4
                                                                        F
                                                                        C
                                                                        C
                                                                        T
                                                                        λf:F.λe1:C.λ:C.λu:T.eq C c3 (CHead e1 (Flat f) u)
                                                                        λf:F.λ:C.λe2:C.λu:T.drop i i c4 (CHead e2 (Flat f) u)
                                                                        λ:F.λe1:C.λe2:C.λ:T.csubst0 i v0 e1 e2
                                                                      ex4_5
                                                                        F
                                                                        C
                                                                        C
                                                                        T
                                                                        T
                                                                        λf:F.λe1:C.λ:C.λu:T.λ:T.eq C c3 (CHead e1 (Flat f) u)
                                                                        λf:F.λ:C.λe2:C.λ:T.λw:T.drop i i c4 (CHead e2 (Flat f) w)
                                                                        λ:F.λ:C.λ:C.λu:T.λw:T.subst0 i v0 u w
                                                                        λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 i v0 e1 e2)
                                                               →∀u:T
                                                                    .eq nat (s k i) O
                                                                      →(or4
                                                                           drop (s k i) (s k i) (CHead c4 k u) (CHead c3 k u)
                                                                           ex3_4
                                                                             F
                                                                             C
                                                                             T
                                                                             T
                                                                             λf:F.λe0:C.λu0:T.λ:T.eq C (CHead c3 k u) (CHead e0 (Flat f) u0)
                                                                             λf:F.λe0:C.λ:T.λw:T.drop (s k i) (s k i) (CHead c4 k u) (CHead e0 (Flat f) w)
                                                                             λ:F.λ:C.λu0:T.λw:T.subst0 (s k i) v0 u0 w
                                                                           ex3_4
                                                                             F
                                                                             C
                                                                             C
                                                                             T
                                                                             λf:F.λe1:C.λ:C.λu0:T.eq C (CHead c3 k u) (CHead e1 (Flat f) u0)
                                                                             λf:F.λ:C.λe2:C.λu0:T.drop (s k i) (s k i) (CHead c4 k u) (CHead e2 (Flat f) u0)
                                                                             λ:F.λe1:C.λe2:C.λ:T.csubst0 (s k i) v0 e1 e2
                                                                           ex4_5
                                                                             F
                                                                             C
                                                                             C
                                                                             T
                                                                             T
                                                                             λf:F.λe1:C.λ:C.λu0:T.λ:T.eq C (CHead c3 k u) (CHead e1 (Flat f) u0)
                                                                             λf:F.λ:C.λe2:C.λ:T.λw:T.drop (s k i) (s k i) (CHead c4 k u) (CHead e2 (Flat f) w)
                                                                             λ:F.λ:C.λ:C.λu0:T.λw:T.subst0 (s k i) v0 u0 w
                                                                             λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 (s k i) v0 e1 e2))
                                     case csubst0_both ⇒
                                        we need to prove 
                                        ∀k:K
                                          .∀i:nat
                                            .∀v0:T
                                              .∀u1:T
                                                .∀u2:T
                                                  .subst0 i v0 u1 u2
                                                    →∀c3:C
                                                         .∀c4:C
                                                           .csubst0 i v0 c3 c4
                                                             →(eq nat i O
                                                                    →(or4
                                                                         drop i i c4 c3
                                                                         ex3_4
                                                                           F
                                                                           C
                                                                           T
                                                                           T
                                                                           λf:F.λe0:C.λu:T.λ:T.eq C c3 (CHead e0 (Flat f) u)
                                                                           λf:F.λe0:C.λ:T.λw:T.drop i i c4 (CHead e0 (Flat f) w)
                                                                           λ:F.λ:C.λu:T.λw:T.subst0 i v0 u w
                                                                         ex3_4
                                                                           F
                                                                           C
                                                                           C
                                                                           T
                                                                           λf:F.λe1:C.λ:C.λu:T.eq C c3 (CHead e1 (Flat f) u)
                                                                           λf:F.λ:C.λe2:C.λu:T.drop i i c4 (CHead e2 (Flat f) u)
                                                                           λ:F.λe1:C.λe2:C.λ:T.csubst0 i v0 e1 e2
                                                                         ex4_5
                                                                           F
                                                                           C
                                                                           C
                                                                           T
                                                                           T
                                                                           λf:F.λe1:C.λ:C.λu:T.λ:T.eq C c3 (CHead e1 (Flat f) u)
                                                                           λf:F.λ:C.λe2:C.λ:T.λw:T.drop i i c4 (CHead e2 (Flat f) w)
                                                                           λ:F.λ:C.λ:C.λu:T.λw:T.subst0 i v0 u w
                                                                           λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 i v0 e1 e2)
                                                                  →(eq nat (s k i) O
                                                                       →(or4
                                                                            drop (s k i) (s k i) (CHead c4 k u2) (CHead c3 k u1)
                                                                            ex3_4
                                                                              F
                                                                              C
                                                                              T
                                                                              T
                                                                              λf:F.λe0:C.λu:T.λ:T.eq C (CHead c3 k u1) (CHead e0 (Flat f) u)
                                                                              λf:F.λe0:C.λ:T.λw:T.drop (s k i) (s k i) (CHead c4 k u2) (CHead e0 (Flat f) w)
                                                                              λ:F.λ:C.λu:T.λw:T.subst0 (s k i) v0 u w
                                                                            ex3_4
                                                                              F
                                                                              C
                                                                              C
                                                                              T
                                                                              λf:F.λe1:C.λ:C.λu:T.eq C (CHead c3 k u1) (CHead e1 (Flat f) u)
                                                                              λf:F.λ:C.λe2:C.λu:T.drop (s k i) (s k i) (CHead c4 k u2) (CHead e2 (Flat f) u)
                                                                              λ:F.λe1:C.λe2:C.λ:T.csubst0 (s k i) v0 e1 e2
                                                                            ex4_5
                                                                              F
                                                                              C
                                                                              C
                                                                              T
                                                                              T
                                                                              λf:F.λe1:C.λ:C.λu:T.λ:T.eq C (CHead c3 k u1) (CHead e1 (Flat f) u)
                                                                              λf:F.λ:C.λe2:C.λ:T.λw:T.drop (s k i) (s k i) (CHead c4 k u2) (CHead e2 (Flat f) w)
                                                                              λ:F.λ:C.λ:C.λu:T.λw:T.subst0 (s k i) v0 u w
                                                                              λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 (s k i) v0 e1 e2)))
                                           assume k: K
                                              we proceed by induction on k to prove 
                                                 ∀i:nat
                                                   .∀v0:T
                                                     .∀u1:T
                                                       .∀u2:T
                                                         .subst0 i v0 u1 u2
                                                           →∀c3:C
                                                                .∀c4:C
                                                                  .csubst0 i v0 c3 c4
                                                                    →(eq nat i O
                                                                           →(or4
                                                                                drop i i c4 c3
                                                                                ex3_4
                                                                                  F
                                                                                  C
                                                                                  T
                                                                                  T
                                                                                  λf:F.λe0:C.λu:T.λ:T.eq C c3 (CHead e0 (Flat f) u)
                                                                                  λf:F.λe0:C.λ:T.λw:T.drop i i c4 (CHead e0 (Flat f) w)
                                                                                  λ:F.λ:C.λu:T.λw:T.subst0 i v0 u w
                                                                                ex3_4
                                                                                  F
                                                                                  C
                                                                                  C
                                                                                  T
                                                                                  λf:F.λe1:C.λ:C.λu:T.eq C c3 (CHead e1 (Flat f) u)
                                                                                  λf:F.λ:C.λe2:C.λu:T.drop i i c4 (CHead e2 (Flat f) u)
                                                                                  λ:F.λe1:C.λe2:C.λ:T.csubst0 i v0 e1 e2
                                                                                ex4_5
                                                                                  F
                                                                                  C
                                                                                  C
                                                                                  T
                                                                                  T
                                                                                  λf:F.λe1:C.λ:C.λu:T.λ:T.eq C c3 (CHead e1 (Flat f) u)
                                                                                  λf:F.λ:C.λe2:C.λ:T.λw:T.drop i i c4 (CHead e2 (Flat f) w)
                                                                                  λ:F.λ:C.λ:C.λu:T.λw:T.subst0 i v0 u w
                                                                                  λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 i v0 e1 e2)
                                                                         →(eq nat (s k i) O
                                                                              →(or4
                                                                                   drop (s k i) (s k i) (CHead c4 k u2) (CHead c3 k u1)
                                                                                   ex3_4
                                                                                     F
                                                                                     C
                                                                                     T
                                                                                     T
                                                                                     λf:F.λe0:C.λu:T.λ:T.eq C (CHead c3 k u1) (CHead e0 (Flat f) u)
                                                                                     λf:F.λe0:C.λ:T.λw:T.drop (s k i) (s k i) (CHead c4 k u2) (CHead e0 (Flat f) w)
                                                                                     λ:F.λ:C.λu:T.λw:T.subst0 (s k i) v0 u w
                                                                                   ex3_4
                                                                                     F
                                                                                     C
                                                                                     C
                                                                                     T
                                                                                     λf:F.λe1:C.λ:C.λu:T.eq C (CHead c3 k u1) (CHead e1 (Flat f) u)
                                                                                     λf:F.λ:C.λe2:C.λu:T.drop (s k i) (s k i) (CHead c4 k u2) (CHead e2 (Flat f) u)
                                                                                     λ:F.λe1:C.λe2:C.λ:T.csubst0 (s k i) v0 e1 e2
                                                                                   ex4_5
                                                                                     F
                                                                                     C
                                                                                     C
                                                                                     T
                                                                                     T
                                                                                     λf:F.λe1:C.λ:C.λu:T.λ:T.eq C (CHead c3 k u1) (CHead e1 (Flat f) u)
                                                                                     λf:F.λ:C.λe2:C.λ:T.λw:T.drop (s k i) (s k i) (CHead c4 k u2) (CHead e2 (Flat f) w)
                                                                                     λ:F.λ:C.λ:C.λu:T.λw:T.subst0 (s k i) v0 u w
                                                                                     λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 (s k i) v0 e1 e2)))
                                                 case Bind : b:B ⇒
                                                    the thesis becomes 
                                                    ∀i:nat
                                                      .∀v0:T
                                                        .∀u1:T
                                                          .∀u2:T
                                                            .subst0 i v0 u1 u2
                                                              →∀c3:C
                                                                   .∀c4:C
                                                                     .csubst0 i v0 c3 c4
                                                                       →(eq nat i O
                                                                              →(or4
                                                                                   drop i i c4 c3
                                                                                   ex3_4
                                                                                     F
                                                                                     C
                                                                                     T
                                                                                     T
                                                                                     λf:F.λe0:C.λu:T.λ:T.eq C c3 (CHead e0 (Flat f) u)
                                                                                     λf:F.λe0:C.λ:T.λw:T.drop i i c4 (CHead e0 (Flat f) w)
                                                                                     λ:F.λ:C.λu:T.λw:T.subst0 i v0 u w
                                                                                   ex3_4
                                                                                     F
                                                                                     C
                                                                                     C
                                                                                     T
                                                                                     λf:F.λe1:C.λ:C.λu:T.eq C c3 (CHead e1 (Flat f) u)
                                                                                     λf:F.λ:C.λe2:C.λu:T.drop i i c4 (CHead e2 (Flat f) u)
                                                                                     λ:F.λe1:C.λe2:C.λ:T.csubst0 i v0 e1 e2
                                                                                   ex4_5
                                                                                     F
                                                                                     C
                                                                                     C
                                                                                     T
                                                                                     T
                                                                                     λf:F.λe1:C.λ:C.λu:T.λ:T.eq C c3 (CHead e1 (Flat f) u)
                                                                                     λf:F.λ:C.λe2:C.λ:T.λw:T.drop i i c4 (CHead e2 (Flat f) w)
                                                                                     λ:F.λ:C.λ:C.λu:T.λw:T.subst0 i v0 u w
                                                                                     λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 i v0 e1 e2)
                                                                            →(eq nat (s (Bind b) i) O
                                                                                 →(or4
                                                                                      drop
                                                                                        s (Bind b) i
                                                                                        s (Bind b) i
                                                                                        CHead c4 (Bind b) u2
                                                                                        CHead c3 (Bind b) u1
                                                                                      ex3_4
                                                                                        F
                                                                                        C
                                                                                        T
                                                                                        T
                                                                                        λf:F.λe0:C.λu:T.λ:T.eq C (CHead c3 (Bind b) u1) (CHead e0 (Flat f) u)
                                                                                        λf:F
                                                                                          .λe0:C
                                                                                            .λ:T
                                                                                              .λw:T
                                                                                                .drop
                                                                                                  s (Bind b) i
                                                                                                  s (Bind b) i
                                                                                                  CHead c4 (Bind b) u2
                                                                                                  CHead e0 (Flat f) w
                                                                                        λ:F.λ:C.λu:T.λw:T.subst0 (s (Bind b) i) v0 u w
                                                                                      ex3_4
                                                                                        F
                                                                                        C
                                                                                        C
                                                                                        T
                                                                                        λf:F.λe1:C.λ:C.λu:T.eq C (CHead c3 (Bind b) u1) (CHead e1 (Flat f) u)
                                                                                        λf:F
                                                                                          .λ:C
                                                                                            .λe2:C
                                                                                              .λu:T
                                                                                                .drop
                                                                                                  s (Bind b) i
                                                                                                  s (Bind b) i
                                                                                                  CHead c4 (Bind b) u2
                                                                                                  CHead e2 (Flat f) u
                                                                                        λ:F.λe1:C.λe2:C.λ:T.csubst0 (s (Bind b) i) v0 e1 e2
                                                                                      ex4_5
                                                                                        F
                                                                                        C
                                                                                        C
                                                                                        T
                                                                                        T
                                                                                        λf:F.λe1:C.λ:C.λu:T.λ:T.eq C (CHead c3 (Bind b) u1) (CHead e1 (Flat f) u)
                                                                                        λf:F
                                                                                          .λ:C
                                                                                            .λe2:C
                                                                                              .λ:T
                                                                                                .λw:T
                                                                                                  .drop
                                                                                                    s (Bind b) i
                                                                                                    s (Bind b) i
                                                                                                    CHead c4 (Bind b) u2
                                                                                                    CHead e2 (Flat f) w
                                                                                        λ:F.λ:C.λ:C.λu:T.λw:T.subst0 (s (Bind b) i) v0 u w
                                                                                        λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 (s (Bind b) i) v0 e1 e2)))
                                                        assume i: nat
                                                        assume v0: T
                                                        assume u1: T
                                                        assume u2: T
                                                        suppose : subst0 i v0 u1 u2
                                                        assume c3: C
                                                        assume c4: C
                                                        suppose : csubst0 i v0 c3 c4
                                                        suppose : 
                                                           eq nat i O
                                                             →(or4
                                                                  drop i i c4 c3
                                                                  ex3_4
                                                                    F
                                                                    C
                                                                    T
                                                                    T
                                                                    λf:F.λe0:C.λu:T.λ:T.eq C c3 (CHead e0 (Flat f) u)
                                                                    λf:F.λe0:C.λ:T.λw:T.drop i i c4 (CHead e0 (Flat f) w)
                                                                    λ:F.λ:C.λu:T.λw:T.subst0 i v0 u w
                                                                  ex3_4
                                                                    F
                                                                    C
                                                                    C
                                                                    T
                                                                    λf:F.λe1:C.λ:C.λu:T.eq C c3 (CHead e1 (Flat f) u)
                                                                    λf:F.λ:C.λe2:C.λu:T.drop i i c4 (CHead e2 (Flat f) u)
                                                                    λ:F.λe1:C.λe2:C.λ:T.csubst0 i v0 e1 e2
                                                                  ex4_5
                                                                    F
                                                                    C
                                                                    C
                                                                    T
                                                                    T
                                                                    λf:F.λe1:C.λ:C.λu:T.λ:T.eq C c3 (CHead e1 (Flat f) u)
                                                                    λf:F.λ:C.λe2:C.λ:T.λw:T.drop i i c4 (CHead e2 (Flat f) w)
                                                                    λ:F.λ:C.λ:C.λu:T.λw:T.subst0 i v0 u w
                                                                    λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 i v0 e1 e2)
                                                          we must prove 
                                                                eq nat (s (Bind b) i) O
                                                                  →(or4
                                                                       drop
                                                                         s (Bind b) i
                                                                         s (Bind b) i
                                                                         CHead c4 (Bind b) u2
                                                                         CHead c3 (Bind b) u1
                                                                       ex3_4
                                                                         F
                                                                         C
                                                                         T
                                                                         T
                                                                         λf:F.λe0:C.λu:T.λ:T.eq C (CHead c3 (Bind b) u1) (CHead e0 (Flat f) u)
                                                                         λf:F
                                                                           .λe0:C
                                                                             .λ:T
                                                                               .λw:T
                                                                                 .drop
                                                                                   s (Bind b) i
                                                                                   s (Bind b) i
                                                                                   CHead c4 (Bind b) u2
                                                                                   CHead e0 (Flat f) w
                                                                         λ:F.λ:C.λu:T.λw:T.subst0 (s (Bind b) i) v0 u w
                                                                       ex3_4
                                                                         F
                                                                         C
                                                                         C
                                                                         T
                                                                         λf:F.λe1:C.λ:C.λu:T.eq C (CHead c3 (Bind b) u1) (CHead e1 (Flat f) u)
                                                                         λf:F
                                                                           .λ:C
                                                                             .λe2:C
                                                                               .λu:T
                                                                                 .drop
                                                                                   s (Bind b) i
                                                                                   s (Bind b) i
                                                                                   CHead c4 (Bind b) u2
                                                                                   CHead e2 (Flat f) u
                                                                         λ:F.λe1:C.λe2:C.λ:T.csubst0 (s (Bind b) i) v0 e1 e2
                                                                       ex4_5
                                                                         F
                                                                         C
                                                                         C
                                                                         T
                                                                         T
                                                                         λf:F.λe1:C.λ:C.λu:T.λ:T.eq C (CHead c3 (Bind b) u1) (CHead e1 (Flat f) u)
                                                                         λf:F
                                                                           .λ:C
                                                                             .λe2:C
                                                                               .λ:T
                                                                                 .λw:T
                                                                                   .drop
                                                                                     s (Bind b) i
                                                                                     s (Bind b) i
                                                                                     CHead c4 (Bind b) u2
                                                                                     CHead e2 (Flat f) w
                                                                         λ:F.λ:C.λ:C.λu:T.λw:T.subst0 (s (Bind b) i) v0 u w
                                                                         λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 (s (Bind b) i) v0 e1 e2)
                                                          or equivalently 
                                                                eq nat (S i) O
                                                                  →(or4
                                                                       drop
                                                                         s (Bind b) i
                                                                         s (Bind b) i
                                                                         CHead c4 (Bind b) u2
                                                                         CHead c3 (Bind b) u1
                                                                       ex3_4
                                                                         F
                                                                         C
                                                                         T
                                                                         T
                                                                         λf:F.λe0:C.λu:T.λ:T.eq C (CHead c3 (Bind b) u1) (CHead e0 (Flat f) u)
                                                                         λf:F
                                                                           .λe0:C
                                                                             .λ:T
                                                                               .λw:T
                                                                                 .drop
                                                                                   s (Bind b) i
                                                                                   s (Bind b) i
                                                                                   CHead c4 (Bind b) u2
                                                                                   CHead e0 (Flat f) w
                                                                         λ:F.λ:C.λu:T.λw:T.subst0 (s (Bind b) i) v0 u w
                                                                       ex3_4
                                                                         F
                                                                         C
                                                                         C
                                                                         T
                                                                         λf:F.λe1:C.λ:C.λu:T.eq C (CHead c3 (Bind b) u1) (CHead e1 (Flat f) u)
                                                                         λf:F
                                                                           .λ:C
                                                                             .λe2:C
                                                                               .λu:T
                                                                                 .drop
                                                                                   s (Bind b) i
                                                                                   s (Bind b) i
                                                                                   CHead c4 (Bind b) u2
                                                                                   CHead e2 (Flat f) u
                                                                         λ:F.λe1:C.λe2:C.λ:T.csubst0 (s (Bind b) i) v0 e1 e2
                                                                       ex4_5
                                                                         F
                                                                         C
                                                                         C
                                                                         T
                                                                         T
                                                                         λf:F.λe1:C.λ:C.λu:T.λ:T.eq C (CHead c3 (Bind b) u1) (CHead e1 (Flat f) u)
                                                                         λf:F
                                                                           .λ:C
                                                                             .λe2:C
                                                                               .λ:T
                                                                                 .λw:T
                                                                                   .drop
                                                                                     s (Bind b) i
                                                                                     s (Bind b) i
                                                                                     CHead c4 (Bind b) u2
                                                                                     CHead e2 (Flat f) w
                                                                         λ:F.λ:C.λ:C.λu:T.λw:T.subst0 (s (Bind b) i) v0 u w
                                                                         λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 (s (Bind b) i) v0 e1 e2)
                                                          suppose H5: eq nat (S i) O
                                                             (H6) 
                                                                we proceed by induction on H5 to prove <λ:nat.Prop> CASE O OF O⇒False | S ⇒True
                                                                   case refl_equal : ⇒
                                                                      the thesis becomes <λ:nat.Prop> CASE S i OF O⇒False | S ⇒True
                                                                         consider I
                                                                         we proved True
<λ:nat.Prop> CASE S i OF O⇒False | S ⇒True
<λ:nat.Prop> CASE O OF O⇒False | S ⇒True
                                                             end of H6
                                                             consider H6
                                                             we proved <λ:nat.Prop> CASE O OF O⇒False | S ⇒True
                                                             that is equivalent to False
                                                             we proceed by induction on the previous result to prove 
                                                                or4
                                                                  drop (S i) (S i) (CHead c4 (Bind b) u2) (CHead c3 (Bind b) u1)
                                                                  ex3_4
                                                                    F
                                                                    C
                                                                    T
                                                                    T
                                                                    λf:F.λe0:C.λu:T.λ:T.eq C (CHead c3 (Bind b) u1) (CHead e0 (Flat f) u)
                                                                    λf:F
                                                                      .λe0:C.λ:T.λw:T.drop (S i) (S i) (CHead c4 (Bind b) u2) (CHead e0 (Flat f) w)
                                                                    λ:F.λ:C.λu:T.λw:T.subst0 (S i) v0 u w
                                                                  ex3_4
                                                                    F
                                                                    C
                                                                    C
                                                                    T
                                                                    λf:F.λe1:C.λ:C.λu:T.eq C (CHead c3 (Bind b) u1) (CHead e1 (Flat f) u)
                                                                    λf:F
                                                                      .λ:C.λe2:C.λu:T.drop (S i) (S i) (CHead c4 (Bind b) u2) (CHead e2 (Flat f) u)
                                                                    λ:F.λe1:C.λe2:C.λ:T.csubst0 (S i) v0 e1 e2
                                                                  ex4_5
                                                                    F
                                                                    C
                                                                    C
                                                                    T
                                                                    T
                                                                    λf:F.λe1:C.λ:C.λu:T.λ:T.eq C (CHead c3 (Bind b) u1) (CHead e1 (Flat f) u)
                                                                    λf:F
                                                                      .λ:C.λe2:C.λ:T.λw:T.drop (S i) (S i) (CHead c4 (Bind b) u2) (CHead e2 (Flat f) w)
                                                                    λ:F.λ:C.λ:C.λu:T.λw:T.subst0 (S i) v0 u w
                                                                    λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 (S i) v0 e1 e2
                                                             we proved 
                                                                or4
                                                                  drop (S i) (S i) (CHead c4 (Bind b) u2) (CHead c3 (Bind b) u1)
                                                                  ex3_4
                                                                    F
                                                                    C
                                                                    T
                                                                    T
                                                                    λf:F.λe0:C.λu:T.λ:T.eq C (CHead c3 (Bind b) u1) (CHead e0 (Flat f) u)
                                                                    λf:F
                                                                      .λe0:C.λ:T.λw:T.drop (S i) (S i) (CHead c4 (Bind b) u2) (CHead e0 (Flat f) w)
                                                                    λ:F.λ:C.λu:T.λw:T.subst0 (S i) v0 u w
                                                                  ex3_4
                                                                    F
                                                                    C
                                                                    C
                                                                    T
                                                                    λf:F.λe1:C.λ:C.λu:T.eq C (CHead c3 (Bind b) u1) (CHead e1 (Flat f) u)
                                                                    λf:F
                                                                      .λ:C.λe2:C.λu:T.drop (S i) (S i) (CHead c4 (Bind b) u2) (CHead e2 (Flat f) u)
                                                                    λ:F.λe1:C.λe2:C.λ:T.csubst0 (S i) v0 e1 e2
                                                                  ex4_5
                                                                    F
                                                                    C
                                                                    C
                                                                    T
                                                                    T
                                                                    λf:F.λe1:C.λ:C.λu:T.λ:T.eq C (CHead c3 (Bind b) u1) (CHead e1 (Flat f) u)
                                                                    λf:F
                                                                      .λ:C.λe2:C.λ:T.λw:T.drop (S i) (S i) (CHead c4 (Bind b) u2) (CHead e2 (Flat f) w)
                                                                    λ:F.λ:C.λ:C.λu:T.λw:T.subst0 (S i) v0 u w
                                                                    λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 (S i) v0 e1 e2
                                                             that is equivalent to 
                                                                or4
                                                                  drop
                                                                    s (Bind b) i
                                                                    s (Bind b) i
                                                                    CHead c4 (Bind b) u2
                                                                    CHead c3 (Bind b) u1
                                                                  ex3_4
                                                                    F
                                                                    C
                                                                    T
                                                                    T
                                                                    λf:F.λe0:C.λu:T.λ:T.eq C (CHead c3 (Bind b) u1) (CHead e0 (Flat f) u)
                                                                    λf:F
                                                                      .λe0:C
                                                                        .λ:T
                                                                          .λw:T
                                                                            .drop
                                                                              s (Bind b) i
                                                                              s (Bind b) i
                                                                              CHead c4 (Bind b) u2
                                                                              CHead e0 (Flat f) w
                                                                    λ:F.λ:C.λu:T.λw:T.subst0 (s (Bind b) i) v0 u w
                                                                  ex3_4
                                                                    F
                                                                    C
                                                                    C
                                                                    T
                                                                    λf:F.λe1:C.λ:C.λu:T.eq C (CHead c3 (Bind b) u1) (CHead e1 (Flat f) u)
                                                                    λf:F
                                                                      .λ:C
                                                                        .λe2:C
                                                                          .λu:T
                                                                            .drop
                                                                              s (Bind b) i
                                                                              s (Bind b) i
                                                                              CHead c4 (Bind b) u2
                                                                              CHead e2 (Flat f) u
                                                                    λ:F.λe1:C.λe2:C.λ:T.csubst0 (s (Bind b) i) v0 e1 e2
                                                                  ex4_5
                                                                    F
                                                                    C
                                                                    C
                                                                    T
                                                                    T
                                                                    λf:F.λe1:C.λ:C.λu:T.λ:T.eq C (CHead c3 (Bind b) u1) (CHead e1 (Flat f) u)
                                                                    λf:F
                                                                      .λ:C
                                                                        .λe2:C
                                                                          .λ:T
                                                                            .λw:T
                                                                              .drop
                                                                                s (Bind b) i
                                                                                s (Bind b) i
                                                                                CHead c4 (Bind b) u2
                                                                                CHead e2 (Flat f) w
                                                                    λ:F.λ:C.λ:C.λu:T.λw:T.subst0 (s (Bind b) i) v0 u w
                                                                    λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 (s (Bind b) i) v0 e1 e2
                                                          we proved 
                                                             eq nat (S i) O
                                                               →(or4
                                                                    drop
                                                                      s (Bind b) i
                                                                      s (Bind b) i
                                                                      CHead c4 (Bind b) u2
                                                                      CHead c3 (Bind b) u1
                                                                    ex3_4
                                                                      F
                                                                      C
                                                                      T
                                                                      T
                                                                      λf:F.λe0:C.λu:T.λ:T.eq C (CHead c3 (Bind b) u1) (CHead e0 (Flat f) u)
                                                                      λf:F
                                                                        .λe0:C
                                                                          .λ:T
                                                                            .λw:T
                                                                              .drop
                                                                                s (Bind b) i
                                                                                s (Bind b) i
                                                                                CHead c4 (Bind b) u2
                                                                                CHead e0 (Flat f) w
                                                                      λ:F.λ:C.λu:T.λw:T.subst0 (s (Bind b) i) v0 u w
                                                                    ex3_4
                                                                      F
                                                                      C
                                                                      C
                                                                      T
                                                                      λf:F.λe1:C.λ:C.λu:T.eq C (CHead c3 (Bind b) u1) (CHead e1 (Flat f) u)
                                                                      λf:F
                                                                        .λ:C
                                                                          .λe2:C
                                                                            .λu:T
                                                                              .drop
                                                                                s (Bind b) i
                                                                                s (Bind b) i
                                                                                CHead c4 (Bind b) u2
                                                                                CHead e2 (Flat f) u
                                                                      λ:F.λe1:C.λe2:C.λ:T.csubst0 (s (Bind b) i) v0 e1 e2
                                                                    ex4_5
                                                                      F
                                                                      C
                                                                      C
                                                                      T
                                                                      T
                                                                      λf:F.λe1:C.λ:C.λu:T.λ:T.eq C (CHead c3 (Bind b) u1) (CHead e1 (Flat f) u)
                                                                      λf:F
                                                                        .λ:C
                                                                          .λe2:C
                                                                            .λ:T
                                                                              .λw:T
                                                                                .drop
                                                                                  s (Bind b) i
                                                                                  s (Bind b) i
                                                                                  CHead c4 (Bind b) u2
                                                                                  CHead e2 (Flat f) w
                                                                      λ:F.λ:C.λ:C.λu:T.λw:T.subst0 (s (Bind b) i) v0 u w
                                                                      λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 (s (Bind b) i) v0 e1 e2)
                                                          that is equivalent to 
                                                             eq nat (s (Bind b) i) O
                                                               →(or4
                                                                    drop
                                                                      s (Bind b) i
                                                                      s (Bind b) i
                                                                      CHead c4 (Bind b) u2
                                                                      CHead c3 (Bind b) u1
                                                                    ex3_4
                                                                      F
                                                                      C
                                                                      T
                                                                      T
                                                                      λf:F.λe0:C.λu:T.λ:T.eq C (CHead c3 (Bind b) u1) (CHead e0 (Flat f) u)
                                                                      λf:F
                                                                        .λe0:C
                                                                          .λ:T
                                                                            .λw:T
                                                                              .drop
                                                                                s (Bind b) i
                                                                                s (Bind b) i
                                                                                CHead c4 (Bind b) u2
                                                                                CHead e0 (Flat f) w
                                                                      λ:F.λ:C.λu:T.λw:T.subst0 (s (Bind b) i) v0 u w
                                                                    ex3_4
                                                                      F
                                                                      C
                                                                      C
                                                                      T
                                                                      λf:F.λe1:C.λ:C.λu:T.eq C (CHead c3 (Bind b) u1) (CHead e1 (Flat f) u)
                                                                      λf:F
                                                                        .λ:C
                                                                          .λe2:C
                                                                            .λu:T
                                                                              .drop
                                                                                s (Bind b) i
                                                                                s (Bind b) i
                                                                                CHead c4 (Bind b) u2
                                                                                CHead e2 (Flat f) u
                                                                      λ:F.λe1:C.λe2:C.λ:T.csubst0 (s (Bind b) i) v0 e1 e2
                                                                    ex4_5
                                                                      F
                                                                      C
                                                                      C
                                                                      T
                                                                      T
                                                                      λf:F.λe1:C.λ:C.λu:T.λ:T.eq C (CHead c3 (Bind b) u1) (CHead e1 (Flat f) u)
                                                                      λf:F
                                                                        .λ:C
                                                                          .λe2:C
                                                                            .λ:T
                                                                              .λw:T
                                                                                .drop
                                                                                  s (Bind b) i
                                                                                  s (Bind b) i
                                                                                  CHead c4 (Bind b) u2
                                                                                  CHead e2 (Flat f) w
                                                                      λ:F.λ:C.λ:C.λu:T.λw:T.subst0 (s (Bind b) i) v0 u w
                                                                      λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 (s (Bind b) i) v0 e1 e2)

                                                          ∀i:nat
                                                            .∀v0:T
                                                              .∀u1:T
                                                                .∀u2:T
                                                                  .subst0 i v0 u1 u2
                                                                    →∀c3:C
                                                                         .∀c4:C
                                                                           .csubst0 i v0 c3 c4
                                                                             →(eq nat i O
                                                                                    →(or4
                                                                                         drop i i c4 c3
                                                                                         ex3_4
                                                                                           F
                                                                                           C
                                                                                           T
                                                                                           T
                                                                                           λf:F.λe0:C.λu:T.λ:T.eq C c3 (CHead e0 (Flat f) u)
                                                                                           λf:F.λe0:C.λ:T.λw:T.drop i i c4 (CHead e0 (Flat f) w)
                                                                                           λ:F.λ:C.λu:T.λw:T.subst0 i v0 u w
                                                                                         ex3_4
                                                                                           F
                                                                                           C
                                                                                           C
                                                                                           T
                                                                                           λf:F.λe1:C.λ:C.λu:T.eq C c3 (CHead e1 (Flat f) u)
                                                                                           λf:F.λ:C.λe2:C.λu:T.drop i i c4 (CHead e2 (Flat f) u)
                                                                                           λ:F.λe1:C.λe2:C.λ:T.csubst0 i v0 e1 e2
                                                                                         ex4_5
                                                                                           F
                                                                                           C
                                                                                           C
                                                                                           T
                                                                                           T
                                                                                           λf:F.λe1:C.λ:C.λu:T.λ:T.eq C c3 (CHead e1 (Flat f) u)
                                                                                           λf:F.λ:C.λe2:C.λ:T.λw:T.drop i i c4 (CHead e2 (Flat f) w)
                                                                                           λ:F.λ:C.λ:C.λu:T.λw:T.subst0 i v0 u w
                                                                                           λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 i v0 e1 e2)
                                                                                  →(eq nat (s (Bind b) i) O
                                                                                       →(or4
                                                                                            drop
                                                                                              s (Bind b) i
                                                                                              s (Bind b) i
                                                                                              CHead c4 (Bind b) u2
                                                                                              CHead c3 (Bind b) u1
                                                                                            ex3_4
                                                                                              F
                                                                                              C
                                                                                              T
                                                                                              T
                                                                                              λf:F.λe0:C.λu:T.λ:T.eq C (CHead c3 (Bind b) u1) (CHead e0 (Flat f) u)
                                                                                              λf:F
                                                                                                .λe0:C
                                                                                                  .λ:T
                                                                                                    .λw:T
                                                                                                      .drop
                                                                                                        s (Bind b) i
                                                                                                        s (Bind b) i
                                                                                                        CHead c4 (Bind b) u2
                                                                                                        CHead e0 (Flat f) w
                                                                                              λ:F.λ:C.λu:T.λw:T.subst0 (s (Bind b) i) v0 u w
                                                                                            ex3_4
                                                                                              F
                                                                                              C
                                                                                              C
                                                                                              T
                                                                                              λf:F.λe1:C.λ:C.λu:T.eq C (CHead c3 (Bind b) u1) (CHead e1 (Flat f) u)
                                                                                              λf:F
                                                                                                .λ:C
                                                                                                  .λe2:C
                                                                                                    .λu:T
                                                                                                      .drop
                                                                                                        s (Bind b) i
                                                                                                        s (Bind b) i
                                                                                                        CHead c4 (Bind b) u2
                                                                                                        CHead e2 (Flat f) u
                                                                                              λ:F.λe1:C.λe2:C.λ:T.csubst0 (s (Bind b) i) v0 e1 e2
                                                                                            ex4_5
                                                                                              F
                                                                                              C
                                                                                              C
                                                                                              T
                                                                                              T
                                                                                              λf:F.λe1:C.λ:C.λu:T.λ:T.eq C (CHead c3 (Bind b) u1) (CHead e1 (Flat f) u)
                                                                                              λf:F
                                                                                                .λ:C
                                                                                                  .λe2:C
                                                                                                    .λ:T
                                                                                                      .λw:T
                                                                                                        .drop
                                                                                                          s (Bind b) i
                                                                                                          s (Bind b) i
                                                                                                          CHead c4 (Bind b) u2
                                                                                                          CHead e2 (Flat f) w
                                                                                              λ:F.λ:C.λ:C.λu:T.λw:T.subst0 (s (Bind b) i) v0 u w
                                                                                              λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 (s (Bind b) i) v0 e1 e2)))
                                                 case Flat : f:F ⇒
                                                    the thesis becomes 
                                                    ∀i:nat
                                                      .∀v0:T
                                                        .∀u1:T
                                                          .∀u2:T
                                                            .∀H2:subst0 i v0 u1 u2
                                                              .∀c3:C
                                                                .∀c4:C
                                                                  .∀H3:csubst0 i v0 c3 c4
                                                                    .∀H4:eq nat i O
                                                                               →(or4
                                                                                    drop i i c4 c3
                                                                                    ex3_4
                                                                                      F
                                                                                      C
                                                                                      T
                                                                                      T
                                                                                      λf0:F.λe0:C.λu:T.λ:T.eq C c3 (CHead e0 (Flat f0) u)
                                                                                      λf0:F.λe0:C.λ:T.λw:T.drop i i c4 (CHead e0 (Flat f0) w)
                                                                                      λ:F.λ:C.λu:T.λw:T.subst0 i v0 u w
                                                                                    ex3_4
                                                                                      F
                                                                                      C
                                                                                      C
                                                                                      T
                                                                                      λf0:F.λe1:C.λ:C.λu:T.eq C c3 (CHead e1 (Flat f0) u)
                                                                                      λf0:F.λ:C.λe2:C.λu:T.drop i i c4 (CHead e2 (Flat f0) u)
                                                                                      λ:F.λe1:C.λe2:C.λ:T.csubst0 i v0 e1 e2
                                                                                    ex4_5
                                                                                      F
                                                                                      C
                                                                                      C
                                                                                      T
                                                                                      T
                                                                                      λf0:F.λe1:C.λ:C.λu:T.λ:T.eq C c3 (CHead e1 (Flat f0) u)
                                                                                      λf0:F.λ:C.λe2:C.λ:T.λw:T.drop i i c4 (CHead e2 (Flat f0) w)
                                                                                      λ:F.λ:C.λ:C.λu:T.λw:T.subst0 i v0 u w
                                                                                      λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 i v0 e1 e2)
                                                                      .eq nat (s (Flat f) i) O
                                                                        →(or4
                                                                             drop
                                                                               s (Flat f) i
                                                                               s (Flat f) i
                                                                               CHead c4 (Flat f) u2
                                                                               CHead c3 (Flat f) u1
                                                                             ex3_4
                                                                               F
                                                                               C
                                                                               T
                                                                               T
                                                                               λf:F.λe0:C.λu:T.λ:T.eq C (CHead c3 (Flat f) u1) (CHead e0 (Flat f) u)
                                                                               λf:F
                                                                                 .λe0:C
                                                                                   .λ:T
                                                                                     .λw:T
                                                                                       .drop
                                                                                         s (Flat f) i
                                                                                         s (Flat f) i
                                                                                         CHead c4 (Flat f) u2
                                                                                         CHead e0 (Flat f) w
                                                                               λ:F.λ:C.λu:T.λw:T.subst0 (s (Flat f) i) v0 u w
                                                                             ex3_4
                                                                               F
                                                                               C
                                                                               C
                                                                               T
                                                                               λf:F.λe1:C.λ:C.λu:T.eq C (CHead c3 (Flat f) u1) (CHead e1 (Flat f) u)
                                                                               λf:F
                                                                                 .λ:C
                                                                                   .λe2:C
                                                                                     .λu:T
                                                                                       .drop
                                                                                         s (Flat f) i
                                                                                         s (Flat f) i
                                                                                         CHead c4 (Flat f) u2
                                                                                         CHead e2 (Flat f) u
                                                                               λ:F.λe1:C.λe2:C.λ:T.csubst0 (s (Flat f) i) v0 e1 e2
                                                                             ex4_5
                                                                               F
                                                                               C
                                                                               C
                                                                               T
                                                                               T
                                                                               λf:F.λe1:C.λ:C.λu:T.λ:T.eq C (CHead c3 (Flat f) u1) (CHead e1 (Flat f) u)
                                                                               λf:F
                                                                                 .λ:C
                                                                                   .λe2:C
                                                                                     .λ:T
                                                                                       .λw:T
                                                                                         .drop
                                                                                           s (Flat f) i
                                                                                           s (Flat f) i
                                                                                           CHead c4 (Flat f) u2
                                                                                           CHead e2 (Flat f) w
                                                                               λ:F.λ:C.λ:C.λu:T.λw:T.subst0 (s (Flat f) i) v0 u w
                                                                               λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 (s (Flat f) i) v0 e1 e2)
                                                        assume i: nat
                                                        assume v0: T
                                                        assume u1: T
                                                        assume u2: T
                                                        suppose H2: subst0 i v0 u1 u2
                                                        assume c3: C
                                                        assume c4: C
                                                        suppose H3: csubst0 i v0 c3 c4
                                                        suppose H4: 
                                                           eq nat i O
                                                             →(or4
                                                                  drop i i c4 c3
                                                                  ex3_4
                                                                    F
                                                                    C
                                                                    T
                                                                    T
                                                                    λf0:F.λe0:C.λu:T.λ:T.eq C c3 (CHead e0 (Flat f0) u)
                                                                    λf0:F.λe0:C.λ:T.λw:T.drop i i c4 (CHead e0 (Flat f0) w)
                                                                    λ:F.λ:C.λu:T.λw:T.subst0 i v0 u w
                                                                  ex3_4
                                                                    F
                                                                    C
                                                                    C
                                                                    T
                                                                    λf0:F.λe1:C.λ:C.λu:T.eq C c3 (CHead e1 (Flat f0) u)
                                                                    λf0:F.λ:C.λe2:C.λu:T.drop i i c4 (CHead e2 (Flat f0) u)
                                                                    λ:F.λe1:C.λe2:C.λ:T.csubst0 i v0 e1 e2
                                                                  ex4_5
                                                                    F
                                                                    C
                                                                    C
                                                                    T
                                                                    T
                                                                    λf0:F.λe1:C.λ:C.λu:T.λ:T.eq C c3 (CHead e1 (Flat f0) u)
                                                                    λf0:F.λ:C.λe2:C.λ:T.λw:T.drop i i c4 (CHead e2 (Flat f0) w)
                                                                    λ:F.λ:C.λ:C.λu:T.λw:T.subst0 i v0 u w
                                                                    λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 i v0 e1 e2)
                                                          we must prove 
                                                                eq nat (s (Flat f) i) O
                                                                  →(or4
                                                                       drop
                                                                         s (Flat f) i
                                                                         s (Flat f) i
                                                                         CHead c4 (Flat f) u2
                                                                         CHead c3 (Flat f) u1
                                                                       ex3_4
                                                                         F
                                                                         C
                                                                         T
                                                                         T
                                                                         λf:F.λe0:C.λu:T.λ:T.eq C (CHead c3 (Flat f) u1) (CHead e0 (Flat f) u)
                                                                         λf:F
                                                                           .λe0:C
                                                                             .λ:T
                                                                               .λw:T
                                                                                 .drop
                                                                                   s (Flat f) i
                                                                                   s (Flat f) i
                                                                                   CHead c4 (Flat f) u2
                                                                                   CHead e0 (Flat f) w
                                                                         λ:F.λ:C.λu:T.λw:T.subst0 (s (Flat f) i) v0 u w
                                                                       ex3_4
                                                                         F
                                                                         C
                                                                         C
                                                                         T
                                                                         λf:F.λe1:C.λ:C.λu:T.eq C (CHead c3 (Flat f) u1) (CHead e1 (Flat f) u)
                                                                         λf:F
                                                                           .λ:C
                                                                             .λe2:C
                                                                               .λu:T
                                                                                 .drop
                                                                                   s (Flat f) i
                                                                                   s (Flat f) i
                                                                                   CHead c4 (Flat f) u2
                                                                                   CHead e2 (Flat f) u
                                                                         λ:F.λe1:C.λe2:C.λ:T.csubst0 (s (Flat f) i) v0 e1 e2
                                                                       ex4_5
                                                                         F
                                                                         C
                                                                         C
                                                                         T
                                                                         T
                                                                         λf:F.λe1:C.λ:C.λu:T.λ:T.eq C (CHead c3 (Flat f) u1) (CHead e1 (Flat f) u)
                                                                         λf:F
                                                                           .λ:C
                                                                             .λe2:C
                                                                               .λ:T
                                                                                 .λw:T
                                                                                   .drop
                                                                                     s (Flat f) i
                                                                                     s (Flat f) i
                                                                                     CHead c4 (Flat f) u2
                                                                                     CHead e2 (Flat f) w
                                                                         λ:F.λ:C.λ:C.λu:T.λw:T.subst0 (s (Flat f) i) v0 u w
                                                                         λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 (s (Flat f) i) v0 e1 e2)
                                                          or equivalently 
                                                                eq nat i O
                                                                  →(or4
                                                                       drop
                                                                         s (Flat f) i
                                                                         s (Flat f) i
                                                                         CHead c4 (Flat f) u2
                                                                         CHead c3 (Flat f) u1
                                                                       ex3_4
                                                                         F
                                                                         C
                                                                         T
                                                                         T
                                                                         λf:F.λe0:C.λu:T.λ:T.eq C (CHead c3 (Flat f) u1) (CHead e0 (Flat f) u)
                                                                         λf:F
                                                                           .λe0:C
                                                                             .λ:T
                                                                               .λw:T
                                                                                 .drop
                                                                                   s (Flat f) i
                                                                                   s (Flat f) i
                                                                                   CHead c4 (Flat f) u2
                                                                                   CHead e0 (Flat f) w
                                                                         λ:F.λ:C.λu:T.λw:T.subst0 (s (Flat f) i) v0 u w
                                                                       ex3_4
                                                                         F
                                                                         C
                                                                         C
                                                                         T
                                                                         λf:F.λe1:C.λ:C.λu:T.eq C (CHead c3 (Flat f) u1) (CHead e1 (Flat f) u)
                                                                         λf:F
                                                                           .λ:C
                                                                             .λe2:C
                                                                               .λu:T
                                                                                 .drop
                                                                                   s (Flat f) i
                                                                                   s (Flat f) i
                                                                                   CHead c4 (Flat f) u2
                                                                                   CHead e2 (Flat f) u
                                                                         λ:F.λe1:C.λe2:C.λ:T.csubst0 (s (Flat f) i) v0 e1 e2
                                                                       ex4_5
                                                                         F
                                                                         C
                                                                         C
                                                                         T
                                                                         T
                                                                         λf:F.λe1:C.λ:C.λu:T.λ:T.eq C (CHead c3 (Flat f) u1) (CHead e1 (Flat f) u)
                                                                         λf:F
                                                                           .λ:C
                                                                             .λe2:C
                                                                               .λ:T
                                                                                 .λw:T
                                                                                   .drop
                                                                                     s (Flat f) i
                                                                                     s (Flat f) i
                                                                                     CHead c4 (Flat f) u2
                                                                                     CHead e2 (Flat f) w
                                                                         λ:F.λ:C.λ:C.λu:T.λw:T.subst0 (s (Flat f) i) v0 u w
                                                                         λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 (s (Flat f) i) v0 e1 e2)
                                                          suppose H5: eq nat i O
                                                             (H7) 
                                                                we proceed by induction on H5 to prove csubst0 O v0 c3 c4
                                                                   case refl_equal : ⇒
                                                                      the thesis becomes the hypothesis H3
csubst0 O v0 c3 c4
                                                             end of H7
                                                             (H8) 
                                                                we proceed by induction on H5 to prove subst0 O v0 u1 u2
                                                                   case refl_equal : ⇒
                                                                      the thesis becomes the hypothesis H2
subst0 O v0 u1 u2
                                                             end of H8
                                                             (h1) 
                                                                by (refl_equal . .)
eq C (CHead c3 (Flat f) u1) (CHead c3 (Flat f) u1)
                                                             end of h1
                                                             (h2) 
                                                                by (drop_refl .)
drop O O (CHead c4 (Flat f) u2) (CHead c4 (Flat f) u2)
                                                             end of h2
                                                             by (ex4_5_intro . . . . . . . . . . . . . . h1 h2 H8 H7)
                                                             we proved 
                                                                ex4_5
                                                                  F
                                                                  C
                                                                  C
                                                                  T
                                                                  T
                                                                  λf0:F.λe1:C.λ:C.λu:T.λ:T.eq C (CHead c3 (Flat f) u1) (CHead e1 (Flat f0) u)
                                                                  λf0:F.λ:C.λe2:C.λ:T.λw:T.drop O O (CHead c4 (Flat f) u2) (CHead e2 (Flat f0) w)
                                                                  λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v0 u w
                                                                  λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v0 e1 e2
                                                             by (or4_intro3 . . . . previous)
                                                             we proved 
                                                                or4
                                                                  drop O O (CHead c4 (Flat f) u2) (CHead c3 (Flat f) u1)
                                                                  ex3_4
                                                                    F
                                                                    C
                                                                    T
                                                                    T
                                                                    λf0:F.λe0:C.λu:T.λ:T.eq C (CHead c3 (Flat f) u1) (CHead e0 (Flat f0) u)
                                                                    λf0:F.λe0:C.λ:T.λw:T.drop O O (CHead c4 (Flat f) u2) (CHead e0 (Flat f0) w)
                                                                    λ:F.λ:C.λu:T.λw:T.subst0 O v0 u w
                                                                  ex3_4
                                                                    F
                                                                    C
                                                                    C
                                                                    T
                                                                    λf0:F.λe1:C.λ:C.λu:T.eq C (CHead c3 (Flat f) u1) (CHead e1 (Flat f0) u)
                                                                    λf0:F.λ:C.λe2:C.λu:T.drop O O (CHead c4 (Flat f) u2) (CHead e2 (Flat f0) u)
                                                                    λ:F.λe1:C.λe2:C.λ:T.csubst0 O v0 e1 e2
                                                                  ex4_5
                                                                    F
                                                                    C
                                                                    C
                                                                    T
                                                                    T
                                                                    λf0:F.λe1:C.λ:C.λu:T.λ:T.eq C (CHead c3 (Flat f) u1) (CHead e1 (Flat f0) u)
                                                                    λf0:F.λ:C.λe2:C.λ:T.λw:T.drop O O (CHead c4 (Flat f) u2) (CHead e2 (Flat f0) w)
                                                                    λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v0 u w
                                                                    λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v0 e1 e2
                                                             by (eq_ind_r . . . previous . H5)
                                                             we proved 
                                                                or4
                                                                  drop i i (CHead c4 (Flat f) u2) (CHead c3 (Flat f) u1)
                                                                  ex3_4
                                                                    F
                                                                    C
                                                                    T
                                                                    T
                                                                    λf0:F.λe0:C.λu:T.λ:T.eq C (CHead c3 (Flat f) u1) (CHead e0 (Flat f0) u)
                                                                    λf0:F.λe0:C.λ:T.λw:T.drop i i (CHead c4 (Flat f) u2) (CHead e0 (Flat f0) w)
                                                                    λ:F.λ:C.λu:T.λw:T.subst0 i v0 u w
                                                                  ex3_4
                                                                    F
                                                                    C
                                                                    C
                                                                    T
                                                                    λf0:F.λe1:C.λ:C.λu:T.eq C (CHead c3 (Flat f) u1) (CHead e1 (Flat f0) u)
                                                                    λf0:F.λ:C.λe2:C.λu:T.drop i i (CHead c4 (Flat f) u2) (CHead e2 (Flat f0) u)
                                                                    λ:F.λe1:C.λe2:C.λ:T.csubst0 i v0 e1 e2
                                                                  ex4_5
                                                                    F
                                                                    C
                                                                    C
                                                                    T
                                                                    T
                                                                    λf0:F.λe1:C.λ:C.λu:T.λ:T.eq C (CHead c3 (Flat f) u1) (CHead e1 (Flat f0) u)
                                                                    λf0:F.λ:C.λe2:C.λ:T.λw:T.drop i i (CHead c4 (Flat f) u2) (CHead e2 (Flat f0) w)
                                                                    λ:F.λ:C.λ:C.λu:T.λw:T.subst0 i v0 u w
                                                                    λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 i v0 e1 e2
                                                             that is equivalent to 
                                                                or4
                                                                  drop
                                                                    s (Flat f) i
                                                                    s (Flat f) i
                                                                    CHead c4 (Flat f) u2
                                                                    CHead c3 (Flat f) u1
                                                                  ex3_4
                                                                    F
                                                                    C
                                                                    T
                                                                    T
                                                                    λf:F.λe0:C.λu:T.λ:T.eq C (CHead c3 (Flat f) u1) (CHead e0 (Flat f) u)
                                                                    λf:F
                                                                      .λe0:C
                                                                        .λ:T
                                                                          .λw:T
                                                                            .drop
                                                                              s (Flat f) i
                                                                              s (Flat f) i
                                                                              CHead c4 (Flat f) u2
                                                                              CHead e0 (Flat f) w
                                                                    λ:F.λ:C.λu:T.λw:T.subst0 (s (Flat f) i) v0 u w
                                                                  ex3_4
                                                                    F
                                                                    C
                                                                    C
                                                                    T
                                                                    λf:F.λe1:C.λ:C.λu:T.eq C (CHead c3 (Flat f) u1) (CHead e1 (Flat f) u)
                                                                    λf:F
                                                                      .λ:C
                                                                        .λe2:C
                                                                          .λu:T
                                                                            .drop
                                                                              s (Flat f) i
                                                                              s (Flat f) i
                                                                              CHead c4 (Flat f) u2
                                                                              CHead e2 (Flat f) u
                                                                    λ:F.λe1:C.λe2:C.λ:T.csubst0 (s (Flat f) i) v0 e1 e2
                                                                  ex4_5
                                                                    F
                                                                    C
                                                                    C
                                                                    T
                                                                    T
                                                                    λf:F.λe1:C.λ:C.λu:T.λ:T.eq C (CHead c3 (Flat f) u1) (CHead e1 (Flat f) u)
                                                                    λf:F
                                                                      .λ:C
                                                                        .λe2:C
                                                                          .λ:T
                                                                            .λw:T
                                                                              .drop
                                                                                s (Flat f) i
                                                                                s (Flat f) i
                                                                                CHead c4 (Flat f) u2
                                                                                CHead e2 (Flat f) w
                                                                    λ:F.λ:C.λ:C.λu:T.λw:T.subst0 (s (Flat f) i) v0 u w
                                                                    λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 (s (Flat f) i) v0 e1 e2
                                                          we proved 
                                                             eq nat i O
                                                               →(or4
                                                                    drop
                                                                      s (Flat f) i
                                                                      s (Flat f) i
                                                                      CHead c4 (Flat f) u2
                                                                      CHead c3 (Flat f) u1
                                                                    ex3_4
                                                                      F
                                                                      C
                                                                      T
                                                                      T
                                                                      λf:F.λe0:C.λu:T.λ:T.eq C (CHead c3 (Flat f) u1) (CHead e0 (Flat f) u)
                                                                      λf:F
                                                                        .λe0:C
                                                                          .λ:T
                                                                            .λw:T
                                                                              .drop
                                                                                s (Flat f) i
                                                                                s (Flat f) i
                                                                                CHead c4 (Flat f) u2
                                                                                CHead e0 (Flat f) w
                                                                      λ:F.λ:C.λu:T.λw:T.subst0 (s (Flat f) i) v0 u w
                                                                    ex3_4
                                                                      F
                                                                      C
                                                                      C
                                                                      T
                                                                      λf:F.λe1:C.λ:C.λu:T.eq C (CHead c3 (Flat f) u1) (CHead e1 (Flat f) u)
                                                                      λf:F
                                                                        .λ:C
                                                                          .λe2:C
                                                                            .λu:T
                                                                              .drop
                                                                                s (Flat f) i
                                                                                s (Flat f) i
                                                                                CHead c4 (Flat f) u2
                                                                                CHead e2 (Flat f) u
                                                                      λ:F.λe1:C.λe2:C.λ:T.csubst0 (s (Flat f) i) v0 e1 e2
                                                                    ex4_5
                                                                      F
                                                                      C
                                                                      C
                                                                      T
                                                                      T
                                                                      λf:F.λe1:C.λ:C.λu:T.λ:T.eq C (CHead c3 (Flat f) u1) (CHead e1 (Flat f) u)
                                                                      λf:F
                                                                        .λ:C
                                                                          .λe2:C
                                                                            .λ:T
                                                                              .λw:T
                                                                                .drop
                                                                                  s (Flat f) i
                                                                                  s (Flat f) i
                                                                                  CHead c4 (Flat f) u2
                                                                                  CHead e2 (Flat f) w
                                                                      λ:F.λ:C.λ:C.λu:T.λw:T.subst0 (s (Flat f) i) v0 u w
                                                                      λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 (s (Flat f) i) v0 e1 e2)
                                                          that is equivalent to 
                                                             eq nat (s (Flat f) i) O
                                                               →(or4
                                                                    drop
                                                                      s (Flat f) i
                                                                      s (Flat f) i
                                                                      CHead c4 (Flat f) u2
                                                                      CHead c3 (Flat f) u1
                                                                    ex3_4
                                                                      F
                                                                      C
                                                                      T
                                                                      T
                                                                      λf:F.λe0:C.λu:T.λ:T.eq C (CHead c3 (Flat f) u1) (CHead e0 (Flat f) u)
                                                                      λf:F
                                                                        .λe0:C
                                                                          .λ:T
                                                                            .λw:T
                                                                              .drop
                                                                                s (Flat f) i
                                                                                s (Flat f) i
                                                                                CHead c4 (Flat f) u2
                                                                                CHead e0 (Flat f) w
                                                                      λ:F.λ:C.λu:T.λw:T.subst0 (s (Flat f) i) v0 u w
                                                                    ex3_4
                                                                      F
                                                                      C
                                                                      C
                                                                      T
                                                                      λf:F.λe1:C.λ:C.λu:T.eq C (CHead c3 (Flat f) u1) (CHead e1 (Flat f) u)
                                                                      λf:F
                                                                        .λ:C
                                                                          .λe2:C
                                                                            .λu:T
                                                                              .drop
                                                                                s (Flat f) i
                                                                                s (Flat f) i
                                                                                CHead c4 (Flat f) u2
                                                                                CHead e2 (Flat f) u
                                                                      λ:F.λe1:C.λe2:C.λ:T.csubst0 (s (Flat f) i) v0 e1 e2
                                                                    ex4_5
                                                                      F
                                                                      C
                                                                      C
                                                                      T
                                                                      T
                                                                      λf:F.λe1:C.λ:C.λu:T.λ:T.eq C (CHead c3 (Flat f) u1) (CHead e1 (Flat f) u)
                                                                      λf:F
                                                                        .λ:C
                                                                          .λe2:C
                                                                            .λ:T
                                                                              .λw:T
                                                                                .drop
                                                                                  s (Flat f) i
                                                                                  s (Flat f) i
                                                                                  CHead c4 (Flat f) u2
                                                                                  CHead e2 (Flat f) w
                                                                      λ:F.λ:C.λ:C.λu:T.λw:T.subst0 (s (Flat f) i) v0 u w
                                                                      λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 (s (Flat f) i) v0 e1 e2)

                                                          ∀i:nat
                                                            .∀v0:T
                                                              .∀u1:T
                                                                .∀u2:T
                                                                  .∀H2:subst0 i v0 u1 u2
                                                                    .∀c3:C
                                                                      .∀c4:C
                                                                        .∀H3:csubst0 i v0 c3 c4
                                                                          .∀H4:eq nat i O
                                                                                     →(or4
                                                                                          drop i i c4 c3
                                                                                          ex3_4
                                                                                            F
                                                                                            C
                                                                                            T
                                                                                            T
                                                                                            λf0:F.λe0:C.λu:T.λ:T.eq C c3 (CHead e0 (Flat f0) u)
                                                                                            λf0:F.λe0:C.λ:T.λw:T.drop i i c4 (CHead e0 (Flat f0) w)
                                                                                            λ:F.λ:C.λu:T.λw:T.subst0 i v0 u w
                                                                                          ex3_4
                                                                                            F
                                                                                            C
                                                                                            C
                                                                                            T
                                                                                            λf0:F.λe1:C.λ:C.λu:T.eq C c3 (CHead e1 (Flat f0) u)
                                                                                            λf0:F.λ:C.λe2:C.λu:T.drop i i c4 (CHead e2 (Flat f0) u)
                                                                                            λ:F.λe1:C.λe2:C.λ:T.csubst0 i v0 e1 e2
                                                                                          ex4_5
                                                                                            F
                                                                                            C
                                                                                            C
                                                                                            T
                                                                                            T
                                                                                            λf0:F.λe1:C.λ:C.λu:T.λ:T.eq C c3 (CHead e1 (Flat f0) u)
                                                                                            λf0:F.λ:C.λe2:C.λ:T.λw:T.drop i i c4 (CHead e2 (Flat f0) w)
                                                                                            λ:F.λ:C.λ:C.λu:T.λw:T.subst0 i v0 u w
                                                                                            λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 i v0 e1 e2)
                                                                            .eq nat (s (Flat f) i) O
                                                                              →(or4
                                                                                   drop
                                                                                     s (Flat f) i
                                                                                     s (Flat f) i
                                                                                     CHead c4 (Flat f) u2
                                                                                     CHead c3 (Flat f) u1
                                                                                   ex3_4
                                                                                     F
                                                                                     C
                                                                                     T
                                                                                     T
                                                                                     λf:F.λe0:C.λu:T.λ:T.eq C (CHead c3 (Flat f) u1) (CHead e0 (Flat f) u)
                                                                                     λf:F
                                                                                       .λe0:C
                                                                                         .λ:T
                                                                                           .λw:T
                                                                                             .drop
                                                                                               s (Flat f) i
                                                                                               s (Flat f) i
                                                                                               CHead c4 (Flat f) u2
                                                                                               CHead e0 (Flat f) w
                                                                                     λ:F.λ:C.λu:T.λw:T.subst0 (s (Flat f) i) v0 u w
                                                                                   ex3_4
                                                                                     F
                                                                                     C
                                                                                     C
                                                                                     T
                                                                                     λf:F.λe1:C.λ:C.λu:T.eq C (CHead c3 (Flat f) u1) (CHead e1 (Flat f) u)
                                                                                     λf:F
                                                                                       .λ:C
                                                                                         .λe2:C
                                                                                           .λu:T
                                                                                             .drop
                                                                                               s (Flat f) i
                                                                                               s (Flat f) i
                                                                                               CHead c4 (Flat f) u2
                                                                                               CHead e2 (Flat f) u
                                                                                     λ:F.λe1:C.λe2:C.λ:T.csubst0 (s (Flat f) i) v0 e1 e2
                                                                                   ex4_5
                                                                                     F
                                                                                     C
                                                                                     C
                                                                                     T
                                                                                     T
                                                                                     λf:F.λe1:C.λ:C.λu:T.λ:T.eq C (CHead c3 (Flat f) u1) (CHead e1 (Flat f) u)
                                                                                     λf:F
                                                                                       .λ:C
                                                                                         .λe2:C
                                                                                           .λ:T
                                                                                             .λw:T
                                                                                               .drop
                                                                                                 s (Flat f) i
                                                                                                 s (Flat f) i
                                                                                                 CHead c4 (Flat f) u2
                                                                                                 CHead e2 (Flat f) w
                                                                                     λ:F.λ:C.λ:C.λu:T.λw:T.subst0 (s (Flat f) i) v0 u w
                                                                                     λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 (s (Flat f) i) v0 e1 e2)
                                              we proved 
                                                 ∀i:nat
                                                   .∀v0:T
                                                     .∀u1:T
                                                       .∀u2:T
                                                         .subst0 i v0 u1 u2
                                                           →∀c3:C
                                                                .∀c4:C
                                                                  .csubst0 i v0 c3 c4
                                                                    →(eq nat i O
                                                                           →(or4
                                                                                drop i i c4 c3
                                                                                ex3_4
                                                                                  F
                                                                                  C
                                                                                  T
                                                                                  T
                                                                                  λf:F.λe0:C.λu:T.λ:T.eq C c3 (CHead e0 (Flat f) u)
                                                                                  λf:F.λe0:C.λ:T.λw:T.drop i i c4 (CHead e0 (Flat f) w)
                                                                                  λ:F.λ:C.λu:T.λw:T.subst0 i v0 u w
                                                                                ex3_4
                                                                                  F
                                                                                  C
                                                                                  C
                                                                                  T
                                                                                  λf:F.λe1:C.λ:C.λu:T.eq C c3 (CHead e1 (Flat f) u)
                                                                                  λf:F.λ:C.λe2:C.λu:T.drop i i c4 (CHead e2 (Flat f) u)
                                                                                  λ:F.λe1:C.λe2:C.λ:T.csubst0 i v0 e1 e2
                                                                                ex4_5
                                                                                  F
                                                                                  C
                                                                                  C
                                                                                  T
                                                                                  T
                                                                                  λf:F.λe1:C.λ:C.λu:T.λ:T.eq C c3 (CHead e1 (Flat f) u)
                                                                                  λf:F.λ:C.λe2:C.λ:T.λw:T.drop i i c4 (CHead e2 (Flat f) w)
                                                                                  λ:F.λ:C.λ:C.λu:T.λw:T.subst0 i v0 u w
                                                                                  λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 i v0 e1 e2)
                                                                         →(eq nat (s k i) O
                                                                              →(or4
                                                                                   drop (s k i) (s k i) (CHead c4 k u2) (CHead c3 k u1)
                                                                                   ex3_4
                                                                                     F
                                                                                     C
                                                                                     T
                                                                                     T
                                                                                     λf:F.λe0:C.λu:T.λ:T.eq C (CHead c3 k u1) (CHead e0 (Flat f) u)
                                                                                     λf:F.λe0:C.λ:T.λw:T.drop (s k i) (s k i) (CHead c4 k u2) (CHead e0 (Flat f) w)
                                                                                     λ:F.λ:C.λu:T.λw:T.subst0 (s k i) v0 u w
                                                                                   ex3_4
                                                                                     F
                                                                                     C
                                                                                     C
                                                                                     T
                                                                                     λf:F.λe1:C.λ:C.λu:T.eq C (CHead c3 k u1) (CHead e1 (Flat f) u)
                                                                                     λf:F.λ:C.λe2:C.λu:T.drop (s k i) (s k i) (CHead c4 k u2) (CHead e2 (Flat f) u)
                                                                                     λ:F.λe1:C.λe2:C.λ:T.csubst0 (s k i) v0 e1 e2
                                                                                   ex4_5
                                                                                     F
                                                                                     C
                                                                                     C
                                                                                     T
                                                                                     T
                                                                                     λf:F.λe1:C.λ:C.λu:T.λ:T.eq C (CHead c3 k u1) (CHead e1 (Flat f) u)
                                                                                     λf:F.λ:C.λe2:C.λ:T.λw:T.drop (s k i) (s k i) (CHead c4 k u2) (CHead e2 (Flat f) w)
                                                                                     λ:F.λ:C.λ:C.λu:T.λw:T.subst0 (s k i) v0 u w
                                                                                     λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 (s k i) v0 e1 e2)))

                                              ∀k:K
                                                .∀i:nat
                                                  .∀v0:T
                                                    .∀u1:T
                                                      .∀u2:T
                                                        .subst0 i v0 u1 u2
                                                          →∀c3:C
                                                               .∀c4:C
                                                                 .csubst0 i v0 c3 c4
                                                                   →(eq nat i O
                                                                          →(or4
                                                                               drop i i c4 c3
                                                                               ex3_4
                                                                                 F
                                                                                 C
                                                                                 T
                                                                                 T
                                                                                 λf:F.λe0:C.λu:T.λ:T.eq C c3 (CHead e0 (Flat f) u)
                                                                                 λf:F.λe0:C.λ:T.λw:T.drop i i c4 (CHead e0 (Flat f) w)
                                                                                 λ:F.λ:C.λu:T.λw:T.subst0 i v0 u w
                                                                               ex3_4
                                                                                 F
                                                                                 C
                                                                                 C
                                                                                 T
                                                                                 λf:F.λe1:C.λ:C.λu:T.eq C c3 (CHead e1 (Flat f) u)
                                                                                 λf:F.λ:C.λe2:C.λu:T.drop i i c4 (CHead e2 (Flat f) u)
                                                                                 λ:F.λe1:C.λe2:C.λ:T.csubst0 i v0 e1 e2
                                                                               ex4_5
                                                                                 F
                                                                                 C
                                                                                 C
                                                                                 T
                                                                                 T
                                                                                 λf:F.λe1:C.λ:C.λu:T.λ:T.eq C c3 (CHead e1 (Flat f) u)
                                                                                 λf:F.λ:C.λe2:C.λ:T.λw:T.drop i i c4 (CHead e2 (Flat f) w)
                                                                                 λ:F.λ:C.λ:C.λu:T.λw:T.subst0 i v0 u w
                                                                                 λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 i v0 e1 e2)
                                                                        →(eq nat (s k i) O
                                                                             →(or4
                                                                                  drop (s k i) (s k i) (CHead c4 k u2) (CHead c3 k u1)
                                                                                  ex3_4
                                                                                    F
                                                                                    C
                                                                                    T
                                                                                    T
                                                                                    λf:F.λe0:C.λu:T.λ:T.eq C (CHead c3 k u1) (CHead e0 (Flat f) u)
                                                                                    λf:F.λe0:C.λ:T.λw:T.drop (s k i) (s k i) (CHead c4 k u2) (CHead e0 (Flat f) w)
                                                                                    λ:F.λ:C.λu:T.λw:T.subst0 (s k i) v0 u w
                                                                                  ex3_4
                                                                                    F
                                                                                    C
                                                                                    C
                                                                                    T
                                                                                    λf:F.λe1:C.λ:C.λu:T.eq C (CHead c3 k u1) (CHead e1 (Flat f) u)
                                                                                    λf:F.λ:C.λe2:C.λu:T.drop (s k i) (s k i) (CHead c4 k u2) (CHead e2 (Flat f) u)
                                                                                    λ:F.λe1:C.λe2:C.λ:T.csubst0 (s k i) v0 e1 e2
                                                                                  ex4_5
                                                                                    F
                                                                                    C
                                                                                    C
                                                                                    T
                                                                                    T
                                                                                    λf:F.λe1:C.λ:C.λu:T.λ:T.eq C (CHead c3 k u1) (CHead e1 (Flat f) u)
                                                                                    λf:F.λ:C.λe2:C.λ:T.λw:T.drop (s k i) (s k i) (CHead c4 k u2) (CHead e2 (Flat f) w)
                                                                                    λ:F.λ:C.λ:C.λu:T.λw:T.subst0 (s k i) v0 u w
                                                                                    λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 (s k i) v0 e1 e2)))
                                  we proved 
                                     eq nat y O
                                       →(or4
                                            drop y y c2 c1
                                            ex3_4
                                              F
                                              C
                                              T
                                              T
                                              λf:F.λe0:C.λu:T.λ:T.eq C c1 (CHead e0 (Flat f) u)
                                              λf:F.λe0:C.λ:T.λw:T.drop y y c2 (CHead e0 (Flat f) w)
                                              λ:F.λ:C.λu:T.λw:T.subst0 y v u w
                                            ex3_4
                                              F
                                              C
                                              C
                                              T
                                              λf:F.λe1:C.λ:C.λu:T.eq C c1 (CHead e1 (Flat f) u)
                                              λf:F.λ:C.λe2:C.λu:T.drop y y c2 (CHead e2 (Flat f) u)
                                              λ:F.λe1:C.λe2:C.λ:T.csubst0 y v e1 e2
                                            ex4_5
                                              F
                                              C
                                              C
                                              T
                                              T
                                              λf:F.λe1:C.λ:C.λu:T.λ:T.eq C c1 (CHead e1 (Flat f) u)
                                              λf:F.λ:C.λe2:C.λ:T.λw:T.drop y y c2 (CHead e2 (Flat f) w)
                                              λ:F.λ:C.λ:C.λu:T.λw:T.subst0 y v u w
                                              λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 y v e1 e2)
                               we proved 
                                  ∀y:nat
                                    .csubst0 y v c1 c2
                                      →(eq nat y O
                                           →(or4
                                                drop y y c2 c1
                                                ex3_4
                                                  F
                                                  C
                                                  T
                                                  T
                                                  λf:F.λe0:C.λu:T.λ:T.eq C c1 (CHead e0 (Flat f) u)
                                                  λf:F.λe0:C.λ:T.λw:T.drop y y c2 (CHead e0 (Flat f) w)
                                                  λ:F.λ:C.λu:T.λw:T.subst0 y v u w
                                                ex3_4
                                                  F
                                                  C
                                                  C
                                                  T
                                                  λf:F.λe1:C.λ:C.λu:T.eq C c1 (CHead e1 (Flat f) u)
                                                  λf:F.λ:C.λe2:C.λu:T.drop y y c2 (CHead e2 (Flat f) u)
                                                  λ:F.λe1:C.λe2:C.λ:T.csubst0 y v e1 e2
                                                ex4_5
                                                  F
                                                  C
                                                  C
                                                  T
                                                  T
                                                  λf:F.λe1:C.λ:C.λu:T.λ:T.eq C c1 (CHead e1 (Flat f) u)
                                                  λf:F.λ:C.λe2:C.λ:T.λw:T.drop y y c2 (CHead e2 (Flat f) w)
                                                  λ:F.λ:C.λ:C.λu:T.λw:T.subst0 y v u w
                                                  λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 y v e1 e2))
                               by (insert_eq . . . . previous H)

                                  or4
                                    drop O O c2 c1
                                    ex3_4
                                      F
                                      C
                                      T
                                      T
                                      λf:F.λe0:C.λu:T.λ:T.eq C c1 (CHead e0 (Flat f) u)
                                      λf:F.λe0:C.λ:T.λw:T.drop O O c2 (CHead e0 (Flat f) w)
                                      λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                    ex3_4
                                      F
                                      C
                                      C
                                      T
                                      λf:F.λe1:C.λ:C.λu:T.eq C c1 (CHead e1 (Flat f) u)
                                      λf:F.λ:C.λe2:C.λu:T.drop O O c2 (CHead e2 (Flat f) u)
                                      λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                    ex4_5
                                      F
                                      C
                                      C
                                      T
                                      T
                                      λf:F.λe1:C.λ:C.λu:T.λ:T.eq C c1 (CHead e1 (Flat f) u)
                                      λf:F.λ:C.λe2:C.λ:T.λw:T.drop O O c2 (CHead e2 (Flat f) w)
                                      λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                      λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2
                      we proved 
                         or4
                           drop O O c2 e
                           ex3_4
                             F
                             C
                             T
                             T
                             λf:F.λe0:C.λu:T.λ:T.eq C e (CHead e0 (Flat f) u)
                             λf:F.λe0:C.λ:T.λw:T.drop O O c2 (CHead e0 (Flat f) w)
                             λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                           ex3_4
                             F
                             C
                             C
                             T
                             λf:F.λe1:C.λ:C.λu:T.eq C e (CHead e1 (Flat f) u)
                             λf:F.λ:C.λe2:C.λu:T.drop O O c2 (CHead e2 (Flat f) u)
                             λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                           ex4_5
                             F
                             C
                             C
                             T
                             T
                             λf:F.λe1:C.λ:C.λu:T.λ:T.eq C e (CHead e1 (Flat f) u)
                             λf:F.λ:C.λe2:C.λ:T.λw:T.drop O O c2 (CHead e2 (Flat f) w)
                             λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                             λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2

                      ∀c1:C
                        .∀c2:C
                          .∀v:T
                            .csubst0 O v c1 c2
                              →∀e:C
                                   .drop O O c1 e
                                     →(or4
                                          drop O O c2 e
                                          ex3_4
                                            F
                                            C
                                            T
                                            T
                                            λf:F.λe0:C.λu:T.λ:T.eq C e (CHead e0 (Flat f) u)
                                            λf:F.λe0:C.λ:T.λw:T.drop O O c2 (CHead e0 (Flat f) w)
                                            λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                          ex3_4
                                            F
                                            C
                                            C
                                            T
                                            λf:F.λe1:C.λ:C.λu:T.eq C e (CHead e1 (Flat f) u)
                                            λf:F.λ:C.λe2:C.λu:T.drop O O c2 (CHead e2 (Flat f) u)
                                            λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                          ex4_5
                                            F
                                            C
                                            C
                                            T
                                            T
                                            λf:F.λe1:C.λ:C.λu:T.λ:T.eq C e (CHead e1 (Flat f) u)
                                            λf:F.λ:C.λe2:C.λ:T.λw:T.drop O O c2 (CHead e2 (Flat f) w)
                                            λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                            λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2)
             case S : n0:nat ⇒
                the thesis becomes 
                ∀c1:C
                  .∀c2:C
                    .∀v:T
                      .csubst0 (S n0) v c1 c2
                        →∀e:C
                             .drop (S n0) O c1 e
                               →(or4
                                    drop (S n0) O c2 e
                                    ex3_4
                                      F
                                      C
                                      T
                                      T
                                      λf:F.λe0:C.λu:T.λ:T.eq C e (CHead e0 (Flat f) u)
                                      λf:F.λe0:C.λ:T.λw:T.drop (S n0) O c2 (CHead e0 (Flat f) w)
                                      λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                    ex3_4
                                      F
                                      C
                                      C
                                      T
                                      λf:F.λe1:C.λ:C.λu:T.eq C e (CHead e1 (Flat f) u)
                                      λf:F.λ:C.λe2:C.λu:T.drop (S n0) O c2 (CHead e2 (Flat f) u)
                                      λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                    ex4_5
                                      F
                                      C
                                      C
                                      T
                                      T
                                      λf:F.λe1:C.λ:C.λu:T.λ:T.eq C e (CHead e1 (Flat f) u)
                                      λf:F.λ:C.λe2:C.λ:T.λw:T.drop (S n0) O c2 (CHead e2 (Flat f) w)
                                      λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                      λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2)
                (IHn) by induction hypothesis we know 
                   ∀c1:C
                     .∀c2:C
                       .∀v:T
                         .csubst0 n0 v c1 c2
                           →∀e:C
                                .drop n0 O c1 e
                                  →(or4
                                       drop n0 O c2 e
                                       ex3_4
                                         F
                                         C
                                         T
                                         T
                                         λf:F.λe0:C.λu:T.λ:T.eq C e (CHead e0 (Flat f) u)
                                         λf:F.λe0:C.λ:T.λw:T.drop n0 O c2 (CHead e0 (Flat f) w)
                                         λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                       ex3_4
                                         F
                                         C
                                         C
                                         T
                                         λf:F.λe1:C.λ:C.λu:T.eq C e (CHead e1 (Flat f) u)
                                         λf:F.λ:C.λe2:C.λu:T.drop n0 O c2 (CHead e2 (Flat f) u)
                                         λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                       ex4_5
                                         F
                                         C
                                         C
                                         T
                                         T
                                         λf:F.λe1:C.λ:C.λu:T.λ:T.eq C e (CHead e1 (Flat f) u)
                                         λf:F.λ:C.λe2:C.λ:T.λw:T.drop n0 O c2 (CHead e2 (Flat f) w)
                                         λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                         λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2)
                   assume c1: C
                      we proceed by induction on c1 to prove 
                         ∀c2:C
                           .∀v:T
                             .csubst0 (S n0) v c1 c2
                               →∀e:C
                                    .drop (S n0) O c1 e
                                      →(or4
                                           drop (S n0) O c2 e
                                           ex3_4
                                             F
                                             C
                                             T
                                             T
                                             λf:F.λe0:C.λu:T.λ:T.eq C e (CHead e0 (Flat f) u)
                                             λf:F.λe0:C.λ:T.λw:T.drop (S n0) O c2 (CHead e0 (Flat f) w)
                                             λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                           ex3_4
                                             F
                                             C
                                             C
                                             T
                                             λf:F.λe1:C.λ:C.λu:T.eq C e (CHead e1 (Flat f) u)
                                             λf:F.λ:C.λe2:C.λu:T.drop (S n0) O c2 (CHead e2 (Flat f) u)
                                             λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                           ex4_5
                                             F
                                             C
                                             C
                                             T
                                             T
                                             λf:F.λe1:C.λ:C.λu:T.λ:T.eq C e (CHead e1 (Flat f) u)
                                             λf:F.λ:C.λe2:C.λ:T.λw:T.drop (S n0) O c2 (CHead e2 (Flat f) w)
                                             λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                             λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2)
                         case CSort : n1:nat ⇒
                            the thesis becomes 
                            ∀c2:C
                              .∀v:T
                                .csubst0 (S n0) v (CSort n1) c2
                                  →∀e:C
                                       .∀H0:drop (S n0) O (CSort n1) e
                                         .or4
                                           drop (S n0) O c2 e
                                           ex3_4
                                             F
                                             C
                                             T
                                             T
                                             λf:F.λe0:C.λu:T.λ:T.eq C e (CHead e0 (Flat f) u)
                                             λf:F.λe0:C.λ:T.λw:T.drop (S n0) O c2 (CHead e0 (Flat f) w)
                                             λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                           ex3_4
                                             F
                                             C
                                             C
                                             T
                                             λf:F.λe1:C.λ:C.λu:T.eq C e (CHead e1 (Flat f) u)
                                             λf:F.λ:C.λe2:C.λu:T.drop (S n0) O c2 (CHead e2 (Flat f) u)
                                             λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                           ex4_5
                                             F
                                             C
                                             C
                                             T
                                             T
                                             λf:F.λe1:C.λ:C.λu:T.λ:T.eq C e (CHead e1 (Flat f) u)
                                             λf:F.λ:C.λe2:C.λ:T.λw:T.drop (S n0) O c2 (CHead e2 (Flat f) w)
                                             λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                             λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2
                                assume c2: C
                                assume v: T
                                suppose : csubst0 (S n0) v (CSort n1) c2
                                assume e: C
                                suppose H0: drop (S n0) O (CSort n1) e
                                  by (drop_gen_sort . . . . H0)
                                  we proved and3 (eq C e (CSort n1)) (eq nat (S n0) O) (eq nat O O)
                                  we proceed by induction on the previous result to prove 
                                     or4
                                       drop (S n0) O c2 e
                                       ex3_4
                                         F
                                         C
                                         T
                                         T
                                         λf:F.λe0:C.λu:T.λ:T.eq C e (CHead e0 (Flat f) u)
                                         λf:F.λe0:C.λ:T.λw:T.drop (S n0) O c2 (CHead e0 (Flat f) w)
                                         λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                       ex3_4
                                         F
                                         C
                                         C
                                         T
                                         λf:F.λe1:C.λ:C.λu:T.eq C e (CHead e1 (Flat f) u)
                                         λf:F.λ:C.λe2:C.λu:T.drop (S n0) O c2 (CHead e2 (Flat f) u)
                                         λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                       ex4_5
                                         F
                                         C
                                         C
                                         T
                                         T
                                         λf:F.λe1:C.λ:C.λu:T.λ:T.eq C e (CHead e1 (Flat f) u)
                                         λf:F.λ:C.λe2:C.λ:T.λw:T.drop (S n0) O c2 (CHead e2 (Flat f) w)
                                         λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                         λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2
                                     case and3_intro : H1:eq C e (CSort n1) H2:eq nat (S n0) O :eq nat O O ⇒
                                        the thesis becomes 
                                        or4
                                          drop (S n0) O c2 e
                                          ex3_4
                                            F
                                            C
                                            T
                                            T
                                            λf:F.λe0:C.λu:T.λ:T.eq C e (CHead e0 (Flat f) u)
                                            λf:F.λe0:C.λ:T.λw:T.drop (S n0) O c2 (CHead e0 (Flat f) w)
                                            λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                          ex3_4
                                            F
                                            C
                                            C
                                            T
                                            λf:F.λe1:C.λ:C.λu:T.eq C e (CHead e1 (Flat f) u)
                                            λf:F.λ:C.λe2:C.λu:T.drop (S n0) O c2 (CHead e2 (Flat f) u)
                                            λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                          ex4_5
                                            F
                                            C
                                            C
                                            T
                                            T
                                            λf:F.λe1:C.λ:C.λu:T.λ:T.eq C e (CHead e1 (Flat f) u)
                                            λf:F.λ:C.λe2:C.λ:T.λw:T.drop (S n0) O c2 (CHead e2 (Flat f) w)
                                            λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                            λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2
                                           (H4) 
                                              we proceed by induction on H2 to prove <λ:nat.Prop> CASE O OF O⇒False | S ⇒True
                                                 case refl_equal : ⇒
                                                    the thesis becomes <λ:nat.Prop> CASE S n0 OF O⇒False | S ⇒True
                                                       consider I
                                                       we proved True
<λ:nat.Prop> CASE S n0 OF O⇒False | S ⇒True
<λ:nat.Prop> CASE O OF O⇒False | S ⇒True
                                           end of H4
                                           consider H4
                                           we proved <λ:nat.Prop> CASE O OF O⇒False | S ⇒True
                                           that is equivalent to False
                                           we proceed by induction on the previous result to prove 
                                              or4
                                                drop (S n0) O c2 (CSort n1)
                                                ex3_4
                                                  F
                                                  C
                                                  T
                                                  T
                                                  λf:F.λe0:C.λu:T.λ:T.eq C (CSort n1) (CHead e0 (Flat f) u)
                                                  λf:F.λe0:C.λ:T.λw:T.drop (S n0) O c2 (CHead e0 (Flat f) w)
                                                  λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                                ex3_4
                                                  F
                                                  C
                                                  C
                                                  T
                                                  λf:F.λe1:C.λ:C.λu:T.eq C (CSort n1) (CHead e1 (Flat f) u)
                                                  λf:F.λ:C.λe2:C.λu:T.drop (S n0) O c2 (CHead e2 (Flat f) u)
                                                  λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                                ex4_5
                                                  F
                                                  C
                                                  C
                                                  T
                                                  T
                                                  λf:F.λe1:C.λ:C.λu:T.λ:T.eq C (CSort n1) (CHead e1 (Flat f) u)
                                                  λf:F.λ:C.λe2:C.λ:T.λw:T.drop (S n0) O c2 (CHead e2 (Flat f) w)
                                                  λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                                  λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2
                                           we proved 
                                              or4
                                                drop (S n0) O c2 (CSort n1)
                                                ex3_4
                                                  F
                                                  C
                                                  T
                                                  T
                                                  λf:F.λe0:C.λu:T.λ:T.eq C (CSort n1) (CHead e0 (Flat f) u)
                                                  λf:F.λe0:C.λ:T.λw:T.drop (S n0) O c2 (CHead e0 (Flat f) w)
                                                  λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                                ex3_4
                                                  F
                                                  C
                                                  C
                                                  T
                                                  λf:F.λe1:C.λ:C.λu:T.eq C (CSort n1) (CHead e1 (Flat f) u)
                                                  λf:F.λ:C.λe2:C.λu:T.drop (S n0) O c2 (CHead e2 (Flat f) u)
                                                  λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                                ex4_5
                                                  F
                                                  C
                                                  C
                                                  T
                                                  T
                                                  λf:F.λe1:C.λ:C.λu:T.λ:T.eq C (CSort n1) (CHead e1 (Flat f) u)
                                                  λf:F.λ:C.λe2:C.λ:T.λw:T.drop (S n0) O c2 (CHead e2 (Flat f) w)
                                                  λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                                  λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2
                                           by (eq_ind_r . . . previous . H1)

                                              or4
                                                drop (S n0) O c2 e
                                                ex3_4
                                                  F
                                                  C
                                                  T
                                                  T
                                                  λf:F.λe0:C.λu:T.λ:T.eq C e (CHead e0 (Flat f) u)
                                                  λf:F.λe0:C.λ:T.λw:T.drop (S n0) O c2 (CHead e0 (Flat f) w)
                                                  λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                                ex3_4
                                                  F
                                                  C
                                                  C
                                                  T
                                                  λf:F.λe1:C.λ:C.λu:T.eq C e (CHead e1 (Flat f) u)
                                                  λf:F.λ:C.λe2:C.λu:T.drop (S n0) O c2 (CHead e2 (Flat f) u)
                                                  λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                                ex4_5
                                                  F
                                                  C
                                                  C
                                                  T
                                                  T
                                                  λf:F.λe1:C.λ:C.λu:T.λ:T.eq C e (CHead e1 (Flat f) u)
                                                  λf:F.λ:C.λe2:C.λ:T.λw:T.drop (S n0) O c2 (CHead e2 (Flat f) w)
                                                  λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                                  λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2
                                  we proved 
                                     or4
                                       drop (S n0) O c2 e
                                       ex3_4
                                         F
                                         C
                                         T
                                         T
                                         λf:F.λe0:C.λu:T.λ:T.eq C e (CHead e0 (Flat f) u)
                                         λf:F.λe0:C.λ:T.λw:T.drop (S n0) O c2 (CHead e0 (Flat f) w)
                                         λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                       ex3_4
                                         F
                                         C
                                         C
                                         T
                                         λf:F.λe1:C.λ:C.λu:T.eq C e (CHead e1 (Flat f) u)
                                         λf:F.λ:C.λe2:C.λu:T.drop (S n0) O c2 (CHead e2 (Flat f) u)
                                         λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                       ex4_5
                                         F
                                         C
                                         C
                                         T
                                         T
                                         λf:F.λe1:C.λ:C.λu:T.λ:T.eq C e (CHead e1 (Flat f) u)
                                         λf:F.λ:C.λe2:C.λ:T.λw:T.drop (S n0) O c2 (CHead e2 (Flat f) w)
                                         λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                         λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2

                                  ∀c2:C
                                    .∀v:T
                                      .csubst0 (S n0) v (CSort n1) c2
                                        →∀e:C
                                             .∀H0:drop (S n0) O (CSort n1) e
                                               .or4
                                                 drop (S n0) O c2 e
                                                 ex3_4
                                                   F
                                                   C
                                                   T
                                                   T
                                                   λf:F.λe0:C.λu:T.λ:T.eq C e (CHead e0 (Flat f) u)
                                                   λf:F.λe0:C.λ:T.λw:T.drop (S n0) O c2 (CHead e0 (Flat f) w)
                                                   λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                                 ex3_4
                                                   F
                                                   C
                                                   C
                                                   T
                                                   λf:F.λe1:C.λ:C.λu:T.eq C e (CHead e1 (Flat f) u)
                                                   λf:F.λ:C.λe2:C.λu:T.drop (S n0) O c2 (CHead e2 (Flat f) u)
                                                   λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                                 ex4_5
                                                   F
                                                   C
                                                   C
                                                   T
                                                   T
                                                   λf:F.λe1:C.λ:C.λu:T.λ:T.eq C e (CHead e1 (Flat f) u)
                                                   λf:F.λ:C.λe2:C.λ:T.λw:T.drop (S n0) O c2 (CHead e2 (Flat f) w)
                                                   λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                                   λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2
                         case CHead : c:C k:K t:T ⇒
                            the thesis becomes 
                            ∀c2:C
                              .∀v:T
                                .∀H0:csubst0 (S n0) v (CHead c k t) c2
                                  .∀e:C
                                    .∀H1:drop (S n0) O (CHead c k t) e
                                      .or4
                                        drop (S n0) O c2 e
                                        ex3_4
                                          F
                                          C
                                          T
                                          T
                                          λf:F.λe0:C.λu:T.λ:T.eq C e (CHead e0 (Flat f) u)
                                          λf:F.λe0:C.λ:T.λw:T.drop (S n0) O c2 (CHead e0 (Flat f) w)
                                          λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                        ex3_4
                                          F
                                          C
                                          C
                                          T
                                          λf:F.λe1:C.λ:C.λu:T.eq C e (CHead e1 (Flat f) u)
                                          λf:F.λ:C.λe2:C.λu:T.drop (S n0) O c2 (CHead e2 (Flat f) u)
                                          λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                        ex4_5
                                          F
                                          C
                                          C
                                          T
                                          T
                                          λf:F.λe1:C.λ:C.λu:T.λ:T.eq C e (CHead e1 (Flat f) u)
                                          λf:F.λ:C.λe2:C.λ:T.λw:T.drop (S n0) O c2 (CHead e2 (Flat f) w)
                                          λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                          λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2
                            (H) by induction hypothesis we know 
                               ∀c2:C
                                 .∀v:T
                                   .csubst0 (S n0) v c c2
                                     →∀e:C
                                          .drop (S n0) O c e
                                            →(or4
                                                 drop (S n0) O c2 e
                                                 ex3_4
                                                   F
                                                   C
                                                   T
                                                   T
                                                   λf:F.λe0:C.λu:T.λ:T.eq C e (CHead e0 (Flat f) u)
                                                   λf:F.λe0:C.λ:T.λw:T.drop (S n0) O c2 (CHead e0 (Flat f) w)
                                                   λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                                 ex3_4
                                                   F
                                                   C
                                                   C
                                                   T
                                                   λf:F.λe1:C.λ:C.λu:T.eq C e (CHead e1 (Flat f) u)
                                                   λf:F.λ:C.λe2:C.λu:T.drop (S n0) O c2 (CHead e2 (Flat f) u)
                                                   λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                                 ex4_5
                                                   F
                                                   C
                                                   C
                                                   T
                                                   T
                                                   λf:F.λe1:C.λ:C.λu:T.λ:T.eq C e (CHead e1 (Flat f) u)
                                                   λf:F.λ:C.λe2:C.λ:T.λw:T.drop (S n0) O c2 (CHead e2 (Flat f) w)
                                                   λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                                   λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2)
                                assume c2: C
                                assume v: T
                                suppose H0: csubst0 (S n0) v (CHead c k t) c2
                                assume e: C
                                suppose H1: drop (S n0) O (CHead c k t) e
                                  by (csubst0_gen_head . . . . . . H0)
                                  we proved 
                                     or3
                                       ex3_2
                                         T
                                         nat
                                         λ:T.λj:nat.eq nat (S n0) (s k j)
                                         λu2:T.λ:nat.eq C c2 (CHead c k u2)
                                         λu2:T.λj:nat.subst0 j v t u2
                                       ex3_2
                                         C
                                         nat
                                         λ:C.λj:nat.eq nat (S n0) (s k j)
                                         λc2:C.λ:nat.eq C c2 (CHead c2 k t)
                                         λc2:C.λj:nat.csubst0 j v c c2
                                       ex4_3
                                         T
                                         C
                                         nat
                                         λ:T.λ:C.λj:nat.eq nat (S n0) (s k j)
                                         λu2:T.λc2:C.λ:nat.eq C c2 (CHead c2 k u2)
                                         λu2:T.λ:C.λj:nat.subst0 j v t u2
                                         λ:T.λc2:C.λj:nat.csubst0 j v c c2
                                  we proceed by induction on the previous result to prove 
                                     or4
                                       drop (S n0) O c2 e
                                       ex3_4
                                         F
                                         C
                                         T
                                         T
                                         λf:F.λe0:C.λu:T.λ:T.eq C e (CHead e0 (Flat f) u)
                                         λf:F.λe0:C.λ:T.λw:T.drop (S n0) O c2 (CHead e0 (Flat f) w)
                                         λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                       ex3_4
                                         F
                                         C
                                         C
                                         T
                                         λf:F.λe1:C.λ:C.λu:T.eq C e (CHead e1 (Flat f) u)
                                         λf:F.λ:C.λe2:C.λu:T.drop (S n0) O c2 (CHead e2 (Flat f) u)
                                         λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                       ex4_5
                                         F
                                         C
                                         C
                                         T
                                         T
                                         λf:F.λe1:C.λ:C.λu:T.λ:T.eq C e (CHead e1 (Flat f) u)
                                         λf:F.λ:C.λe2:C.λ:T.λw:T.drop (S n0) O c2 (CHead e2 (Flat f) w)
                                         λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                         λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2
                                     case or3_intro0 : H2:ex3_2 T nat λ:T.λj:nat.eq nat (S n0) (s k j) λu2:T.λ:nat.eq C c2 (CHead c k u2) λu2:T.λj:nat.subst0 j v t u2 ⇒
                                        the thesis becomes 
                                        or4
                                          drop (S n0) O c2 e
                                          ex3_4
                                            F
                                            C
                                            T
                                            T
                                            λf:F.λe0:C.λu:T.λ:T.eq C e (CHead e0 (Flat f) u)
                                            λf:F.λe0:C.λ:T.λw:T.drop (S n0) O c2 (CHead e0 (Flat f) w)
                                            λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                          ex3_4
                                            F
                                            C
                                            C
                                            T
                                            λf:F.λe1:C.λ:C.λu:T.eq C e (CHead e1 (Flat f) u)
                                            λf:F.λ:C.λe2:C.λu:T.drop (S n0) O c2 (CHead e2 (Flat f) u)
                                            λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                          ex4_5
                                            F
                                            C
                                            C
                                            T
                                            T
                                            λf:F.λe1:C.λ:C.λu:T.λ:T.eq C e (CHead e1 (Flat f) u)
                                            λf:F.λ:C.λe2:C.λ:T.λw:T.drop (S n0) O c2 (CHead e2 (Flat f) w)
                                            λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                            λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2
                                           we proceed by induction on H2 to prove 
                                              or4
                                                drop (S n0) O c2 e
                                                ex3_4
                                                  F
                                                  C
                                                  T
                                                  T
                                                  λf:F.λe0:C.λu:T.λ:T.eq C e (CHead e0 (Flat f) u)
                                                  λf:F.λe0:C.λ:T.λw:T.drop (S n0) O c2 (CHead e0 (Flat f) w)
                                                  λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                                ex3_4
                                                  F
                                                  C
                                                  C
                                                  T
                                                  λf:F.λe1:C.λ:C.λu:T.eq C e (CHead e1 (Flat f) u)
                                                  λf:F.λ:C.λe2:C.λu:T.drop (S n0) O c2 (CHead e2 (Flat f) u)
                                                  λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                                ex4_5
                                                  F
                                                  C
                                                  C
                                                  T
                                                  T
                                                  λf:F.λe1:C.λ:C.λu:T.λ:T.eq C e (CHead e1 (Flat f) u)
                                                  λf:F.λ:C.λe2:C.λ:T.λw:T.drop (S n0) O c2 (CHead e2 (Flat f) w)
                                                  λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                                  λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2
                                              case ex3_2_intro : x0:T x1:nat H3:eq nat (S n0) (s k x1) H4:eq C c2 (CHead c k x0) H5:subst0 x1 v t x0 ⇒
                                                 the thesis becomes 
                                                 or4
                                                   drop (S n0) O c2 e
                                                   ex3_4
                                                     F
                                                     C
                                                     T
                                                     T
                                                     λf:F.λe0:C.λu:T.λ:T.eq C e (CHead e0 (Flat f) u)
                                                     λf:F.λe0:C.λ:T.λw:T.drop (S n0) O c2 (CHead e0 (Flat f) w)
                                                     λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                                   ex3_4
                                                     F
                                                     C
                                                     C
                                                     T
                                                     λf:F.λe1:C.λ:C.λu:T.eq C e (CHead e1 (Flat f) u)
                                                     λf:F.λ:C.λe2:C.λu:T.drop (S n0) O c2 (CHead e2 (Flat f) u)
                                                     λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                                   ex4_5
                                                     F
                                                     C
                                                     C
                                                     T
                                                     T
                                                     λf:F.λe1:C.λ:C.λu:T.λ:T.eq C e (CHead e1 (Flat f) u)
                                                     λf:F.λ:C.λe2:C.λ:T.λw:T.drop (S n0) O c2 (CHead e2 (Flat f) w)
                                                     λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                                     λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2
                                                    by (drop_gen_drop . . . . . H1)
                                                    we proved drop (r k n0) O c e
                                                       assume b: B
                                                        suppose H6: drop (r (Bind b) n0) O c e
                                                        suppose H7: eq nat (S n0) (s (Bind b) x1)
                                                          (H8) 
                                                             consider H7
                                                             we proved eq nat (S n0) (s (Bind b) x1)
                                                             that is equivalent to eq nat (S n0) (S x1)
                                                             by (f_equal . . . . . previous)
                                                             we proved eq nat <λ:nat.nat> CASE S n0 OF O⇒n0 | S n1⇒n1 <λ:nat.nat> CASE S x1 OF O⇒n0 | S n1⇒n1

                                                                eq
                                                                  nat
                                                                  λe0:nat.<λ:nat.nat> CASE e0 OF O⇒n0 | S n1⇒n1 (S n0)
                                                                  λe0:nat.<λ:nat.nat> CASE e0 OF O⇒n0 | S n1⇒n1 (S x1)
                                                          end of H8
                                                          by (drop_drop . . . . H6 .)
                                                          we proved drop (S n0) O (CHead c (Bind b) x0) e
                                                          by (or4_intro0 . . . . previous)
                                                          we proved 
                                                             or4
                                                               drop (S n0) O (CHead c (Bind b) x0) e
                                                               ex3_4
                                                                 F
                                                                 C
                                                                 T
                                                                 T
                                                                 λf:F.λe0:C.λu:T.λ:T.eq C e (CHead e0 (Flat f) u)
                                                                 λf:F.λe0:C.λ:T.λw:T.drop (S n0) O (CHead c (Bind b) x0) (CHead e0 (Flat f) w)
                                                                 λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                                               ex3_4
                                                                 F
                                                                 C
                                                                 C
                                                                 T
                                                                 λf:F.λe1:C.λ:C.λu:T.eq C e (CHead e1 (Flat f) u)
                                                                 λf:F.λ:C.λe2:C.λu:T.drop (S n0) O (CHead c (Bind b) x0) (CHead e2 (Flat f) u)
                                                                 λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                                               ex4_5
                                                                 F
                                                                 C
                                                                 C
                                                                 T
                                                                 T
                                                                 λf:F.λe1:C.λ:C.λu:T.λ:T.eq C e (CHead e1 (Flat f) u)
                                                                 λf:F.λ:C.λe2:C.λ:T.λw:T.drop (S n0) O (CHead c (Bind b) x0) (CHead e2 (Flat f) w)
                                                                 λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                                                 λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2

                                                          ∀H6:drop (r (Bind b) n0) O c e
                                                            .∀H7:eq nat (S n0) (s (Bind b) x1)
                                                              .or4
                                                                drop (S n0) O (CHead c (Bind b) x0) e
                                                                ex3_4
                                                                  F
                                                                  C
                                                                  T
                                                                  T
                                                                  λf:F.λe0:C.λu:T.λ:T.eq C e (CHead e0 (Flat f) u)
                                                                  λf:F.λe0:C.λ:T.λw:T.drop (S n0) O (CHead c (Bind b) x0) (CHead e0 (Flat f) w)
                                                                  λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                                                ex3_4
                                                                  F
                                                                  C
                                                                  C
                                                                  T
                                                                  λf:F.λe1:C.λ:C.λu:T.eq C e (CHead e1 (Flat f) u)
                                                                  λf:F.λ:C.λe2:C.λu:T.drop (S n0) O (CHead c (Bind b) x0) (CHead e2 (Flat f) u)
                                                                  λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                                                ex4_5
                                                                  F
                                                                  C
                                                                  C
                                                                  T
                                                                  T
                                                                  λf:F.λe1:C.λ:C.λu:T.λ:T.eq C e (CHead e1 (Flat f) u)
                                                                  λf:F.λ:C.λe2:C.λ:T.λw:T.drop (S n0) O (CHead c (Bind b) x0) (CHead e2 (Flat f) w)
                                                                  λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                                                  λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2
                                                       assume f: F
                                                        suppose H6: drop (r (Flat f) n0) O c e
                                                        suppose H7: eq nat (S n0) (s (Flat f) x1)
                                                          (H8) 
                                                             by (f_equal . . . . . H7)
                                                             we proved eq nat (S n0) (s (Flat f) x1)
eq nat (λe0:nat.e0 (S n0)) (λe0:nat.e0 (s (Flat f) x1))
                                                          end of H8
                                                          by (drop_drop . . . . H6 .)
                                                          we proved drop (S n0) O (CHead c (Flat f) x0) e
                                                          by (or4_intro0 . . . . previous)
                                                          we proved 
                                                             or4
                                                               drop (S n0) O (CHead c (Flat f) x0) e
                                                               ex3_4
                                                                 F
                                                                 C
                                                                 T
                                                                 T
                                                                 λf0:F.λe0:C.λu:T.λ:T.eq C e (CHead e0 (Flat f0) u)
                                                                 λf0:F.λe0:C.λ:T.λw:T.drop (S n0) O (CHead c (Flat f) x0) (CHead e0 (Flat f0) w)
                                                                 λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                                               ex3_4
                                                                 F
                                                                 C
                                                                 C
                                                                 T
                                                                 λf0:F.λe1:C.λ:C.λu:T.eq C e (CHead e1 (Flat f0) u)
                                                                 λf0:F.λ:C.λe2:C.λu:T.drop (S n0) O (CHead c (Flat f) x0) (CHead e2 (Flat f0) u)
                                                                 λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                                               ex4_5
                                                                 F
                                                                 C
                                                                 C
                                                                 T
                                                                 T
                                                                 λf0:F.λe1:C.λ:C.λu:T.λ:T.eq C e (CHead e1 (Flat f0) u)
                                                                 λf0:F.λ:C.λe2:C.λ:T.λw:T.drop (S n0) O (CHead c (Flat f) x0) (CHead e2 (Flat f0) w)
                                                                 λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                                                 λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2

                                                          ∀H6:drop (r (Flat f) n0) O c e
                                                            .∀H7:eq nat (S n0) (s (Flat f) x1)
                                                              .or4
                                                                drop (S n0) O (CHead c (Flat f) x0) e
                                                                ex3_4
                                                                  F
                                                                  C
                                                                  T
                                                                  T
                                                                  λf0:F.λe0:C.λu:T.λ:T.eq C e (CHead e0 (Flat f0) u)
                                                                  λf0:F.λe0:C.λ:T.λw:T.drop (S n0) O (CHead c (Flat f) x0) (CHead e0 (Flat f0) w)
                                                                  λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                                                ex3_4
                                                                  F
                                                                  C
                                                                  C
                                                                  T
                                                                  λf0:F.λe1:C.λ:C.λu:T.eq C e (CHead e1 (Flat f0) u)
                                                                  λf0:F.λ:C.λe2:C.λu:T.drop (S n0) O (CHead c (Flat f) x0) (CHead e2 (Flat f0) u)
                                                                  λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                                                ex4_5
                                                                  F
                                                                  C
                                                                  C
                                                                  T
                                                                  T
                                                                  λf0:F.λe1:C.λ:C.λu:T.λ:T.eq C e (CHead e1 (Flat f0) u)
                                                                  λf0:F.λ:C.λe2:C.λ:T.λw:T.drop (S n0) O (CHead c (Flat f) x0) (CHead e2 (Flat f0) w)
                                                                  λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                                                  λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2
                                                    by (previous . previous H3)
                                                    we proved 
                                                       or4
                                                         drop (S n0) O (CHead c k x0) e
                                                         ex3_4
                                                           F
                                                           C
                                                           T
                                                           T
                                                           λf:F.λe0:C.λu:T.λ:T.eq C e (CHead e0 (Flat f) u)
                                                           λf:F.λe0:C.λ:T.λw:T.drop (S n0) O (CHead c k x0) (CHead e0 (Flat f) w)
                                                           λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                                         ex3_4
                                                           F
                                                           C
                                                           C
                                                           T
                                                           λf:F.λe1:C.λ:C.λu:T.eq C e (CHead e1 (Flat f) u)
                                                           λf:F.λ:C.λe2:C.λu:T.drop (S n0) O (CHead c k x0) (CHead e2 (Flat f) u)
                                                           λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                                         ex4_5
                                                           F
                                                           C
                                                           C
                                                           T
                                                           T
                                                           λf:F.λe1:C.λ:C.λu:T.λ:T.eq C e (CHead e1 (Flat f) u)
                                                           λf:F.λ:C.λe2:C.λ:T.λw:T.drop (S n0) O (CHead c k x0) (CHead e2 (Flat f) w)
                                                           λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                                           λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2
                                                    by (eq_ind_r . . . previous . H4)

                                                       or4
                                                         drop (S n0) O c2 e
                                                         ex3_4
                                                           F
                                                           C
                                                           T
                                                           T
                                                           λf:F.λe0:C.λu:T.λ:T.eq C e (CHead e0 (Flat f) u)
                                                           λf:F.λe0:C.λ:T.λw:T.drop (S n0) O c2 (CHead e0 (Flat f) w)
                                                           λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                                         ex3_4
                                                           F
                                                           C
                                                           C
                                                           T
                                                           λf:F.λe1:C.λ:C.λu:T.eq C e (CHead e1 (Flat f) u)
                                                           λf:F.λ:C.λe2:C.λu:T.drop (S n0) O c2 (CHead e2 (Flat f) u)
                                                           λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                                         ex4_5
                                                           F
                                                           C
                                                           C
                                                           T
                                                           T
                                                           λf:F.λe1:C.λ:C.λu:T.λ:T.eq C e (CHead e1 (Flat f) u)
                                                           λf:F.λ:C.λe2:C.λ:T.λw:T.drop (S n0) O c2 (CHead e2 (Flat f) w)
                                                           λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                                           λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2

                                              or4
                                                drop (S n0) O c2 e
                                                ex3_4
                                                  F
                                                  C
                                                  T
                                                  T
                                                  λf:F.λe0:C.λu:T.λ:T.eq C e (CHead e0 (Flat f) u)
                                                  λf:F.λe0:C.λ:T.λw:T.drop (S n0) O c2 (CHead e0 (Flat f) w)
                                                  λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                                ex3_4
                                                  F
                                                  C
                                                  C
                                                  T
                                                  λf:F.λe1:C.λ:C.λu:T.eq C e (CHead e1 (Flat f) u)
                                                  λf:F.λ:C.λe2:C.λu:T.drop (S n0) O c2 (CHead e2 (Flat f) u)
                                                  λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                                ex4_5
                                                  F
                                                  C
                                                  C
                                                  T
                                                  T
                                                  λf:F.λe1:C.λ:C.λu:T.λ:T.eq C e (CHead e1 (Flat f) u)
                                                  λf:F.λ:C.λe2:C.λ:T.λw:T.drop (S n0) O c2 (CHead e2 (Flat f) w)
                                                  λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                                  λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2
                                     case or3_intro1 : H2:ex3_2 C nat λ:C.λj:nat.eq nat (S n0) (s k j) λc3:C.λ:nat.eq C c2 (CHead c3 k t) λc3:C.λj:nat.csubst0 j v c c3 ⇒
                                        the thesis becomes 
                                        or4
                                          drop (S n0) O c2 e
                                          ex3_4
                                            F
                                            C
                                            T
                                            T
                                            λf:F.λe0:C.λu:T.λ:T.eq C e (CHead e0 (Flat f) u)
                                            λf:F.λe0:C.λ:T.λw:T.drop (S n0) O c2 (CHead e0 (Flat f) w)
                                            λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                          ex3_4
                                            F
                                            C
                                            C
                                            T
                                            λf:F.λe1:C.λ:C.λu:T.eq C e (CHead e1 (Flat f) u)
                                            λf:F.λ:C.λe2:C.λu:T.drop (S n0) O c2 (CHead e2 (Flat f) u)
                                            λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                          ex4_5
                                            F
                                            C
                                            C
                                            T
                                            T
                                            λf:F.λe1:C.λ:C.λu:T.λ:T.eq C e (CHead e1 (Flat f) u)
                                            λf:F.λ:C.λe2:C.λ:T.λw:T.drop (S n0) O c2 (CHead e2 (Flat f) w)
                                            λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                            λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2
                                           we proceed by induction on H2 to prove 
                                              or4
                                                drop (S n0) O c2 e
                                                ex3_4
                                                  F
                                                  C
                                                  T
                                                  T
                                                  λf:F.λe0:C.λu:T.λ:T.eq C e (CHead e0 (Flat f) u)
                                                  λf:F.λe0:C.λ:T.λw:T.drop (S n0) O c2 (CHead e0 (Flat f) w)
                                                  λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                                ex3_4
                                                  F
                                                  C
                                                  C
                                                  T
                                                  λf:F.λe1:C.λ:C.λu:T.eq C e (CHead e1 (Flat f) u)
                                                  λf:F.λ:C.λe2:C.λu:T.drop (S n0) O c2 (CHead e2 (Flat f) u)
                                                  λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                                ex4_5
                                                  F
                                                  C
                                                  C
                                                  T
                                                  T
                                                  λf:F.λe1:C.λ:C.λu:T.λ:T.eq C e (CHead e1 (Flat f) u)
                                                  λf:F.λ:C.λe2:C.λ:T.λw:T.drop (S n0) O c2 (CHead e2 (Flat f) w)
                                                  λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                                  λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2
                                              case ex3_2_intro : x0:C x1:nat H3:eq nat (S n0) (s k x1) H4:eq C c2 (CHead x0 k t) H5:csubst0 x1 v c x0 ⇒
                                                 the thesis becomes 
                                                 or4
                                                   drop (S n0) O c2 e
                                                   ex3_4
                                                     F
                                                     C
                                                     T
                                                     T
                                                     λf:F.λe0:C.λu:T.λ:T.eq C e (CHead e0 (Flat f) u)
                                                     λf:F.λe0:C.λ:T.λw:T.drop (S n0) O c2 (CHead e0 (Flat f) w)
                                                     λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                                   ex3_4
                                                     F
                                                     C
                                                     C
                                                     T
                                                     λf:F.λe1:C.λ:C.λu:T.eq C e (CHead e1 (Flat f) u)
                                                     λf:F.λ:C.λe2:C.λu:T.drop (S n0) O c2 (CHead e2 (Flat f) u)
                                                     λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                                   ex4_5
                                                     F
                                                     C
                                                     C
                                                     T
                                                     T
                                                     λf:F.λe1:C.λ:C.λu:T.λ:T.eq C e (CHead e1 (Flat f) u)
                                                     λf:F.λ:C.λe2:C.λ:T.λw:T.drop (S n0) O c2 (CHead e2 (Flat f) w)
                                                     λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                                     λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2
                                                    by (drop_gen_drop . . . . . H1)
                                                    we proved drop (r k n0) O c e
                                                       assume b: B
                                                        suppose H6: drop (r (Bind b) n0) O c e
                                                        suppose H7: eq nat (S n0) (s (Bind b) x1)
                                                          (H8) 
                                                             consider H7
                                                             we proved eq nat (S n0) (s (Bind b) x1)
                                                             that is equivalent to eq nat (S n0) (S x1)
                                                             by (f_equal . . . . . previous)
                                                             we proved eq nat <λ:nat.nat> CASE S n0 OF O⇒n0 | S n1⇒n1 <λ:nat.nat> CASE S x1 OF O⇒n0 | S n1⇒n1

                                                                eq
                                                                  nat
                                                                  λe0:nat.<λ:nat.nat> CASE e0 OF O⇒n0 | S n1⇒n1 (S n0)
                                                                  λe0:nat.<λ:nat.nat> CASE e0 OF O⇒n0 | S n1⇒n1 (S x1)
                                                          end of H8
                                                          (H9) 
                                                             consider H8
                                                             we proved eq nat <λ:nat.nat> CASE S n0 OF O⇒n0 | S n1⇒n1 <λ:nat.nat> CASE S x1 OF O⇒n0 | S n1⇒n1
                                                             that is equivalent to eq nat n0 x1
                                                             by (eq_ind_r . . . H5 . previous)
csubst0 n0 v c x0
                                                          end of H9
                                                          (H10) 
                                                             consider H6
                                                             we proved drop (r (Bind b) n0) O c e
                                                             that is equivalent to drop n0 O c e
                                                             by (IHn . . . H9 . previous)

                                                                or4
                                                                  drop n0 O x0 e
                                                                  ex3_4
                                                                    F
                                                                    C
                                                                    T
                                                                    T
                                                                    λf:F.λe0:C.λu:T.λ:T.eq C e (CHead e0 (Flat f) u)
                                                                    λf:F.λe0:C.λ:T.λw:T.drop n0 O x0 (CHead e0 (Flat f) w)
                                                                    λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                                                  ex3_4
                                                                    F
                                                                    C
                                                                    C
                                                                    T
                                                                    λf:F.λe1:C.λ:C.λu:T.eq C e (CHead e1 (Flat f) u)
                                                                    λf:F.λ:C.λe2:C.λu:T.drop n0 O x0 (CHead e2 (Flat f) u)
                                                                    λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                                                  ex4_5
                                                                    F
                                                                    C
                                                                    C
                                                                    T
                                                                    T
                                                                    λf:F.λe1:C.λ:C.λu:T.λ:T.eq C e (CHead e1 (Flat f) u)
                                                                    λf:F.λ:C.λe2:C.λ:T.λw:T.drop n0 O x0 (CHead e2 (Flat f) w)
                                                                    λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                                                    λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2
                                                          end of H10
                                                          we proceed by induction on H10 to prove 
                                                             or4
                                                               drop (S n0) O (CHead x0 (Bind b) t) e
                                                               ex3_4
                                                                 F
                                                                 C
                                                                 T
                                                                 T
                                                                 λf:F.λe0:C.λu:T.λ:T.eq C e (CHead e0 (Flat f) u)
                                                                 λf:F.λe0:C.λ:T.λw:T.drop (S n0) O (CHead x0 (Bind b) t) (CHead e0 (Flat f) w)
                                                                 λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                                               ex3_4
                                                                 F
                                                                 C
                                                                 C
                                                                 T
                                                                 λf:F.λe1:C.λ:C.λu:T.eq C e (CHead e1 (Flat f) u)
                                                                 λf:F.λ:C.λe2:C.λu:T.drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f) u)
                                                                 λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                                               ex4_5
                                                                 F
                                                                 C
                                                                 C
                                                                 T
                                                                 T
                                                                 λf:F.λe1:C.λ:C.λu:T.λ:T.eq C e (CHead e1 (Flat f) u)
                                                                 λf:F.λ:C.λe2:C.λ:T.λw:T.drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f) w)
                                                                 λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                                                 λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2
                                                             case or4_intro0 : H11:drop n0 O x0 e ⇒
                                                                the thesis becomes 
                                                                or4
                                                                  drop (S n0) O (CHead x0 (Bind b) t) e
                                                                  ex3_4
                                                                    F
                                                                    C
                                                                    T
                                                                    T
                                                                    λf:F.λe0:C.λu:T.λ:T.eq C e (CHead e0 (Flat f) u)
                                                                    λf:F.λe0:C.λ:T.λw:T.drop (S n0) O (CHead x0 (Bind b) t) (CHead e0 (Flat f) w)
                                                                    λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                                                  ex3_4
                                                                    F
                                                                    C
                                                                    C
                                                                    T
                                                                    λf:F.λe1:C.λ:C.λu:T.eq C e (CHead e1 (Flat f) u)
                                                                    λf:F.λ:C.λe2:C.λu:T.drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f) u)
                                                                    λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                                                  ex4_5
                                                                    F
                                                                    C
                                                                    C
                                                                    T
                                                                    T
                                                                    λf:F.λe1:C.λ:C.λu:T.λ:T.eq C e (CHead e1 (Flat f) u)
                                                                    λf:F.λ:C.λe2:C.λ:T.λw:T.drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f) w)
                                                                    λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                                                    λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2
                                                                   consider H11
                                                                   we proved drop n0 O x0 e
                                                                   that is equivalent to drop (r (Bind b) n0) O x0 e
                                                                   by (drop_drop . . . . previous .)
                                                                   we proved drop (S n0) O (CHead x0 (Bind b) t) e
                                                                   by (or4_intro0 . . . . previous)

                                                                      or4
                                                                        drop (S n0) O (CHead x0 (Bind b) t) e
                                                                        ex3_4
                                                                          F
                                                                          C
                                                                          T
                                                                          T
                                                                          λf:F.λe0:C.λu:T.λ:T.eq C e (CHead e0 (Flat f) u)
                                                                          λf:F.λe0:C.λ:T.λw:T.drop (S n0) O (CHead x0 (Bind b) t) (CHead e0 (Flat f) w)
                                                                          λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                                                        ex3_4
                                                                          F
                                                                          C
                                                                          C
                                                                          T
                                                                          λf:F.λe1:C.λ:C.λu:T.eq C e (CHead e1 (Flat f) u)
                                                                          λf:F.λ:C.λe2:C.λu:T.drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f) u)
                                                                          λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                                                        ex4_5
                                                                          F
                                                                          C
                                                                          C
                                                                          T
                                                                          T
                                                                          λf:F.λe1:C.λ:C.λu:T.λ:T.eq C e (CHead e1 (Flat f) u)
                                                                          λf:F.λ:C.λe2:C.λ:T.λw:T.drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f) w)
                                                                          λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                                                          λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2
                                                             case or4_intro1 : H11:ex3_4 F C T T λf:F.λe0:C.λu:T.λ:T.eq C e (CHead e0 (Flat f) u) λf:F.λe0:C.λ:T.λw:T.drop n0 O x0 (CHead e0 (Flat f) w) λ:F.λ:C.λu:T.λw:T.subst0 O v u w ⇒
                                                                the thesis becomes 
                                                                or4
                                                                  drop (S n0) O (CHead x0 (Bind b) t) e
                                                                  ex3_4
                                                                    F
                                                                    C
                                                                    T
                                                                    T
                                                                    λf:F.λe0:C.λu:T.λ:T.eq C e (CHead e0 (Flat f) u)
                                                                    λf:F.λe0:C.λ:T.λw:T.drop (S n0) O (CHead x0 (Bind b) t) (CHead e0 (Flat f) w)
                                                                    λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                                                  ex3_4
                                                                    F
                                                                    C
                                                                    C
                                                                    T
                                                                    λf:F.λe1:C.λ:C.λu:T.eq C e (CHead e1 (Flat f) u)
                                                                    λf:F.λ:C.λe2:C.λu:T.drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f) u)
                                                                    λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                                                  ex4_5
                                                                    F
                                                                    C
                                                                    C
                                                                    T
                                                                    T
                                                                    λf:F.λe1:C.λ:C.λu:T.λ:T.eq C e (CHead e1 (Flat f) u)
                                                                    λf:F.λ:C.λe2:C.λ:T.λw:T.drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f) w)
                                                                    λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                                                    λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2
                                                                   we proceed by induction on H11 to prove 
                                                                      or4
                                                                        drop (S n0) O (CHead x0 (Bind b) t) e
                                                                        ex3_4
                                                                          F
                                                                          C
                                                                          T
                                                                          T
                                                                          λf:F.λe0:C.λu:T.λ:T.eq C e (CHead e0 (Flat f) u)
                                                                          λf:F.λe0:C.λ:T.λw:T.drop (S n0) O (CHead x0 (Bind b) t) (CHead e0 (Flat f) w)
                                                                          λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                                                        ex3_4
                                                                          F
                                                                          C
                                                                          C
                                                                          T
                                                                          λf:F.λe1:C.λ:C.λu:T.eq C e (CHead e1 (Flat f) u)
                                                                          λf:F.λ:C.λe2:C.λu:T.drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f) u)
                                                                          λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                                                        ex4_5
                                                                          F
                                                                          C
                                                                          C
                                                                          T
                                                                          T
                                                                          λf:F.λe1:C.λ:C.λu:T.λ:T.eq C e (CHead e1 (Flat f) u)
                                                                          λf:F.λ:C.λe2:C.λ:T.λw:T.drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f) w)
                                                                          λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                                                          λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2
                                                                      case ex3_4_intro : x2:F x3:C x4:T x5:T H12:eq C e (CHead x3 (Flat x2) x4) H13:drop n0 O x0 (CHead x3 (Flat x2) x5) H14:subst0 O v x4 x5 ⇒
                                                                         the thesis becomes 
                                                                         or4
                                                                           drop (S n0) O (CHead x0 (Bind b) t) e
                                                                           ex3_4
                                                                             F
                                                                             C
                                                                             T
                                                                             T
                                                                             λf:F.λe0:C.λu:T.λ:T.eq C e (CHead e0 (Flat f) u)
                                                                             λf:F.λe0:C.λ:T.λw:T.drop (S n0) O (CHead x0 (Bind b) t) (CHead e0 (Flat f) w)
                                                                             λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                                                           ex3_4
                                                                             F
                                                                             C
                                                                             C
                                                                             T
                                                                             λf:F.λe1:C.λ:C.λu:T.eq C e (CHead e1 (Flat f) u)
                                                                             λf:F.λ:C.λe2:C.λu:T.drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f) u)
                                                                             λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                                                           ex4_5
                                                                             F
                                                                             C
                                                                             C
                                                                             T
                                                                             T
                                                                             λf:F.λe1:C.λ:C.λu:T.λ:T.eq C e (CHead e1 (Flat f) u)
                                                                             λf:F.λ:C.λe2:C.λ:T.λw:T.drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f) w)
                                                                             λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                                                             λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2
                                                                            (h1) 
                                                                               by (refl_equal . .)
eq C (CHead x3 (Flat x2) x4) (CHead x3 (Flat x2) x4)
                                                                            end of h1
                                                                            (h2) 
                                                                               consider H13
                                                                               we proved drop n0 O x0 (CHead x3 (Flat x2) x5)
                                                                               that is equivalent to drop (r (Bind b) n0) O x0 (CHead x3 (Flat x2) x5)
                                                                               by (drop_drop . . . . previous .)
drop (S n0) O (CHead x0 (Bind b) t) (CHead x3 (Flat x2) x5)
                                                                            end of h2
                                                                            by (ex3_4_intro . . . . . . . . . . . h1 h2 H14)
                                                                            we proved 
                                                                               ex3_4
                                                                                 F
                                                                                 C
                                                                                 T
                                                                                 T
                                                                                 λf:F.λe0:C.λu:T.λ:T.eq C (CHead x3 (Flat x2) x4) (CHead e0 (Flat f) u)
                                                                                 λf:F.λe0:C.λ:T.λw:T.drop (S n0) O (CHead x0 (Bind b) t) (CHead e0 (Flat f) w)
                                                                                 λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                                                            by (or4_intro1 . . . . previous)
                                                                            we proved 
                                                                               or4
                                                                                 drop (S n0) O (CHead x0 (Bind b) t) (CHead x3 (Flat x2) x4)
                                                                                 ex3_4
                                                                                   F
                                                                                   C
                                                                                   T
                                                                                   T
                                                                                   λf:F.λe0:C.λu:T.λ:T.eq C (CHead x3 (Flat x2) x4) (CHead e0 (Flat f) u)
                                                                                   λf:F.λe0:C.λ:T.λw:T.drop (S n0) O (CHead x0 (Bind b) t) (CHead e0 (Flat f) w)
                                                                                   λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                                                                 ex3_4
                                                                                   F
                                                                                   C
                                                                                   C
                                                                                   T
                                                                                   λf:F.λe1:C.λ:C.λu:T.eq C (CHead x3 (Flat x2) x4) (CHead e1 (Flat f) u)
                                                                                   λf:F.λ:C.λe2:C.λu:T.drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f) u)
                                                                                   λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                                                                 ex4_5
                                                                                   F
                                                                                   C
                                                                                   C
                                                                                   T
                                                                                   T
                                                                                   λf:F.λe1:C.λ:C.λu:T.λ:T.eq C (CHead x3 (Flat x2) x4) (CHead e1 (Flat f) u)
                                                                                   λf:F.λ:C.λe2:C.λ:T.λw:T.drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f) w)
                                                                                   λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                                                                   λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2
                                                                            by (eq_ind_r . . . previous . H12)

                                                                               or4
                                                                                 drop (S n0) O (CHead x0 (Bind b) t) e
                                                                                 ex3_4
                                                                                   F
                                                                                   C
                                                                                   T
                                                                                   T
                                                                                   λf:F.λe0:C.λu:T.λ:T.eq C e (CHead e0 (Flat f) u)
                                                                                   λf:F.λe0:C.λ:T.λw:T.drop (S n0) O (CHead x0 (Bind b) t) (CHead e0 (Flat f) w)
                                                                                   λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                                                                 ex3_4
                                                                                   F
                                                                                   C
                                                                                   C
                                                                                   T
                                                                                   λf:F.λe1:C.λ:C.λu:T.eq C e (CHead e1 (Flat f) u)
                                                                                   λf:F.λ:C.λe2:C.λu:T.drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f) u)
                                                                                   λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                                                                 ex4_5
                                                                                   F
                                                                                   C
                                                                                   C
                                                                                   T
                                                                                   T
                                                                                   λf:F.λe1:C.λ:C.λu:T.λ:T.eq C e (CHead e1 (Flat f) u)
                                                                                   λf:F.λ:C.λe2:C.λ:T.λw:T.drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f) w)
                                                                                   λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                                                                   λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2

                                                                      or4
                                                                        drop (S n0) O (CHead x0 (Bind b) t) e
                                                                        ex3_4
                                                                          F
                                                                          C
                                                                          T
                                                                          T
                                                                          λf:F.λe0:C.λu:T.λ:T.eq C e (CHead e0 (Flat f) u)
                                                                          λf:F.λe0:C.λ:T.λw:T.drop (S n0) O (CHead x0 (Bind b) t) (CHead e0 (Flat f) w)
                                                                          λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                                                        ex3_4
                                                                          F
                                                                          C
                                                                          C
                                                                          T
                                                                          λf:F.λe1:C.λ:C.λu:T.eq C e (CHead e1 (Flat f) u)
                                                                          λf:F.λ:C.λe2:C.λu:T.drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f) u)
                                                                          λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                                                        ex4_5
                                                                          F
                                                                          C
                                                                          C
                                                                          T
                                                                          T
                                                                          λf:F.λe1:C.λ:C.λu:T.λ:T.eq C e (CHead e1 (Flat f) u)
                                                                          λf:F.λ:C.λe2:C.λ:T.λw:T.drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f) w)
                                                                          λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                                                          λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2
                                                             case or4_intro2 : H11:ex3_4 F C C T λf:F.λe1:C.λ:C.λu:T.eq C e (CHead e1 (Flat f) u) λf:F.λ:C.λe2:C.λu:T.drop n0 O x0 (CHead e2 (Flat f) u) λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2 ⇒
                                                                the thesis becomes 
                                                                or4
                                                                  drop (S n0) O (CHead x0 (Bind b) t) e
                                                                  ex3_4
                                                                    F
                                                                    C
                                                                    T
                                                                    T
                                                                    λf:F.λe0:C.λu:T.λ:T.eq C e (CHead e0 (Flat f) u)
                                                                    λf:F.λe0:C.λ:T.λw:T.drop (S n0) O (CHead x0 (Bind b) t) (CHead e0 (Flat f) w)
                                                                    λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                                                  ex3_4
                                                                    F
                                                                    C
                                                                    C
                                                                    T
                                                                    λf:F.λe1:C.λ:C.λu:T.eq C e (CHead e1 (Flat f) u)
                                                                    λf:F.λ:C.λe2:C.λu:T.drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f) u)
                                                                    λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                                                  ex4_5
                                                                    F
                                                                    C
                                                                    C
                                                                    T
                                                                    T
                                                                    λf:F.λe1:C.λ:C.λu:T.λ:T.eq C e (CHead e1 (Flat f) u)
                                                                    λf:F.λ:C.λe2:C.λ:T.λw:T.drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f) w)
                                                                    λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                                                    λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2
                                                                   we proceed by induction on H11 to prove 
                                                                      or4
                                                                        drop (S n0) O (CHead x0 (Bind b) t) e
                                                                        ex3_4
                                                                          F
                                                                          C
                                                                          T
                                                                          T
                                                                          λf:F.λe0:C.λu:T.λ:T.eq C e (CHead e0 (Flat f) u)
                                                                          λf:F.λe0:C.λ:T.λw:T.drop (S n0) O (CHead x0 (Bind b) t) (CHead e0 (Flat f) w)
                                                                          λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                                                        ex3_4
                                                                          F
                                                                          C
                                                                          C
                                                                          T
                                                                          λf:F.λe1:C.λ:C.λu:T.eq C e (CHead e1 (Flat f) u)
                                                                          λf:F.λ:C.λe2:C.λu:T.drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f) u)
                                                                          λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                                                        ex4_5
                                                                          F
                                                                          C
                                                                          C
                                                                          T
                                                                          T
                                                                          λf:F.λe1:C.λ:C.λu:T.λ:T.eq C e (CHead e1 (Flat f) u)
                                                                          λf:F.λ:C.λe2:C.λ:T.λw:T.drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f) w)
                                                                          λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                                                          λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2
                                                                      case ex3_4_intro : x2:F x3:C x4:C x5:T H12:eq C e (CHead x3 (Flat x2) x5) H13:drop n0 O x0 (CHead x4 (Flat x2) x5) H14:csubst0 O v x3 x4 ⇒
                                                                         the thesis becomes 
                                                                         or4
                                                                           drop (S n0) O (CHead x0 (Bind b) t) e
                                                                           ex3_4
                                                                             F
                                                                             C
                                                                             T
                                                                             T
                                                                             λf:F.λe0:C.λu:T.λ:T.eq C e (CHead e0 (Flat f) u)
                                                                             λf:F.λe0:C.λ:T.λw:T.drop (S n0) O (CHead x0 (Bind b) t) (CHead e0 (Flat f) w)
                                                                             λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                                                           ex3_4
                                                                             F
                                                                             C
                                                                             C
                                                                             T
                                                                             λf:F.λe1:C.λ:C.λu:T.eq C e (CHead e1 (Flat f) u)
                                                                             λf:F.λ:C.λe2:C.λu:T.drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f) u)
                                                                             λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                                                           ex4_5
                                                                             F
                                                                             C
                                                                             C
                                                                             T
                                                                             T
                                                                             λf:F.λe1:C.λ:C.λu:T.λ:T.eq C e (CHead e1 (Flat f) u)
                                                                             λf:F.λ:C.λe2:C.λ:T.λw:T.drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f) w)
                                                                             λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                                                             λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2
                                                                            (h1) 
                                                                               by (refl_equal . .)
eq C (CHead x3 (Flat x2) x5) (CHead x3 (Flat x2) x5)
                                                                            end of h1
                                                                            (h2) 
                                                                               consider H13
                                                                               we proved drop n0 O x0 (CHead x4 (Flat x2) x5)
                                                                               that is equivalent to drop (r (Bind b) n0) O x0 (CHead x4 (Flat x2) x5)
                                                                               by (drop_drop . . . . previous .)
drop (S n0) O (CHead x0 (Bind b) t) (CHead x4 (Flat x2) x5)
                                                                            end of h2
                                                                            by (ex3_4_intro . . . . . . . . . . . h1 h2 H14)
                                                                            we proved 
                                                                               ex3_4
                                                                                 F
                                                                                 C
                                                                                 C
                                                                                 T
                                                                                 λf:F.λe1:C.λ:C.λu:T.eq C (CHead x3 (Flat x2) x5) (CHead e1 (Flat f) u)
                                                                                 λf:F.λ:C.λe2:C.λu:T.drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f) u)
                                                                                 λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                                                            by (or4_intro2 . . . . previous)
                                                                            we proved 
                                                                               or4
                                                                                 drop (S n0) O (CHead x0 (Bind b) t) (CHead x3 (Flat x2) x5)
                                                                                 ex3_4
                                                                                   F
                                                                                   C
                                                                                   T
                                                                                   T
                                                                                   λf:F.λe0:C.λu:T.λ:T.eq C (CHead x3 (Flat x2) x5) (CHead e0 (Flat f) u)
                                                                                   λf:F.λe0:C.λ:T.λw:T.drop (S n0) O (CHead x0 (Bind b) t) (CHead e0 (Flat f) w)
                                                                                   λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                                                                 ex3_4
                                                                                   F
                                                                                   C
                                                                                   C
                                                                                   T
                                                                                   λf:F.λe1:C.λ:C.λu:T.eq C (CHead x3 (Flat x2) x5) (CHead e1 (Flat f) u)
                                                                                   λf:F.λ:C.λe2:C.λu:T.drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f) u)
                                                                                   λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                                                                 ex4_5
                                                                                   F
                                                                                   C
                                                                                   C
                                                                                   T
                                                                                   T
                                                                                   λf:F.λe1:C.λ:C.λu:T.λ:T.eq C (CHead x3 (Flat x2) x5) (CHead e1 (Flat f) u)
                                                                                   λf:F.λ:C.λe2:C.λ:T.λw:T.drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f) w)
                                                                                   λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                                                                   λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2
                                                                            by (eq_ind_r . . . previous . H12)

                                                                               or4
                                                                                 drop (S n0) O (CHead x0 (Bind b) t) e
                                                                                 ex3_4
                                                                                   F
                                                                                   C
                                                                                   T
                                                                                   T
                                                                                   λf:F.λe0:C.λu:T.λ:T.eq C e (CHead e0 (Flat f) u)
                                                                                   λf:F.λe0:C.λ:T.λw:T.drop (S n0) O (CHead x0 (Bind b) t) (CHead e0 (Flat f) w)
                                                                                   λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                                                                 ex3_4
                                                                                   F
                                                                                   C
                                                                                   C
                                                                                   T
                                                                                   λf:F.λe1:C.λ:C.λu:T.eq C e (CHead e1 (Flat f) u)
                                                                                   λf:F.λ:C.λe2:C.λu:T.drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f) u)
                                                                                   λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                                                                 ex4_5
                                                                                   F
                                                                                   C
                                                                                   C
                                                                                   T
                                                                                   T
                                                                                   λf:F.λe1:C.λ:C.λu:T.λ:T.eq C e (CHead e1 (Flat f) u)
                                                                                   λf:F.λ:C.λe2:C.λ:T.λw:T.drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f) w)
                                                                                   λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                                                                   λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2

                                                                      or4
                                                                        drop (S n0) O (CHead x0 (Bind b) t) e
                                                                        ex3_4
                                                                          F
                                                                          C
                                                                          T
                                                                          T
                                                                          λf:F.λe0:C.λu:T.λ:T.eq C e (CHead e0 (Flat f) u)
                                                                          λf:F.λe0:C.λ:T.λw:T.drop (S n0) O (CHead x0 (Bind b) t) (CHead e0 (Flat f) w)
                                                                          λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                                                        ex3_4
                                                                          F
                                                                          C
                                                                          C
                                                                          T
                                                                          λf:F.λe1:C.λ:C.λu:T.eq C e (CHead e1 (Flat f) u)
                                                                          λf:F.λ:C.λe2:C.λu:T.drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f) u)
                                                                          λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                                                        ex4_5
                                                                          F
                                                                          C
                                                                          C
                                                                          T
                                                                          T
                                                                          λf:F.λe1:C.λ:C.λu:T.λ:T.eq C e (CHead e1 (Flat f) u)
                                                                          λf:F.λ:C.λe2:C.λ:T.λw:T.drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f) w)
                                                                          λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                                                          λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2
                                                             case or4_intro3 : H11:ex4_5 F C C T T λf:F.λe1:C.λ:C.λu:T.λ:T.eq C e (CHead e1 (Flat f) u) λf:F.λ:C.λe2:C.λ:T.λw:T.drop n0 O x0 (CHead e2 (Flat f) w) λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2 ⇒
                                                                the thesis becomes 
                                                                or4
                                                                  drop (S n0) O (CHead x0 (Bind b) t) e
                                                                  ex3_4
                                                                    F
                                                                    C
                                                                    T
                                                                    T
                                                                    λf:F.λe0:C.λu:T.λ:T.eq C e (CHead e0 (Flat f) u)
                                                                    λf:F.λe0:C.λ:T.λw:T.drop (S n0) O (CHead x0 (Bind b) t) (CHead e0 (Flat f) w)
                                                                    λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                                                  ex3_4
                                                                    F
                                                                    C
                                                                    C
                                                                    T
                                                                    λf:F.λe1:C.λ:C.λu:T.eq C e (CHead e1 (Flat f) u)
                                                                    λf:F.λ:C.λe2:C.λu:T.drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f) u)
                                                                    λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                                                  ex4_5
                                                                    F
                                                                    C
                                                                    C
                                                                    T
                                                                    T
                                                                    λf:F.λe1:C.λ:C.λu:T.λ:T.eq C e (CHead e1 (Flat f) u)
                                                                    λf:F.λ:C.λe2:C.λ:T.λw:T.drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f) w)
                                                                    λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                                                    λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2
                                                                   we proceed by induction on H11 to prove 
                                                                      or4
                                                                        drop (S n0) O (CHead x0 (Bind b) t) e
                                                                        ex3_4
                                                                          F
                                                                          C
                                                                          T
                                                                          T
                                                                          λf:F.λe0:C.λu:T.λ:T.eq C e (CHead e0 (Flat f) u)
                                                                          λf:F.λe0:C.λ:T.λw:T.drop (S n0) O (CHead x0 (Bind b) t) (CHead e0 (Flat f) w)
                                                                          λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                                                        ex3_4
                                                                          F
                                                                          C
                                                                          C
                                                                          T
                                                                          λf:F.λe1:C.λ:C.λu:T.eq C e (CHead e1 (Flat f) u)
                                                                          λf:F.λ:C.λe2:C.λu:T.drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f) u)
                                                                          λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                                                        ex4_5
                                                                          F
                                                                          C
                                                                          C
                                                                          T
                                                                          T
                                                                          λf:F.λe1:C.λ:C.λu:T.λ:T.eq C e (CHead e1 (Flat f) u)
                                                                          λf:F.λ:C.λe2:C.λ:T.λw:T.drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f) w)
                                                                          λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                                                          λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2
                                                                      case ex4_5_intro : x2:F x3:C x4:C x5:T x6:T H12:eq C e (CHead x3 (Flat x2) x5) H13:drop n0 O x0 (CHead x4 (Flat x2) x6) H14:subst0 O v x5 x6 H15:csubst0 O v x3 x4 ⇒
                                                                         the thesis becomes 
                                                                         or4
                                                                           drop (S n0) O (CHead x0 (Bind b) t) e
                                                                           ex3_4
                                                                             F
                                                                             C
                                                                             T
                                                                             T
                                                                             λf:F.λe0:C.λu:T.λ:T.eq C e (CHead e0 (Flat f) u)
                                                                             λf:F.λe0:C.λ:T.λw:T.drop (S n0) O (CHead x0 (Bind b) t) (CHead e0 (Flat f) w)
                                                                             λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                                                           ex3_4
                                                                             F
                                                                             C
                                                                             C
                                                                             T
                                                                             λf:F.λe1:C.λ:C.λu:T.eq C e (CHead e1 (Flat f) u)
                                                                             λf:F.λ:C.λe2:C.λu:T.drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f) u)
                                                                             λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                                                           ex4_5
                                                                             F
                                                                             C
                                                                             C
                                                                             T
                                                                             T
                                                                             λf:F.λe1:C.λ:C.λu:T.λ:T.eq C e (CHead e1 (Flat f) u)
                                                                             λf:F.λ:C.λe2:C.λ:T.λw:T.drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f) w)
                                                                             λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                                                             λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2
                                                                            (h1) 
                                                                               by (refl_equal . .)
eq C (CHead x3 (Flat x2) x5) (CHead x3 (Flat x2) x5)
                                                                            end of h1
                                                                            (h2) 
                                                                               consider H13
                                                                               we proved drop n0 O x0 (CHead x4 (Flat x2) x6)
                                                                               that is equivalent to drop (r (Bind b) n0) O x0 (CHead x4 (Flat x2) x6)
                                                                               by (drop_drop . . . . previous .)
drop (S n0) O (CHead x0 (Bind b) t) (CHead x4 (Flat x2) x6)
                                                                            end of h2
                                                                            by (ex4_5_intro . . . . . . . . . . . . . . h1 h2 H14 H15)
                                                                            we proved 
                                                                               ex4_5
                                                                                 F
                                                                                 C
                                                                                 C
                                                                                 T
                                                                                 T
                                                                                 λf:F.λe1:C.λ:C.λu:T.λ:T.eq C (CHead x3 (Flat x2) x5) (CHead e1 (Flat f) u)
                                                                                 λf:F.λ:C.λe2:C.λ:T.λw:T.drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f) w)
                                                                                 λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                                                                 λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2
                                                                            by (or4_intro3 . . . . previous)
                                                                            we proved 
                                                                               or4
                                                                                 drop (S n0) O (CHead x0 (Bind b) t) (CHead x3 (Flat x2) x5)
                                                                                 ex3_4
                                                                                   F
                                                                                   C
                                                                                   T
                                                                                   T
                                                                                   λf:F.λe0:C.λu:T.λ:T.eq C (CHead x3 (Flat x2) x5) (CHead e0 (Flat f) u)
                                                                                   λf:F.λe0:C.λ:T.λw:T.drop (S n0) O (CHead x0 (Bind b) t) (CHead e0 (Flat f) w)
                                                                                   λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                                                                 ex3_4
                                                                                   F
                                                                                   C
                                                                                   C
                                                                                   T
                                                                                   λf:F.λe1:C.λ:C.λu:T.eq C (CHead x3 (Flat x2) x5) (CHead e1 (Flat f) u)
                                                                                   λf:F.λ:C.λe2:C.λu:T.drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f) u)
                                                                                   λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                                                                 ex4_5
                                                                                   F
                                                                                   C
                                                                                   C
                                                                                   T
                                                                                   T
                                                                                   λf:F.λe1:C.λ:C.λu:T.λ:T.eq C (CHead x3 (Flat x2) x5) (CHead e1 (Flat f) u)
                                                                                   λf:F.λ:C.λe2:C.λ:T.λw:T.drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f) w)
                                                                                   λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                                                                   λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2
                                                                            by (eq_ind_r . . . previous . H12)

                                                                               or4
                                                                                 drop (S n0) O (CHead x0 (Bind b) t) e
                                                                                 ex3_4
                                                                                   F
                                                                                   C
                                                                                   T
                                                                                   T
                                                                                   λf:F.λe0:C.λu:T.λ:T.eq C e (CHead e0 (Flat f) u)
                                                                                   λf:F.λe0:C.λ:T.λw:T.drop (S n0) O (CHead x0 (Bind b) t) (CHead e0 (Flat f) w)
                                                                                   λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                                                                 ex3_4
                                                                                   F
                                                                                   C
                                                                                   C
                                                                                   T
                                                                                   λf:F.λe1:C.λ:C.λu:T.eq C e (CHead e1 (Flat f) u)
                                                                                   λf:F.λ:C.λe2:C.λu:T.drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f) u)
                                                                                   λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                                                                 ex4_5
                                                                                   F
                                                                                   C
                                                                                   C
                                                                                   T
                                                                                   T
                                                                                   λf:F.λe1:C.λ:C.λu:T.λ:T.eq C e (CHead e1 (Flat f) u)
                                                                                   λf:F.λ:C.λe2:C.λ:T.λw:T.drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f) w)
                                                                                   λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                                                                   λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2

                                                                      or4
                                                                        drop (S n0) O (CHead x0 (Bind b) t) e
                                                                        ex3_4
                                                                          F
                                                                          C
                                                                          T
                                                                          T
                                                                          λf:F.λe0:C.λu:T.λ:T.eq C e (CHead e0 (Flat f) u)
                                                                          λf:F.λe0:C.λ:T.λw:T.drop (S n0) O (CHead x0 (Bind b) t) (CHead e0 (Flat f) w)
                                                                          λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                                                        ex3_4
                                                                          F
                                                                          C
                                                                          C
                                                                          T
                                                                          λf:F.λe1:C.λ:C.λu:T.eq C e (CHead e1 (Flat f) u)
                                                                          λf:F.λ:C.λe2:C.λu:T.drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f) u)
                                                                          λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                                                        ex4_5
                                                                          F
                                                                          C
                                                                          C
                                                                          T
                                                                          T
                                                                          λf:F.λe1:C.λ:C.λu:T.λ:T.eq C e (CHead e1 (Flat f) u)
                                                                          λf:F.λ:C.λe2:C.λ:T.λw:T.drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f) w)
                                                                          λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                                                          λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2
                                                          we proved 
                                                             or4
                                                               drop (S n0) O (CHead x0 (Bind b) t) e
                                                               ex3_4
                                                                 F
                                                                 C
                                                                 T
                                                                 T
                                                                 λf:F.λe0:C.λu:T.λ:T.eq C e (CHead e0 (Flat f) u)
                                                                 λf:F.λe0:C.λ:T.λw:T.drop (S n0) O (CHead x0 (Bind b) t) (CHead e0 (Flat f) w)
                                                                 λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                                               ex3_4
                                                                 F
                                                                 C
                                                                 C
                                                                 T
                                                                 λf:F.λe1:C.λ:C.λu:T.eq C e (CHead e1 (Flat f) u)
                                                                 λf:F.λ:C.λe2:C.λu:T.drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f) u)
                                                                 λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                                               ex4_5
                                                                 F
                                                                 C
                                                                 C
                                                                 T
                                                                 T
                                                                 λf:F.λe1:C.λ:C.λu:T.λ:T.eq C e (CHead e1 (Flat f) u)
                                                                 λf:F.λ:C.λe2:C.λ:T.λw:T.drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f) w)
                                                                 λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                                                 λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2

                                                          ∀H6:drop (r (Bind b) n0) O c e
                                                            .∀H7:eq nat (S n0) (s (Bind b) x1)
                                                              .or4
                                                                drop (S n0) O (CHead x0 (Bind b) t) e
                                                                ex3_4
                                                                  F
                                                                  C
                                                                  T
                                                                  T
                                                                  λf:F.λe0:C.λu:T.λ:T.eq C e (CHead e0 (Flat f) u)
                                                                  λf:F.λe0:C.λ:T.λw:T.drop (S n0) O (CHead x0 (Bind b) t) (CHead e0 (Flat f) w)
                                                                  λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                                                ex3_4
                                                                  F
                                                                  C
                                                                  C
                                                                  T
                                                                  λf:F.λe1:C.λ:C.λu:T.eq C e (CHead e1 (Flat f) u)
                                                                  λf:F.λ:C.λe2:C.λu:T.drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f) u)
                                                                  λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                                                ex4_5
                                                                  F
                                                                  C
                                                                  C
                                                                  T
                                                                  T
                                                                  λf:F.λe1:C.λ:C.λu:T.λ:T.eq C e (CHead e1 (Flat f) u)
                                                                  λf:F.λ:C.λe2:C.λ:T.λw:T.drop (S n0) O (CHead x0 (Bind b) t) (CHead e2 (Flat f) w)
                                                                  λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                                                  λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2
                                                       assume f: F
                                                        suppose H6: drop (r (Flat f) n0) O c e
                                                        suppose H7: eq nat (S n0) (s (Flat f) x1)
                                                          (H8) 
                                                             by (f_equal . . . . . H7)
                                                             we proved eq nat (S n0) (s (Flat f) x1)
eq nat (λe0:nat.e0 (S n0)) (λe0:nat.e0 (s (Flat f) x1))
                                                          end of H8
                                                          (H9) 
                                                             consider H8
                                                             we proved eq nat (S n0) (s (Flat f) x1)
                                                             that is equivalent to eq nat (S n0) x1
                                                             by (eq_ind_r . . . H5 . previous)
csubst0 (S n0) v c x0
                                                          end of H9
                                                          (H10) 
                                                             consider H6
                                                             we proved drop (r (Flat f) n0) O c e
                                                             that is equivalent to drop (S n0) O c e
                                                             by (H . . H9 . previous)

                                                                or4
                                                                  drop (S n0) O x0 e
                                                                  ex3_4
                                                                    F
                                                                    C
                                                                    T
                                                                    T
                                                                    λf:F.λe0:C.λu:T.λ:T.eq C e (CHead e0 (Flat f) u)
                                                                    λf:F.λe0:C.λ:T.λw:T.drop (S n0) O x0 (CHead e0 (Flat f) w)
                                                                    λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                                                  ex3_4
                                                                    F
                                                                    C
                                                                    C
                                                                    T
                                                                    λf:F.λe1:C.λ:C.λu:T.eq C e (CHead e1 (Flat f) u)
                                                                    λf:F.λ:C.λe2:C.λu:T.drop (S n0) O x0 (CHead e2 (Flat f) u)
                                                                    λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                                                  ex4_5
                                                                    F
                                                                    C
                                                                    C
                                                                    T
                                                                    T
                                                                    λf:F.λe1:C.λ:C.λu:T.λ:T.eq C e (CHead e1 (Flat f) u)
                                                                    λf:F.λ:C.λe2:C.λ:T.λw:T.drop (S n0) O x0 (CHead e2 (Flat f) w)
                                                                    λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                                                    λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2
                                                          end of H10
                                                          we proceed by induction on H10 to prove 
                                                             or4
                                                               drop (S n0) O (CHead x0 (Flat f) t) e
                                                               ex3_4
                                                                 F
                                                                 C
                                                                 T
                                                                 T
                                                                 λf0:F.λe0:C.λu:T.λ:T.eq C e (CHead e0 (Flat f0) u)
                                                                 λf0:F.λe0:C.λ:T.λw:T.drop (S n0) O (CHead x0 (Flat f) t) (CHead e0 (Flat f0) w)
                                                                 λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                                               ex3_4
                                                                 F
                                                                 C
                                                                 C
                                                                 T
                                                                 λf0:F.λe1:C.λ:C.λu:T.eq C e (CHead e1 (Flat f0) u)
                                                                 λf0:F.λ:C.λe2:C.λu:T.drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) u)
                                                                 λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                                               ex4_5
                                                                 F
                                                                 C
                                                                 C
                                                                 T
                                                                 T
                                                                 λf0:F.λe1:C.λ:C.λu:T.λ:T.eq C e (CHead e1 (Flat f0) u)
                                                                 λf0:F.λ:C.λe2:C.λ:T.λw:T.drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) w)
                                                                 λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                                                 λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2
                                                             case or4_intro0 : H11:drop (S n0) O x0 e ⇒
                                                                the thesis becomes 
                                                                or4
                                                                  drop (S n0) O (CHead x0 (Flat f) t) e
                                                                  ex3_4
                                                                    F
                                                                    C
                                                                    T
                                                                    T
                                                                    λf0:F.λe0:C.λu:T.λ:T.eq C e (CHead e0 (Flat f0) u)
                                                                    λf0:F.λe0:C.λ:T.λw:T.drop (S n0) O (CHead x0 (Flat f) t) (CHead e0 (Flat f0) w)
                                                                    λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                                                  ex3_4
                                                                    F
                                                                    C
                                                                    C
                                                                    T
                                                                    λf0:F.λe1:C.λ:C.λu:T.eq C e (CHead e1 (Flat f0) u)
                                                                    λf0:F.λ:C.λe2:C.λu:T.drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) u)
                                                                    λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                                                  ex4_5
                                                                    F
                                                                    C
                                                                    C
                                                                    T
                                                                    T
                                                                    λf0:F.λe1:C.λ:C.λu:T.λ:T.eq C e (CHead e1 (Flat f0) u)
                                                                    λf0:F.λ:C.λe2:C.λ:T.λw:T.drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) w)
                                                                    λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                                                    λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2
                                                                   consider H11
                                                                   we proved drop (S n0) O x0 e
                                                                   that is equivalent to drop (r (Flat f) n0) O x0 e
                                                                   by (drop_drop . . . . previous .)
                                                                   we proved drop (S n0) O (CHead x0 (Flat f) t) e
                                                                   by (or4_intro0 . . . . previous)

                                                                      or4
                                                                        drop (S n0) O (CHead x0 (Flat f) t) e
                                                                        ex3_4
                                                                          F
                                                                          C
                                                                          T
                                                                          T
                                                                          λf0:F.λe0:C.λu:T.λ:T.eq C e (CHead e0 (Flat f0) u)
                                                                          λf0:F.λe0:C.λ:T.λw:T.drop (S n0) O (CHead x0 (Flat f) t) (CHead e0 (Flat f0) w)
                                                                          λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                                                        ex3_4
                                                                          F
                                                                          C
                                                                          C
                                                                          T
                                                                          λf0:F.λe1:C.λ:C.λu:T.eq C e (CHead e1 (Flat f0) u)
                                                                          λf0:F.λ:C.λe2:C.λu:T.drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) u)
                                                                          λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                                                        ex4_5
                                                                          F
                                                                          C
                                                                          C
                                                                          T
                                                                          T
                                                                          λf0:F.λe1:C.λ:C.λu:T.λ:T.eq C e (CHead e1 (Flat f0) u)
                                                                          λf0:F.λ:C.λe2:C.λ:T.λw:T.drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) w)
                                                                          λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                                                          λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2
                                                             case or4_intro1 : H11:ex3_4 F C T T λf0:F.λe0:C.λu:T.λ:T.eq C e (CHead e0 (Flat f0) u) λf0:F.λe0:C.λ:T.λw:T.drop (S n0) O x0 (CHead e0 (Flat f0) w) λ:F.λ:C.λu:T.λw:T.subst0 O v u w ⇒
                                                                the thesis becomes 
                                                                or4
                                                                  drop (S n0) O (CHead x0 (Flat f) t) e
                                                                  ex3_4
                                                                    F
                                                                    C
                                                                    T
                                                                    T
                                                                    λf0:F.λe0:C.λu:T.λ:T.eq C e (CHead e0 (Flat f0) u)
                                                                    λf0:F.λe0:C.λ:T.λw:T.drop (S n0) O (CHead x0 (Flat f) t) (CHead e0 (Flat f0) w)
                                                                    λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                                                  ex3_4
                                                                    F
                                                                    C
                                                                    C
                                                                    T
                                                                    λf0:F.λe1:C.λ:C.λu:T.eq C e (CHead e1 (Flat f0) u)
                                                                    λf0:F.λ:C.λe2:C.λu:T.drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) u)
                                                                    λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                                                  ex4_5
                                                                    F
                                                                    C
                                                                    C
                                                                    T
                                                                    T
                                                                    λf0:F.λe1:C.λ:C.λu:T.λ:T.eq C e (CHead e1 (Flat f0) u)
                                                                    λf0:F.λ:C.λe2:C.λ:T.λw:T.drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) w)
                                                                    λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                                                    λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2
                                                                   we proceed by induction on H11 to prove 
                                                                      or4
                                                                        drop (S n0) O (CHead x0 (Flat f) t) e
                                                                        ex3_4
                                                                          F
                                                                          C
                                                                          T
                                                                          T
                                                                          λf0:F.λe0:C.λu:T.λ:T.eq C e (CHead e0 (Flat f0) u)
                                                                          λf0:F.λe0:C.λ:T.λw:T.drop (S n0) O (CHead x0 (Flat f) t) (CHead e0 (Flat f0) w)
                                                                          λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                                                        ex3_4
                                                                          F
                                                                          C
                                                                          C
                                                                          T
                                                                          λf0:F.λe1:C.λ:C.λu:T.eq C e (CHead e1 (Flat f0) u)
                                                                          λf0:F.λ:C.λe2:C.λu:T.drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) u)
                                                                          λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                                                        ex4_5
                                                                          F
                                                                          C
                                                                          C
                                                                          T
                                                                          T
                                                                          λf0:F.λe1:C.λ:C.λu:T.λ:T.eq C e (CHead e1 (Flat f0) u)
                                                                          λf0:F.λ:C.λe2:C.λ:T.λw:T.drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) w)
                                                                          λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                                                          λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2
                                                                      case ex3_4_intro : x2:F x3:C x4:T x5:T H12:eq C e (CHead x3 (Flat x2) x4) H13:drop (S n0) O x0 (CHead x3 (Flat x2) x5) H14:subst0 O v x4 x5 ⇒
                                                                         the thesis becomes 
                                                                         or4
                                                                           drop (S n0) O (CHead x0 (Flat f) t) e
                                                                           ex3_4
                                                                             F
                                                                             C
                                                                             T
                                                                             T
                                                                             λf0:F.λe0:C.λu:T.λ:T.eq C e (CHead e0 (Flat f0) u)
                                                                             λf0:F.λe0:C.λ:T.λw:T.drop (S n0) O (CHead x0 (Flat f) t) (CHead e0 (Flat f0) w)
                                                                             λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                                                           ex3_4
                                                                             F
                                                                             C
                                                                             C
                                                                             T
                                                                             λf0:F.λe1:C.λ:C.λu:T.eq C e (CHead e1 (Flat f0) u)
                                                                             λf0:F.λ:C.λe2:C.λu:T.drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) u)
                                                                             λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                                                           ex4_5
                                                                             F
                                                                             C
                                                                             C
                                                                             T
                                                                             T
                                                                             λf0:F.λe1:C.λ:C.λu:T.λ:T.eq C e (CHead e1 (Flat f0) u)
                                                                             λf0:F.λ:C.λe2:C.λ:T.λw:T.drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) w)
                                                                             λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                                                             λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2
                                                                            (h1) 
                                                                               by (refl_equal . .)
eq C (CHead x3 (Flat x2) x4) (CHead x3 (Flat x2) x4)
                                                                            end of h1
                                                                            (h2) 
                                                                               consider H13
                                                                               we proved drop (S n0) O x0 (CHead x3 (Flat x2) x5)
                                                                               that is equivalent to drop (r (Flat f) n0) O x0 (CHead x3 (Flat x2) x5)
                                                                               by (drop_drop . . . . previous .)
drop (S n0) O (CHead x0 (Flat f) t) (CHead x3 (Flat x2) x5)
                                                                            end of h2
                                                                            by (ex3_4_intro . . . . . . . . . . . h1 h2 H14)
                                                                            we proved 
                                                                               ex3_4
                                                                                 F
                                                                                 C
                                                                                 T
                                                                                 T
                                                                                 λf0:F.λe0:C.λu:T.λ:T.eq C (CHead x3 (Flat x2) x4) (CHead e0 (Flat f0) u)
                                                                                 λf0:F.λe0:C.λ:T.λw:T.drop (S n0) O (CHead x0 (Flat f) t) (CHead e0 (Flat f0) w)
                                                                                 λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                                                            by (or4_intro1 . . . . previous)
                                                                            we proved 
                                                                               or4
                                                                                 drop (S n0) O (CHead x0 (Flat f) t) (CHead x3 (Flat x2) x4)
                                                                                 ex3_4
                                                                                   F
                                                                                   C
                                                                                   T
                                                                                   T
                                                                                   λf0:F.λe0:C.λu:T.λ:T.eq C (CHead x3 (Flat x2) x4) (CHead e0 (Flat f0) u)
                                                                                   λf0:F.λe0:C.λ:T.λw:T.drop (S n0) O (CHead x0 (Flat f) t) (CHead e0 (Flat f0) w)
                                                                                   λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                                                                 ex3_4
                                                                                   F
                                                                                   C
                                                                                   C
                                                                                   T
                                                                                   λf0:F.λe1:C.λ:C.λu:T.eq C (CHead x3 (Flat x2) x4) (CHead e1 (Flat f0) u)
                                                                                   λf0:F.λ:C.λe2:C.λu:T.drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) u)
                                                                                   λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                                                                 ex4_5
                                                                                   F
                                                                                   C
                                                                                   C
                                                                                   T
                                                                                   T
                                                                                   λf0:F.λe1:C.λ:C.λu:T.λ:T.eq C (CHead x3 (Flat x2) x4) (CHead e1 (Flat f0) u)
                                                                                   λf0:F.λ:C.λe2:C.λ:T.λw:T.drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) w)
                                                                                   λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                                                                   λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2
                                                                            by (eq_ind_r . . . previous . H12)

                                                                               or4
                                                                                 drop (S n0) O (CHead x0 (Flat f) t) e
                                                                                 ex3_4
                                                                                   F
                                                                                   C
                                                                                   T
                                                                                   T
                                                                                   λf0:F.λe0:C.λu:T.λ:T.eq C e (CHead e0 (Flat f0) u)
                                                                                   λf0:F.λe0:C.λ:T.λw:T.drop (S n0) O (CHead x0 (Flat f) t) (CHead e0 (Flat f0) w)
                                                                                   λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                                                                 ex3_4
                                                                                   F
                                                                                   C
                                                                                   C
                                                                                   T
                                                                                   λf0:F.λe1:C.λ:C.λu:T.eq C e (CHead e1 (Flat f0) u)
                                                                                   λf0:F.λ:C.λe2:C.λu:T.drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) u)
                                                                                   λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                                                                 ex4_5
                                                                                   F
                                                                                   C
                                                                                   C
                                                                                   T
                                                                                   T
                                                                                   λf0:F.λe1:C.λ:C.λu:T.λ:T.eq C e (CHead e1 (Flat f0) u)
                                                                                   λf0:F.λ:C.λe2:C.λ:T.λw:T.drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) w)
                                                                                   λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                                                                   λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2

                                                                      or4
                                                                        drop (S n0) O (CHead x0 (Flat f) t) e
                                                                        ex3_4
                                                                          F
                                                                          C
                                                                          T
                                                                          T
                                                                          λf0:F.λe0:C.λu:T.λ:T.eq C e (CHead e0 (Flat f0) u)
                                                                          λf0:F.λe0:C.λ:T.λw:T.drop (S n0) O (CHead x0 (Flat f) t) (CHead e0 (Flat f0) w)
                                                                          λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                                                        ex3_4
                                                                          F
                                                                          C
                                                                          C
                                                                          T
                                                                          λf0:F.λe1:C.λ:C.λu:T.eq C e (CHead e1 (Flat f0) u)
                                                                          λf0:F.λ:C.λe2:C.λu:T.drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) u)
                                                                          λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                                                        ex4_5
                                                                          F
                                                                          C
                                                                          C
                                                                          T
                                                                          T
                                                                          λf0:F.λe1:C.λ:C.λu:T.λ:T.eq C e (CHead e1 (Flat f0) u)
                                                                          λf0:F.λ:C.λe2:C.λ:T.λw:T.drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) w)
                                                                          λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                                                          λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2
                                                             case or4_intro2 : H11:ex3_4 F C C T λf0:F.λe1:C.λ:C.λu:T.eq C e (CHead e1 (Flat f0) u) λf0:F.λ:C.λe2:C.λu:T.drop (S n0) O x0 (CHead e2 (Flat f0) u) λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2 ⇒
                                                                the thesis becomes 
                                                                or4
                                                                  drop (S n0) O (CHead x0 (Flat f) t) e
                                                                  ex3_4
                                                                    F
                                                                    C
                                                                    T
                                                                    T
                                                                    λf0:F.λe0:C.λu:T.λ:T.eq C e (CHead e0 (Flat f0) u)
                                                                    λf0:F.λe0:C.λ:T.λw:T.drop (S n0) O (CHead x0 (Flat f) t) (CHead e0 (Flat f0) w)
                                                                    λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                                                  ex3_4
                                                                    F
                                                                    C
                                                                    C
                                                                    T
                                                                    λf0:F.λe1:C.λ:C.λu:T.eq C e (CHead e1 (Flat f0) u)
                                                                    λf0:F.λ:C.λe2:C.λu:T.drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) u)
                                                                    λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                                                  ex4_5
                                                                    F
                                                                    C
                                                                    C
                                                                    T
                                                                    T
                                                                    λf0:F.λe1:C.λ:C.λu:T.λ:T.eq C e (CHead e1 (Flat f0) u)
                                                                    λf0:F.λ:C.λe2:C.λ:T.λw:T.drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) w)
                                                                    λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                                                    λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2
                                                                   we proceed by induction on H11 to prove 
                                                                      or4
                                                                        drop (S n0) O (CHead x0 (Flat f) t) e
                                                                        ex3_4
                                                                          F
                                                                          C
                                                                          T
                                                                          T
                                                                          λf0:F.λe0:C.λu:T.λ:T.eq C e (CHead e0 (Flat f0) u)
                                                                          λf0:F.λe0:C.λ:T.λw:T.drop (S n0) O (CHead x0 (Flat f) t) (CHead e0 (Flat f0) w)
                                                                          λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                                                        ex3_4
                                                                          F
                                                                          C
                                                                          C
                                                                          T
                                                                          λf0:F.λe1:C.λ:C.λu:T.eq C e (CHead e1 (Flat f0) u)
                                                                          λf0:F.λ:C.λe2:C.λu:T.drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) u)
                                                                          λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                                                        ex4_5
                                                                          F
                                                                          C
                                                                          C
                                                                          T
                                                                          T
                                                                          λf0:F.λe1:C.λ:C.λu:T.λ:T.eq C e (CHead e1 (Flat f0) u)
                                                                          λf0:F.λ:C.λe2:C.λ:T.λw:T.drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) w)
                                                                          λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                                                          λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2
                                                                      case ex3_4_intro : x2:F x3:C x4:C x5:T H12:eq C e (CHead x3 (Flat x2) x5) H13:drop (S n0) O x0 (CHead x4 (Flat x2) x5) H14:csubst0 O v x3 x4 ⇒
                                                                         the thesis becomes 
                                                                         or4
                                                                           drop (S n0) O (CHead x0 (Flat f) t) e
                                                                           ex3_4
                                                                             F
                                                                             C
                                                                             T
                                                                             T
                                                                             λf0:F.λe0:C.λu:T.λ:T.eq C e (CHead e0 (Flat f0) u)
                                                                             λf0:F.λe0:C.λ:T.λw:T.drop (S n0) O (CHead x0 (Flat f) t) (CHead e0 (Flat f0) w)
                                                                             λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                                                           ex3_4
                                                                             F
                                                                             C
                                                                             C
                                                                             T
                                                                             λf0:F.λe1:C.λ:C.λu:T.eq C e (CHead e1 (Flat f0) u)
                                                                             λf0:F.λ:C.λe2:C.λu:T.drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) u)
                                                                             λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                                                           ex4_5
                                                                             F
                                                                             C
                                                                             C
                                                                             T
                                                                             T
                                                                             λf0:F.λe1:C.λ:C.λu:T.λ:T.eq C e (CHead e1 (Flat f0) u)
                                                                             λf0:F.λ:C.λe2:C.λ:T.λw:T.drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) w)
                                                                             λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                                                             λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2
                                                                            (h1) 
                                                                               by (refl_equal . .)
eq C (CHead x3 (Flat x2) x5) (CHead x3 (Flat x2) x5)
                                                                            end of h1
                                                                            (h2) 
                                                                               consider H13
                                                                               we proved drop (S n0) O x0 (CHead x4 (Flat x2) x5)
                                                                               that is equivalent to drop (r (Flat f) n0) O x0 (CHead x4 (Flat x2) x5)
                                                                               by (drop_drop . . . . previous .)
drop (S n0) O (CHead x0 (Flat f) t) (CHead x4 (Flat x2) x5)
                                                                            end of h2
                                                                            by (ex3_4_intro . . . . . . . . . . . h1 h2 H14)
                                                                            we proved 
                                                                               ex3_4
                                                                                 F
                                                                                 C
                                                                                 C
                                                                                 T
                                                                                 λf0:F.λe1:C.λ:C.λu:T.eq C (CHead x3 (Flat x2) x5) (CHead e1 (Flat f0) u)
                                                                                 λf0:F.λ:C.λe2:C.λu:T.drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) u)
                                                                                 λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                                                            by (or4_intro2 . . . . previous)
                                                                            we proved 
                                                                               or4
                                                                                 drop (S n0) O (CHead x0 (Flat f) t) (CHead x3 (Flat x2) x5)
                                                                                 ex3_4
                                                                                   F
                                                                                   C
                                                                                   T
                                                                                   T
                                                                                   λf0:F.λe0:C.λu:T.λ:T.eq C (CHead x3 (Flat x2) x5) (CHead e0 (Flat f0) u)
                                                                                   λf0:F.λe0:C.λ:T.λw:T.drop (S n0) O (CHead x0 (Flat f) t) (CHead e0 (Flat f0) w)
                                                                                   λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                                                                 ex3_4
                                                                                   F
                                                                                   C
                                                                                   C
                                                                                   T
                                                                                   λf0:F.λe1:C.λ:C.λu:T.eq C (CHead x3 (Flat x2) x5) (CHead e1 (Flat f0) u)
                                                                                   λf0:F.λ:C.λe2:C.λu:T.drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) u)
                                                                                   λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                                                                 ex4_5
                                                                                   F
                                                                                   C
                                                                                   C
                                                                                   T
                                                                                   T
                                                                                   λf0:F.λe1:C.λ:C.λu:T.λ:T.eq C (CHead x3 (Flat x2) x5) (CHead e1 (Flat f0) u)
                                                                                   λf0:F.λ:C.λe2:C.λ:T.λw:T.drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) w)
                                                                                   λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                                                                   λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2
                                                                            by (eq_ind_r . . . previous . H12)

                                                                               or4
                                                                                 drop (S n0) O (CHead x0 (Flat f) t) e
                                                                                 ex3_4
                                                                                   F
                                                                                   C
                                                                                   T
                                                                                   T
                                                                                   λf0:F.λe0:C.λu:T.λ:T.eq C e (CHead e0 (Flat f0) u)
                                                                                   λf0:F.λe0:C.λ:T.λw:T.drop (S n0) O (CHead x0 (Flat f) t) (CHead e0 (Flat f0) w)
                                                                                   λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                                                                 ex3_4
                                                                                   F
                                                                                   C
                                                                                   C
                                                                                   T
                                                                                   λf0:F.λe1:C.λ:C.λu:T.eq C e (CHead e1 (Flat f0) u)
                                                                                   λf0:F.λ:C.λe2:C.λu:T.drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) u)
                                                                                   λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                                                                 ex4_5
                                                                                   F
                                                                                   C
                                                                                   C
                                                                                   T
                                                                                   T
                                                                                   λf0:F.λe1:C.λ:C.λu:T.λ:T.eq C e (CHead e1 (Flat f0) u)
                                                                                   λf0:F.λ:C.λe2:C.λ:T.λw:T.drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) w)
                                                                                   λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                                                                   λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2

                                                                      or4
                                                                        drop (S n0) O (CHead x0 (Flat f) t) e
                                                                        ex3_4
                                                                          F
                                                                          C
                                                                          T
                                                                          T
                                                                          λf0:F.λe0:C.λu:T.λ:T.eq C e (CHead e0 (Flat f0) u)
                                                                          λf0:F.λe0:C.λ:T.λw:T.drop (S n0) O (CHead x0 (Flat f) t) (CHead e0 (Flat f0) w)
                                                                          λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                                                        ex3_4
                                                                          F
                                                                          C
                                                                          C
                                                                          T
                                                                          λf0:F.λe1:C.λ:C.λu:T.eq C e (CHead e1 (Flat f0) u)
                                                                          λf0:F.λ:C.λe2:C.λu:T.drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) u)
                                                                          λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                                                        ex4_5
                                                                          F
                                                                          C
                                                                          C
                                                                          T
                                                                          T
                                                                          λf0:F.λe1:C.λ:C.λu:T.λ:T.eq C e (CHead e1 (Flat f0) u)
                                                                          λf0:F.λ:C.λe2:C.λ:T.λw:T.drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) w)
                                                                          λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                                                          λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2
                                                             case or4_intro3 : H11:ex4_5 F C C T T λf0:F.λe1:C.λ:C.λu:T.λ:T.eq C e (CHead e1 (Flat f0) u) λf0:F.λ:C.λe2:C.λ:T.λw:T.drop (S n0) O x0 (CHead e2 (Flat f0) w) λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2 ⇒
                                                                the thesis becomes 
                                                                or4
                                                                  drop (S n0) O (CHead x0 (Flat f) t) e
                                                                  ex3_4
                                                                    F
                                                                    C
                                                                    T
                                                                    T
                                                                    λf0:F.λe0:C.λu:T.λ:T.eq C e (CHead e0 (Flat f0) u)
                                                                    λf0:F.λe0:C.λ:T.λw:T.drop (S n0) O (CHead x0 (Flat f) t) (CHead e0 (Flat f0) w)
                                                                    λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                                                  ex3_4
                                                                    F
                                                                    C
                                                                    C
                                                                    T
                                                                    λf0:F.λe1:C.λ:C.λu:T.eq C e (CHead e1 (Flat f0) u)
                                                                    λf0:F.λ:C.λe2:C.λu:T.drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) u)
                                                                    λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                                                  ex4_5
                                                                    F
                                                                    C
                                                                    C
                                                                    T
                                                                    T
                                                                    λf0:F.λe1:C.λ:C.λu:T.λ:T.eq C e (CHead e1 (Flat f0) u)
                                                                    λf0:F.λ:C.λe2:C.λ:T.λw:T.drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) w)
                                                                    λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                                                    λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2
                                                                   we proceed by induction on H11 to prove 
                                                                      or4
                                                                        drop (S n0) O (CHead x0 (Flat f) t) e
                                                                        ex3_4
                                                                          F
                                                                          C
                                                                          T
                                                                          T
                                                                          λf0:F.λe0:C.λu:T.λ:T.eq C e (CHead e0 (Flat f0) u)
                                                                          λf0:F.λe0:C.λ:T.λw:T.drop (S n0) O (CHead x0 (Flat f) t) (CHead e0 (Flat f0) w)
                                                                          λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                                                        ex3_4
                                                                          F
                                                                          C
                                                                          C
                                                                          T
                                                                          λf0:F.λe1:C.λ:C.λu:T.eq C e (CHead e1 (Flat f0) u)
                                                                          λf0:F.λ:C.λe2:C.λu:T.drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) u)
                                                                          λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                                                        ex4_5
                                                                          F
                                                                          C
                                                                          C
                                                                          T
                                                                          T
                                                                          λf0:F.λe1:C.λ:C.λu:T.λ:T.eq C e (CHead e1 (Flat f0) u)
                                                                          λf0:F.λ:C.λe2:C.λ:T.λw:T.drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) w)
                                                                          λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                                                          λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2
                                                                      case ex4_5_intro : x2:F x3:C x4:C x5:T x6:T H12:eq C e (CHead x3 (Flat x2) x5) H13:drop (S n0) O x0 (CHead x4 (Flat x2) x6) H14:subst0 O v x5 x6 H15:csubst0 O v x3 x4 ⇒
                                                                         the thesis becomes 
                                                                         or4
                                                                           drop (S n0) O (CHead x0 (Flat f) t) e
                                                                           ex3_4
                                                                             F
                                                                             C
                                                                             T
                                                                             T
                                                                             λf0:F.λe0:C.λu:T.λ:T.eq C e (CHead e0 (Flat f0) u)
                                                                             λf0:F.λe0:C.λ:T.λw:T.drop (S n0) O (CHead x0 (Flat f) t) (CHead e0 (Flat f0) w)
                                                                             λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                                                           ex3_4
                                                                             F
                                                                             C
                                                                             C
                                                                             T
                                                                             λf0:F.λe1:C.λ:C.λu:T.eq C e (CHead e1 (Flat f0) u)
                                                                             λf0:F.λ:C.λe2:C.λu:T.drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) u)
                                                                             λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                                                           ex4_5
                                                                             F
                                                                             C
                                                                             C
                                                                             T
                                                                             T
                                                                             λf0:F.λe1:C.λ:C.λu:T.λ:T.eq C e (CHead e1 (Flat f0) u)
                                                                             λf0:F.λ:C.λe2:C.λ:T.λw:T.drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) w)
                                                                             λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                                                             λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2
                                                                            (h1) 
                                                                               by (refl_equal . .)
eq C (CHead x3 (Flat x2) x5) (CHead x3 (Flat x2) x5)
                                                                            end of h1
                                                                            (h2) 
                                                                               consider H13
                                                                               we proved drop (S n0) O x0 (CHead x4 (Flat x2) x6)
                                                                               that is equivalent to drop (r (Flat f) n0) O x0 (CHead x4 (Flat x2) x6)
                                                                               by (drop_drop . . . . previous .)
drop (S n0) O (CHead x0 (Flat f) t) (CHead x4 (Flat x2) x6)
                                                                            end of h2
                                                                            by (ex4_5_intro . . . . . . . . . . . . . . h1 h2 H14 H15)
                                                                            we proved 
                                                                               ex4_5
                                                                                 F
                                                                                 C
                                                                                 C
                                                                                 T
                                                                                 T
                                                                                 λf0:F.λe1:C.λ:C.λu:T.λ:T.eq C (CHead x3 (Flat x2) x5) (CHead e1 (Flat f0) u)
                                                                                 λf0:F.λ:C.λe2:C.λ:T.λw:T.drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) w)
                                                                                 λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                                                                 λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2
                                                                            by (or4_intro3 . . . . previous)
                                                                            we proved 
                                                                               or4
                                                                                 drop (S n0) O (CHead x0 (Flat f) t) (CHead x3 (Flat x2) x5)
                                                                                 ex3_4
                                                                                   F
                                                                                   C
                                                                                   T
                                                                                   T
                                                                                   λf0:F.λe0:C.λu:T.λ:T.eq C (CHead x3 (Flat x2) x5) (CHead e0 (Flat f0) u)
                                                                                   λf0:F.λe0:C.λ:T.λw:T.drop (S n0) O (CHead x0 (Flat f) t) (CHead e0 (Flat f0) w)
                                                                                   λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                                                                 ex3_4
                                                                                   F
                                                                                   C
                                                                                   C
                                                                                   T
                                                                                   λf0:F.λe1:C.λ:C.λu:T.eq C (CHead x3 (Flat x2) x5) (CHead e1 (Flat f0) u)
                                                                                   λf0:F.λ:C.λe2:C.λu:T.drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) u)
                                                                                   λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                                                                 ex4_5
                                                                                   F
                                                                                   C
                                                                                   C
                                                                                   T
                                                                                   T
                                                                                   λf0:F.λe1:C.λ:C.λu:T.λ:T.eq C (CHead x3 (Flat x2) x5) (CHead e1 (Flat f0) u)
                                                                                   λf0:F.λ:C.λe2:C.λ:T.λw:T.drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) w)
                                                                                   λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                                                                   λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2
                                                                            by (eq_ind_r . . . previous . H12)

                                                                               or4
                                                                                 drop (S n0) O (CHead x0 (Flat f) t) e
                                                                                 ex3_4
                                                                                   F
                                                                                   C
                                                                                   T
                                                                                   T
                                                                                   λf0:F.λe0:C.λu:T.λ:T.eq C e (CHead e0 (Flat f0) u)
                                                                                   λf0:F.λe0:C.λ:T.λw:T.drop (S n0) O (CHead x0 (Flat f) t) (CHead e0 (Flat f0) w)
                                                                                   λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                                                                 ex3_4
                                                                                   F
                                                                                   C
                                                                                   C
                                                                                   T
                                                                                   λf0:F.λe1:C.λ:C.λu:T.eq C e (CHead e1 (Flat f0) u)
                                                                                   λf0:F.λ:C.λe2:C.λu:T.drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) u)
                                                                                   λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                                                                 ex4_5
                                                                                   F
                                                                                   C
                                                                                   C
                                                                                   T
                                                                                   T
                                                                                   λf0:F.λe1:C.λ:C.λu:T.λ:T.eq C e (CHead e1 (Flat f0) u)
                                                                                   λf0:F.λ:C.λe2:C.λ:T.λw:T.drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) w)
                                                                                   λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                                                                   λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2

                                                                      or4
                                                                        drop (S n0) O (CHead x0 (Flat f) t) e
                                                                        ex3_4
                                                                          F
                                                                          C
                                                                          T
                                                                          T
                                                                          λf0:F.λe0:C.λu:T.λ:T.eq C e (CHead e0 (Flat f0) u)
                                                                          λf0:F.λe0:C.λ:T.λw:T.drop (S n0) O (CHead x0 (Flat f) t) (CHead e0 (Flat f0) w)
                                                                          λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                                                        ex3_4
                                                                          F
                                                                          C
                                                                          C
                                                                          T
                                                                          λf0:F.λe1:C.λ:C.λu:T.eq C e (CHead e1 (Flat f0) u)
                                                                          λf0:F.λ:C.λe2:C.λu:T.drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) u)
                                                                          λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                                                        ex4_5
                                                                          F
                                                                          C
                                                                          C
                                                                          T
                                                                          T
                                                                          λf0:F.λe1:C.λ:C.λu:T.λ:T.eq C e (CHead e1 (Flat f0) u)
                                                                          λf0:F.λ:C.λe2:C.λ:T.λw:T.drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) w)
                                                                          λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                                                          λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2
                                                          we proved 
                                                             or4
                                                               drop (S n0) O (CHead x0 (Flat f) t) e
                                                               ex3_4
                                                                 F
                                                                 C
                                                                 T
                                                                 T
                                                                 λf0:F.λe0:C.λu:T.λ:T.eq C e (CHead e0 (Flat f0) u)
                                                                 λf0:F.λe0:C.λ:T.λw:T.drop (S n0) O (CHead x0 (Flat f) t) (CHead e0 (Flat f0) w)
                                                                 λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                                               ex3_4
                                                                 F
                                                                 C
                                                                 C
                                                                 T
                                                                 λf0:F.λe1:C.λ:C.λu:T.eq C e (CHead e1 (Flat f0) u)
                                                                 λf0:F.λ:C.λe2:C.λu:T.drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) u)
                                                                 λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                                               ex4_5
                                                                 F
                                                                 C
                                                                 C
                                                                 T
                                                                 T
                                                                 λf0:F.λe1:C.λ:C.λu:T.λ:T.eq C e (CHead e1 (Flat f0) u)
                                                                 λf0:F.λ:C.λe2:C.λ:T.λw:T.drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) w)
                                                                 λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                                                 λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2

                                                          ∀H6:drop (r (Flat f) n0) O c e
                                                            .∀H7:eq nat (S n0) (s (Flat f) x1)
                                                              .or4
                                                                drop (S n0) O (CHead x0 (Flat f) t) e
                                                                ex3_4
                                                                  F
                                                                  C
                                                                  T
                                                                  T
                                                                  λf0:F.λe0:C.λu:T.λ:T.eq C e (CHead e0 (Flat f0) u)
                                                                  λf0:F.λe0:C.λ:T.λw:T.drop (S n0) O (CHead x0 (Flat f) t) (CHead e0 (Flat f0) w)
                                                                  λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                                                ex3_4
                                                                  F
                                                                  C
                                                                  C
                                                                  T
                                                                  λf0:F.λe1:C.λ:C.λu:T.eq C e (CHead e1 (Flat f0) u)
                                                                  λf0:F.λ:C.λe2:C.λu:T.drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) u)
                                                                  λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                                                ex4_5
                                                                  F
                                                                  C
                                                                  C
                                                                  T
                                                                  T
                                                                  λf0:F.λe1:C.λ:C.λu:T.λ:T.eq C e (CHead e1 (Flat f0) u)
                                                                  λf0:F.λ:C.λe2:C.λ:T.λw:T.drop (S n0) O (CHead x0 (Flat f) t) (CHead e2 (Flat f0) w)
                                                                  λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                                                  λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2
                                                    by (previous . previous H3)
                                                    we proved 
                                                       or4
                                                         drop (S n0) O (CHead x0 k t) e
                                                         ex3_4
                                                           F
                                                           C
                                                           T
                                                           T
                                                           λf:F.λe0:C.λu:T.λ:T.eq C e (CHead e0 (Flat f) u)
                                                           λf:F.λe0:C.λ:T.λw:T.drop (S n0) O (CHead x0 k t) (CHead e0 (Flat f) w)
                                                           λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                                         ex3_4
                                                           F
                                                           C
                                                           C
                                                           T
                                                           λf:F.λe1:C.λ:C.λu:T.eq C e (CHead e1 (Flat f) u)
                                                           λf:F.λ:C.λe2:C.λu:T.drop (S n0) O (CHead x0 k t) (CHead e2 (Flat f) u)
                                                           λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                                         ex4_5
                                                           F
                                                           C
                                                           C
                                                           T
                                                           T
                                                           λf:F.λe1:C.λ:C.λu:T.λ:T.eq C e (CHead e1 (Flat f) u)
                                                           λf:F.λ:C.λe2:C.λ:T.λw:T.drop (S n0) O (CHead x0 k t) (CHead e2 (Flat f) w)
                                                           λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                                           λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2
                                                    by (eq_ind_r . . . previous . H4)

                                                       or4
                                                         drop (S n0) O c2 e
                                                         ex3_4
                                                           F
                                                           C
                                                           T
                                                           T
                                                           λf:F.λe0:C.λu:T.λ:T.eq C e (CHead e0 (Flat f) u)
                                                           λf:F.λe0:C.λ:T.λw:T.drop (S n0) O c2 (CHead e0 (Flat f) w)
                                                           λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                                         ex3_4
                                                           F
                                                           C
                                                           C
                                                           T
                                                           λf:F.λe1:C.λ:C.λu:T.eq C e (CHead e1 (Flat f) u)
                                                           λf:F.λ:C.λe2:C.λu:T.drop (S n0) O c2 (CHead e2 (Flat f) u)
                                                           λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                                         ex4_5
                                                           F
                                                           C
                                                           C
                                                           T
                                                           T
                                                           λf:F.λe1:C.λ:C.λu:T.λ:T.eq C e (CHead e1 (Flat f) u)
                                                           λf:F.λ:C.λe2:C.λ:T.λw:T.drop (S n0) O c2 (CHead e2 (Flat f) w)
                                                           λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                                           λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2

                                              or4
                                                drop (S n0) O c2 e
                                                ex3_4
                                                  F
                                                  C
                                                  T
                                                  T
                                                  λf:F.λe0:C.λu:T.λ:T.eq C e (CHead e0 (Flat f) u)
                                                  λf:F.λe0:C.λ:T.λw:T.drop (S n0) O c2 (CHead e0 (Flat f) w)
                                                  λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                                ex3_4
                                                  F
                                                  C
                                                  C
                                                  T
                                                  λf:F.λe1:C.λ:C.λu:T.eq C e (CHead e1 (Flat f) u)
                                                  λf:F.λ:C.λe2:C.λu:T.drop (S n0) O c2 (CHead e2 (Flat f) u)
                                                  λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                                ex4_5
                                                  F
                                                  C
                                                  C
                                                  T
                                                  T
                                                  λf:F.λe1:C.λ:C.λu:T.λ:T.eq C e (CHead e1 (Flat f) u)
                                                  λf:F.λ:C.λe2:C.λ:T.λw:T.drop (S n0) O c2 (CHead e2 (Flat f) w)
                                                  λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                                  λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2
                                     case or3_intro2 : H2:ex4_3 T C nat λ:T.λ:C.λj:nat.eq nat (S n0) (s k j) λu2:T.λc3:C.λ:nat.eq C c2 (CHead c3 k u2) λu2:T.λ:C.λj:nat.subst0 j v t u2 λ:T.λc3:C.λj:nat.csubst0 j v c c3 ⇒
                                        the thesis becomes 
                                        or4
                                          drop (S n0) O c2 e
                                          ex3_4
                                            F
                                            C
                                            T
                                            T
                                            λf:F.λe0:C.λu:T.λ:T.eq C e (CHead e0 (Flat f) u)
                                            λf:F.λe0:C.λ:T.λw:T.drop (S n0) O c2 (CHead e0 (Flat f) w)
                                            λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                          ex3_4
                                            F
                                            C
                                            C
                                            T
                                            λf:F.λe1:C.λ:C.λu:T.eq C e (CHead e1 (Flat f) u)
                                            λf:F.λ:C.λe2:C.λu:T.drop (S n0) O c2 (CHead e2 (Flat f) u)
                                            λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                          ex4_5
                                            F
                                            C
                                            C
                                            T
                                            T
                                            λf:F.λe1:C.λ:C.λu:T.λ:T.eq C e (CHead e1 (Flat f) u)
                                            λf:F.λ:C.λe2:C.λ:T.λw:T.drop (S n0) O c2 (CHead e2 (Flat f) w)
                                            λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                            λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2
                                           we proceed by induction on H2 to prove 
                                              or4
                                                drop (S n0) O c2 e
                                                ex3_4
                                                  F
                                                  C
                                                  T
                                                  T
                                                  λf:F.λe0:C.λu:T.λ:T.eq C e (CHead e0 (Flat f) u)
                                                  λf:F.λe0:C.λ:T.λw:T.drop (S n0) O c2 (CHead e0 (Flat f) w)
                                                  λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                                ex3_4
                                                  F
                                                  C
                                                  C
                                                  T
                                                  λf:F.λe1:C.λ:C.λu:T.eq C e (CHead e1 (Flat f) u)
                                                  λf:F.λ:C.λe2:C.λu:T.drop (S n0) O c2 (CHead e2 (Flat f) u)
                                                  λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                                ex4_5
                                                  F
                                                  C
                                                  C
                                                  T
                                                  T
                                                  λf:F.λe1:C.λ:C.λu:T.λ:T.eq C e (CHead e1 (Flat f) u)
                                                  λf:F.λ:C.λe2:C.λ:T.λw:T.drop (S n0) O c2 (CHead e2 (Flat f) w)
                                                  λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                                  λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2
                                              case ex4_3_intro : x0:T x1:C x2:nat H3:eq nat (S n0) (s k x2) H4:eq C c2 (CHead x1 k x0) H5:subst0 x2 v t x0 H6:csubst0 x2 v c x1 ⇒
                                                 the thesis becomes 
                                                 or4
                                                   drop (S n0) O c2 e
                                                   ex3_4
                                                     F
                                                     C
                                                     T
                                                     T
                                                     λf:F.λe0:C.λu:T.λ:T.eq C e (CHead e0 (Flat f) u)
                                                     λf:F.λe0:C.λ:T.λw:T.drop (S n0) O c2 (CHead e0 (Flat f) w)
                                                     λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                                   ex3_4
                                                     F
                                                     C
                                                     C
                                                     T
                                                     λf:F.λe1:C.λ:C.λu:T.eq C e (CHead e1 (Flat f) u)
                                                     λf:F.λ:C.λe2:C.λu:T.drop (S n0) O c2 (CHead e2 (Flat f) u)
                                                     λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                                   ex4_5
                                                     F
                                                     C
                                                     C
                                                     T
                                                     T
                                                     λf:F.λe1:C.λ:C.λu:T.λ:T.eq C e (CHead e1 (Flat f) u)
                                                     λf:F.λ:C.λe2:C.λ:T.λw:T.drop (S n0) O c2 (CHead e2 (Flat f) w)
                                                     λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                                     λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2
                                                    by (drop_gen_drop . . . . . H1)
                                                    we proved drop (r k n0) O c e
                                                       assume b: B
                                                        suppose H7: drop (r (Bind b) n0) O c e
                                                        suppose H8: eq nat (S n0) (s (Bind b) x2)
                                                          (H9) 
                                                             consider H8
                                                             we proved eq nat (S n0) (s (Bind b) x2)
                                                             that is equivalent to eq nat (S n0) (S x2)
                                                             by (f_equal . . . . . previous)
                                                             we proved eq nat <λ:nat.nat> CASE S n0 OF O⇒n0 | S n1⇒n1 <λ:nat.nat> CASE S x2 OF O⇒n0 | S n1⇒n1

                                                                eq
                                                                  nat
                                                                  λe0:nat.<λ:nat.nat> CASE e0 OF O⇒n0 | S n1⇒n1 (S n0)
                                                                  λe0:nat.<λ:nat.nat> CASE e0 OF O⇒n0 | S n1⇒n1 (S x2)
                                                          end of H9
                                                          (H10) 
                                                             consider H9
                                                             we proved eq nat <λ:nat.nat> CASE S n0 OF O⇒n0 | S n1⇒n1 <λ:nat.nat> CASE S x2 OF O⇒n0 | S n1⇒n1
                                                             that is equivalent to eq nat n0 x2
                                                             by (eq_ind_r . . . H6 . previous)
csubst0 n0 v c x1
                                                          end of H10
                                                          (H12) 
                                                             consider H7
                                                             we proved drop (r (Bind b) n0) O c e
                                                             that is equivalent to drop n0 O c e
                                                             by (IHn . . . H10 . previous)

                                                                or4
                                                                  drop n0 O x1 e
                                                                  ex3_4
                                                                    F
                                                                    C
                                                                    T
                                                                    T
                                                                    λf:F.λe0:C.λu:T.λ:T.eq C e (CHead e0 (Flat f) u)
                                                                    λf:F.λe0:C.λ:T.λw:T.drop n0 O x1 (CHead e0 (Flat f) w)
                                                                    λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                                                  ex3_4
                                                                    F
                                                                    C
                                                                    C
                                                                    T
                                                                    λf:F.λe1:C.λ:C.λu:T.eq C e (CHead e1 (Flat f) u)
                                                                    λf:F.λ:C.λe2:C.λu:T.drop n0 O x1 (CHead e2 (Flat f) u)
                                                                    λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                                                  ex4_5
                                                                    F
                                                                    C
                                                                    C
                                                                    T
                                                                    T
                                                                    λf:F.λe1:C.λ:C.λu:T.λ:T.eq C e (CHead e1 (Flat f) u)
                                                                    λf:F.λ:C.λe2:C.λ:T.λw:T.drop n0 O x1 (CHead e2 (Flat f) w)
                                                                    λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                                                    λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2
                                                          end of H12
                                                          we proceed by induction on H12 to prove 
                                                             or4
                                                               drop (S n0) O (CHead x1 (Bind b) x0) e
                                                               ex3_4
                                                                 F
                                                                 C
                                                                 T
                                                                 T
                                                                 λf:F.λe0:C.λu:T.λ:T.eq C e (CHead e0 (Flat f) u)
                                                                 λf:F.λe0:C.λ:T.λw:T.drop (S n0) O (CHead x1 (Bind b) x0) (CHead e0 (Flat f) w)
                                                                 λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                                               ex3_4
                                                                 F
                                                                 C
                                                                 C
                                                                 T
                                                                 λf:F.λe1:C.λ:C.λu:T.eq C e (CHead e1 (Flat f) u)
                                                                 λf:F.λ:C.λe2:C.λu:T.drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) u)
                                                                 λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                                               ex4_5
                                                                 F
                                                                 C
                                                                 C
                                                                 T
                                                                 T
                                                                 λf:F.λe1:C.λ:C.λu:T.λ:T.eq C e (CHead e1 (Flat f) u)
                                                                 λf:F.λ:C.λe2:C.λ:T.λw:T.drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) w)
                                                                 λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                                                 λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2
                                                             case or4_intro0 : H13:drop n0 O x1 e ⇒
                                                                the thesis becomes 
                                                                or4
                                                                  drop (S n0) O (CHead x1 (Bind b) x0) e
                                                                  ex3_4
                                                                    F
                                                                    C
                                                                    T
                                                                    T
                                                                    λf:F.λe0:C.λu:T.λ:T.eq C e (CHead e0 (Flat f) u)
                                                                    λf:F.λe0:C.λ:T.λw:T.drop (S n0) O (CHead x1 (Bind b) x0) (CHead e0 (Flat f) w)
                                                                    λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                                                  ex3_4
                                                                    F
                                                                    C
                                                                    C
                                                                    T
                                                                    λf:F.λe1:C.λ:C.λu:T.eq C e (CHead e1 (Flat f) u)
                                                                    λf:F.λ:C.λe2:C.λu:T.drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) u)
                                                                    λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                                                  ex4_5
                                                                    F
                                                                    C
                                                                    C
                                                                    T
                                                                    T
                                                                    λf:F.λe1:C.λ:C.λu:T.λ:T.eq C e (CHead e1 (Flat f) u)
                                                                    λf:F.λ:C.λe2:C.λ:T.λw:T.drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) w)
                                                                    λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                                                    λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2
                                                                   consider H13
                                                                   we proved drop n0 O x1 e
                                                                   that is equivalent to drop (r (Bind b) n0) O x1 e
                                                                   by (drop_drop . . . . previous .)
                                                                   we proved drop (S n0) O (CHead x1 (Bind b) x0) e
                                                                   by (or4_intro0 . . . . previous)

                                                                      or4
                                                                        drop (S n0) O (CHead x1 (Bind b) x0) e
                                                                        ex3_4
                                                                          F
                                                                          C
                                                                          T
                                                                          T
                                                                          λf:F.λe0:C.λu:T.λ:T.eq C e (CHead e0 (Flat f) u)
                                                                          λf:F.λe0:C.λ:T.λw:T.drop (S n0) O (CHead x1 (Bind b) x0) (CHead e0 (Flat f) w)
                                                                          λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                                                        ex3_4
                                                                          F
                                                                          C
                                                                          C
                                                                          T
                                                                          λf:F.λe1:C.λ:C.λu:T.eq C e (CHead e1 (Flat f) u)
                                                                          λf:F.λ:C.λe2:C.λu:T.drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) u)
                                                                          λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                                                        ex4_5
                                                                          F
                                                                          C
                                                                          C
                                                                          T
                                                                          T
                                                                          λf:F.λe1:C.λ:C.λu:T.λ:T.eq C e (CHead e1 (Flat f) u)
                                                                          λf:F.λ:C.λe2:C.λ:T.λw:T.drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) w)
                                                                          λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                                                          λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2
                                                             case or4_intro1 : H13:ex3_4 F C T T λf:F.λe0:C.λu:T.λ:T.eq C e (CHead e0 (Flat f) u) λf:F.λe0:C.λ:T.λw:T.drop n0 O x1 (CHead e0 (Flat f) w) λ:F.λ:C.λu:T.λw:T.subst0 O v u w ⇒
                                                                the thesis becomes 
                                                                or4
                                                                  drop (S n0) O (CHead x1 (Bind b) x0) e
                                                                  ex3_4
                                                                    F
                                                                    C
                                                                    T
                                                                    T
                                                                    λf:F.λe0:C.λu:T.λ:T.eq C e (CHead e0 (Flat f) u)
                                                                    λf:F.λe0:C.λ:T.λw:T.drop (S n0) O (CHead x1 (Bind b) x0) (CHead e0 (Flat f) w)
                                                                    λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                                                  ex3_4
                                                                    F
                                                                    C
                                                                    C
                                                                    T
                                                                    λf:F.λe1:C.λ:C.λu:T.eq C e (CHead e1 (Flat f) u)
                                                                    λf:F.λ:C.λe2:C.λu:T.drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) u)
                                                                    λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                                                  ex4_5
                                                                    F
                                                                    C
                                                                    C
                                                                    T
                                                                    T
                                                                    λf:F.λe1:C.λ:C.λu:T.λ:T.eq C e (CHead e1 (Flat f) u)
                                                                    λf:F.λ:C.λe2:C.λ:T.λw:T.drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) w)
                                                                    λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                                                    λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2
                                                                   we proceed by induction on H13 to prove 
                                                                      or4
                                                                        drop (S n0) O (CHead x1 (Bind b) x0) e
                                                                        ex3_4
                                                                          F
                                                                          C
                                                                          T
                                                                          T
                                                                          λf:F.λe0:C.λu:T.λ:T.eq C e (CHead e0 (Flat f) u)
                                                                          λf:F.λe0:C.λ:T.λw:T.drop (S n0) O (CHead x1 (Bind b) x0) (CHead e0 (Flat f) w)
                                                                          λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                                                        ex3_4
                                                                          F
                                                                          C
                                                                          C
                                                                          T
                                                                          λf:F.λe1:C.λ:C.λu:T.eq C e (CHead e1 (Flat f) u)
                                                                          λf:F.λ:C.λe2:C.λu:T.drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) u)
                                                                          λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                                                        ex4_5
                                                                          F
                                                                          C
                                                                          C
                                                                          T
                                                                          T
                                                                          λf:F.λe1:C.λ:C.λu:T.λ:T.eq C e (CHead e1 (Flat f) u)
                                                                          λf:F.λ:C.λe2:C.λ:T.λw:T.drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) w)
                                                                          λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                                                          λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2
                                                                      case ex3_4_intro : x3:F x4:C x5:T x6:T H14:eq C e (CHead x4 (Flat x3) x5) H15:drop n0 O x1 (CHead x4 (Flat x3) x6) H16:subst0 O v x5 x6 ⇒
                                                                         the thesis becomes 
                                                                         or4
                                                                           drop (S n0) O (CHead x1 (Bind b) x0) e
                                                                           ex3_4
                                                                             F
                                                                             C
                                                                             T
                                                                             T
                                                                             λf:F.λe0:C.λu:T.λ:T.eq C e (CHead e0 (Flat f) u)
                                                                             λf:F.λe0:C.λ:T.λw:T.drop (S n0) O (CHead x1 (Bind b) x0) (CHead e0 (Flat f) w)
                                                                             λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                                                           ex3_4
                                                                             F
                                                                             C
                                                                             C
                                                                             T
                                                                             λf:F.λe1:C.λ:C.λu:T.eq C e (CHead e1 (Flat f) u)
                                                                             λf:F.λ:C.λe2:C.λu:T.drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) u)
                                                                             λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                                                           ex4_5
                                                                             F
                                                                             C
                                                                             C
                                                                             T
                                                                             T
                                                                             λf:F.λe1:C.λ:C.λu:T.λ:T.eq C e (CHead e1 (Flat f) u)
                                                                             λf:F.λ:C.λe2:C.λ:T.λw:T.drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) w)
                                                                             λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                                                             λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2
                                                                            (h1) 
                                                                               by (refl_equal . .)
eq C (CHead x4 (Flat x3) x5) (CHead x4 (Flat x3) x5)
                                                                            end of h1
                                                                            (h2) 
                                                                               consider H15
                                                                               we proved drop n0 O x1 (CHead x4 (Flat x3) x6)
                                                                               that is equivalent to drop (r (Bind b) n0) O x1 (CHead x4 (Flat x3) x6)
                                                                               by (drop_drop . . . . previous .)
drop (S n0) O (CHead x1 (Bind b) x0) (CHead x4 (Flat x3) x6)
                                                                            end of h2
                                                                            by (ex3_4_intro . . . . . . . . . . . h1 h2 H16)
                                                                            we proved 
                                                                               ex3_4
                                                                                 F
                                                                                 C
                                                                                 T
                                                                                 T
                                                                                 λf:F.λe0:C.λu:T.λ:T.eq C (CHead x4 (Flat x3) x5) (CHead e0 (Flat f) u)
                                                                                 λf:F.λe0:C.λ:T.λw:T.drop (S n0) O (CHead x1 (Bind b) x0) (CHead e0 (Flat f) w)
                                                                                 λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                                                            by (or4_intro1 . . . . previous)
                                                                            we proved 
                                                                               or4
                                                                                 drop (S n0) O (CHead x1 (Bind b) x0) (CHead x4 (Flat x3) x5)
                                                                                 ex3_4
                                                                                   F
                                                                                   C
                                                                                   T
                                                                                   T
                                                                                   λf:F.λe0:C.λu:T.λ:T.eq C (CHead x4 (Flat x3) x5) (CHead e0 (Flat f) u)
                                                                                   λf:F.λe0:C.λ:T.λw:T.drop (S n0) O (CHead x1 (Bind b) x0) (CHead e0 (Flat f) w)
                                                                                   λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                                                                 ex3_4
                                                                                   F
                                                                                   C
                                                                                   C
                                                                                   T
                                                                                   λf:F.λe1:C.λ:C.λu:T.eq C (CHead x4 (Flat x3) x5) (CHead e1 (Flat f) u)
                                                                                   λf:F.λ:C.λe2:C.λu:T.drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) u)
                                                                                   λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                                                                 ex4_5
                                                                                   F
                                                                                   C
                                                                                   C
                                                                                   T
                                                                                   T
                                                                                   λf:F.λe1:C.λ:C.λu:T.λ:T.eq C (CHead x4 (Flat x3) x5) (CHead e1 (Flat f) u)
                                                                                   λf:F.λ:C.λe2:C.λ:T.λw:T.drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) w)
                                                                                   λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                                                                   λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2
                                                                            by (eq_ind_r . . . previous . H14)

                                                                               or4
                                                                                 drop (S n0) O (CHead x1 (Bind b) x0) e
                                                                                 ex3_4
                                                                                   F
                                                                                   C
                                                                                   T
                                                                                   T
                                                                                   λf:F.λe0:C.λu:T.λ:T.eq C e (CHead e0 (Flat f) u)
                                                                                   λf:F.λe0:C.λ:T.λw:T.drop (S n0) O (CHead x1 (Bind b) x0) (CHead e0 (Flat f) w)
                                                                                   λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                                                                 ex3_4
                                                                                   F
                                                                                   C
                                                                                   C
                                                                                   T
                                                                                   λf:F.λe1:C.λ:C.λu:T.eq C e (CHead e1 (Flat f) u)
                                                                                   λf:F.λ:C.λe2:C.λu:T.drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) u)
                                                                                   λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                                                                 ex4_5
                                                                                   F
                                                                                   C
                                                                                   C
                                                                                   T
                                                                                   T
                                                                                   λf:F.λe1:C.λ:C.λu:T.λ:T.eq C e (CHead e1 (Flat f) u)
                                                                                   λf:F.λ:C.λe2:C.λ:T.λw:T.drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) w)
                                                                                   λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                                                                   λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2

                                                                      or4
                                                                        drop (S n0) O (CHead x1 (Bind b) x0) e
                                                                        ex3_4
                                                                          F
                                                                          C
                                                                          T
                                                                          T
                                                                          λf:F.λe0:C.λu:T.λ:T.eq C e (CHead e0 (Flat f) u)
                                                                          λf:F.λe0:C.λ:T.λw:T.drop (S n0) O (CHead x1 (Bind b) x0) (CHead e0 (Flat f) w)
                                                                          λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                                                        ex3_4
                                                                          F
                                                                          C
                                                                          C
                                                                          T
                                                                          λf:F.λe1:C.λ:C.λu:T.eq C e (CHead e1 (Flat f) u)
                                                                          λf:F.λ:C.λe2:C.λu:T.drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) u)
                                                                          λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                                                        ex4_5
                                                                          F
                                                                          C
                                                                          C
                                                                          T
                                                                          T
                                                                          λf:F.λe1:C.λ:C.λu:T.λ:T.eq C e (CHead e1 (Flat f) u)
                                                                          λf:F.λ:C.λe2:C.λ:T.λw:T.drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) w)
                                                                          λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                                                          λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2
                                                             case or4_intro2 : H13:ex3_4 F C C T λf:F.λe1:C.λ:C.λu:T.eq C e (CHead e1 (Flat f) u) λf:F.λ:C.λe2:C.λu:T.drop n0 O x1 (CHead e2 (Flat f) u) λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2 ⇒
                                                                the thesis becomes 
                                                                or4
                                                                  drop (S n0) O (CHead x1 (Bind b) x0) e
                                                                  ex3_4
                                                                    F
                                                                    C
                                                                    T
                                                                    T
                                                                    λf:F.λe0:C.λu:T.λ:T.eq C e (CHead e0 (Flat f) u)
                                                                    λf:F.λe0:C.λ:T.λw:T.drop (S n0) O (CHead x1 (Bind b) x0) (CHead e0 (Flat f) w)
                                                                    λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                                                  ex3_4
                                                                    F
                                                                    C
                                                                    C
                                                                    T
                                                                    λf:F.λe1:C.λ:C.λu:T.eq C e (CHead e1 (Flat f) u)
                                                                    λf:F.λ:C.λe2:C.λu:T.drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) u)
                                                                    λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                                                  ex4_5
                                                                    F
                                                                    C
                                                                    C
                                                                    T
                                                                    T
                                                                    λf:F.λe1:C.λ:C.λu:T.λ:T.eq C e (CHead e1 (Flat f) u)
                                                                    λf:F.λ:C.λe2:C.λ:T.λw:T.drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) w)
                                                                    λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                                                    λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2
                                                                   we proceed by induction on H13 to prove 
                                                                      or4
                                                                        drop (S n0) O (CHead x1 (Bind b) x0) e
                                                                        ex3_4
                                                                          F
                                                                          C
                                                                          T
                                                                          T
                                                                          λf:F.λe0:C.λu:T.λ:T.eq C e (CHead e0 (Flat f) u)
                                                                          λf:F.λe0:C.λ:T.λw:T.drop (S n0) O (CHead x1 (Bind b) x0) (CHead e0 (Flat f) w)
                                                                          λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                                                        ex3_4
                                                                          F
                                                                          C
                                                                          C
                                                                          T
                                                                          λf:F.λe1:C.λ:C.λu:T.eq C e (CHead e1 (Flat f) u)
                                                                          λf:F.λ:C.λe2:C.λu:T.drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) u)
                                                                          λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                                                        ex4_5
                                                                          F
                                                                          C
                                                                          C
                                                                          T
                                                                          T
                                                                          λf:F.λe1:C.λ:C.λu:T.λ:T.eq C e (CHead e1 (Flat f) u)
                                                                          λf:F.λ:C.λe2:C.λ:T.λw:T.drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) w)
                                                                          λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                                                          λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2
                                                                      case ex3_4_intro : x3:F x4:C x5:C x6:T H14:eq C e (CHead x4 (Flat x3) x6) H15:drop n0 O x1 (CHead x5 (Flat x3) x6) H16:csubst0 O v x4 x5 ⇒
                                                                         the thesis becomes 
                                                                         or4
                                                                           drop (S n0) O (CHead x1 (Bind b) x0) e
                                                                           ex3_4
                                                                             F
                                                                             C
                                                                             T
                                                                             T
                                                                             λf:F.λe0:C.λu:T.λ:T.eq C e (CHead e0 (Flat f) u)
                                                                             λf:F.λe0:C.λ:T.λw:T.drop (S n0) O (CHead x1 (Bind b) x0) (CHead e0 (Flat f) w)
                                                                             λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                                                           ex3_4
                                                                             F
                                                                             C
                                                                             C
                                                                             T
                                                                             λf:F.λe1:C.λ:C.λu:T.eq C e (CHead e1 (Flat f) u)
                                                                             λf:F.λ:C.λe2:C.λu:T.drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) u)
                                                                             λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                                                           ex4_5
                                                                             F
                                                                             C
                                                                             C
                                                                             T
                                                                             T
                                                                             λf:F.λe1:C.λ:C.λu:T.λ:T.eq C e (CHead e1 (Flat f) u)
                                                                             λf:F.λ:C.λe2:C.λ:T.λw:T.drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) w)
                                                                             λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                                                             λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2
                                                                            (h1) 
                                                                               by (refl_equal . .)
eq C (CHead x4 (Flat x3) x6) (CHead x4 (Flat x3) x6)
                                                                            end of h1
                                                                            (h2) 
                                                                               consider H15
                                                                               we proved drop n0 O x1 (CHead x5 (Flat x3) x6)
                                                                               that is equivalent to drop (r (Bind b) n0) O x1 (CHead x5 (Flat x3) x6)
                                                                               by (drop_drop . . . . previous .)
drop (S n0) O (CHead x1 (Bind b) x0) (CHead x5 (Flat x3) x6)
                                                                            end of h2
                                                                            by (ex3_4_intro . . . . . . . . . . . h1 h2 H16)
                                                                            we proved 
                                                                               ex3_4
                                                                                 F
                                                                                 C
                                                                                 C
                                                                                 T
                                                                                 λf:F.λe1:C.λ:C.λu:T.eq C (CHead x4 (Flat x3) x6) (CHead e1 (Flat f) u)
                                                                                 λf:F.λ:C.λe2:C.λu:T.drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) u)
                                                                                 λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                                                            by (or4_intro2 . . . . previous)
                                                                            we proved 
                                                                               or4
                                                                                 drop (S n0) O (CHead x1 (Bind b) x0) (CHead x4 (Flat x3) x6)
                                                                                 ex3_4
                                                                                   F
                                                                                   C
                                                                                   T
                                                                                   T
                                                                                   λf:F.λe0:C.λu:T.λ:T.eq C (CHead x4 (Flat x3) x6) (CHead e0 (Flat f) u)
                                                                                   λf:F.λe0:C.λ:T.λw:T.drop (S n0) O (CHead x1 (Bind b) x0) (CHead e0 (Flat f) w)
                                                                                   λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                                                                 ex3_4
                                                                                   F
                                                                                   C
                                                                                   C
                                                                                   T
                                                                                   λf:F.λe1:C.λ:C.λu:T.eq C (CHead x4 (Flat x3) x6) (CHead e1 (Flat f) u)
                                                                                   λf:F.λ:C.λe2:C.λu:T.drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) u)
                                                                                   λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                                                                 ex4_5
                                                                                   F
                                                                                   C
                                                                                   C
                                                                                   T
                                                                                   T
                                                                                   λf:F.λe1:C.λ:C.λu:T.λ:T.eq C (CHead x4 (Flat x3) x6) (CHead e1 (Flat f) u)
                                                                                   λf:F.λ:C.λe2:C.λ:T.λw:T.drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) w)
                                                                                   λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                                                                   λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2
                                                                            by (eq_ind_r . . . previous . H14)

                                                                               or4
                                                                                 drop (S n0) O (CHead x1 (Bind b) x0) e
                                                                                 ex3_4
                                                                                   F
                                                                                   C
                                                                                   T
                                                                                   T
                                                                                   λf:F.λe0:C.λu:T.λ:T.eq C e (CHead e0 (Flat f) u)
                                                                                   λf:F.λe0:C.λ:T.λw:T.drop (S n0) O (CHead x1 (Bind b) x0) (CHead e0 (Flat f) w)
                                                                                   λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                                                                 ex3_4
                                                                                   F
                                                                                   C
                                                                                   C
                                                                                   T
                                                                                   λf:F.λe1:C.λ:C.λu:T.eq C e (CHead e1 (Flat f) u)
                                                                                   λf:F.λ:C.λe2:C.λu:T.drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) u)
                                                                                   λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                                                                 ex4_5
                                                                                   F
                                                                                   C
                                                                                   C
                                                                                   T
                                                                                   T
                                                                                   λf:F.λe1:C.λ:C.λu:T.λ:T.eq C e (CHead e1 (Flat f) u)
                                                                                   λf:F.λ:C.λe2:C.λ:T.λw:T.drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) w)
                                                                                   λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                                                                   λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2

                                                                      or4
                                                                        drop (S n0) O (CHead x1 (Bind b) x0) e
                                                                        ex3_4
                                                                          F
                                                                          C
                                                                          T
                                                                          T
                                                                          λf:F.λe0:C.λu:T.λ:T.eq C e (CHead e0 (Flat f) u)
                                                                          λf:F.λe0:C.λ:T.λw:T.drop (S n0) O (CHead x1 (Bind b) x0) (CHead e0 (Flat f) w)
                                                                          λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                                                        ex3_4
                                                                          F
                                                                          C
                                                                          C
                                                                          T
                                                                          λf:F.λe1:C.λ:C.λu:T.eq C e (CHead e1 (Flat f) u)
                                                                          λf:F.λ:C.λe2:C.λu:T.drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) u)
                                                                          λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                                                        ex4_5
                                                                          F
                                                                          C
                                                                          C
                                                                          T
                                                                          T
                                                                          λf:F.λe1:C.λ:C.λu:T.λ:T.eq C e (CHead e1 (Flat f) u)
                                                                          λf:F.λ:C.λe2:C.λ:T.λw:T.drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) w)
                                                                          λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                                                          λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2
                                                             case or4_intro3 : H13:ex4_5 F C C T T λf:F.λe1:C.λ:C.λu:T.λ:T.eq C e (CHead e1 (Flat f) u) λf:F.λ:C.λe2:C.λ:T.λw:T.drop n0 O x1 (CHead e2 (Flat f) w) λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2 ⇒
                                                                the thesis becomes 
                                                                or4
                                                                  drop (S n0) O (CHead x1 (Bind b) x0) e
                                                                  ex3_4
                                                                    F
                                                                    C
                                                                    T
                                                                    T
                                                                    λf:F.λe0:C.λu:T.λ:T.eq C e (CHead e0 (Flat f) u)
                                                                    λf:F.λe0:C.λ:T.λw:T.drop (S n0) O (CHead x1 (Bind b) x0) (CHead e0 (Flat f) w)
                                                                    λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                                                  ex3_4
                                                                    F
                                                                    C
                                                                    C
                                                                    T
                                                                    λf:F.λe1:C.λ:C.λu:T.eq C e (CHead e1 (Flat f) u)
                                                                    λf:F.λ:C.λe2:C.λu:T.drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) u)
                                                                    λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                                                  ex4_5
                                                                    F
                                                                    C
                                                                    C
                                                                    T
                                                                    T
                                                                    λf:F.λe1:C.λ:C.λu:T.λ:T.eq C e (CHead e1 (Flat f) u)
                                                                    λf:F.λ:C.λe2:C.λ:T.λw:T.drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) w)
                                                                    λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                                                    λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2
                                                                   we proceed by induction on H13 to prove 
                                                                      or4
                                                                        drop (S n0) O (CHead x1 (Bind b) x0) e
                                                                        ex3_4
                                                                          F
                                                                          C
                                                                          T
                                                                          T
                                                                          λf:F.λe0:C.λu:T.λ:T.eq C e (CHead e0 (Flat f) u)
                                                                          λf:F.λe0:C.λ:T.λw:T.drop (S n0) O (CHead x1 (Bind b) x0) (CHead e0 (Flat f) w)
                                                                          λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                                                        ex3_4
                                                                          F
                                                                          C
                                                                          C
                                                                          T
                                                                          λf:F.λe1:C.λ:C.λu:T.eq C e (CHead e1 (Flat f) u)
                                                                          λf:F.λ:C.λe2:C.λu:T.drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) u)
                                                                          λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                                                        ex4_5
                                                                          F
                                                                          C
                                                                          C
                                                                          T
                                                                          T
                                                                          λf:F.λe1:C.λ:C.λu:T.λ:T.eq C e (CHead e1 (Flat f) u)
                                                                          λf:F.λ:C.λe2:C.λ:T.λw:T.drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) w)
                                                                          λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                                                          λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2
                                                                      case ex4_5_intro : x3:F x4:C x5:C x6:T x7:T H14:eq C e (CHead x4 (Flat x3) x6) H15:drop n0 O x1 (CHead x5 (Flat x3) x7) H16:subst0 O v x6 x7 H17:csubst0 O v x4 x5 ⇒
                                                                         the thesis becomes 
                                                                         or4
                                                                           drop (S n0) O (CHead x1 (Bind b) x0) e
                                                                           ex3_4
                                                                             F
                                                                             C
                                                                             T
                                                                             T
                                                                             λf:F.λe0:C.λu:T.λ:T.eq C e (CHead e0 (Flat f) u)
                                                                             λf:F.λe0:C.λ:T.λw:T.drop (S n0) O (CHead x1 (Bind b) x0) (CHead e0 (Flat f) w)
                                                                             λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                                                           ex3_4
                                                                             F
                                                                             C
                                                                             C
                                                                             T
                                                                             λf:F.λe1:C.λ:C.λu:T.eq C e (CHead e1 (Flat f) u)
                                                                             λf:F.λ:C.λe2:C.λu:T.drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) u)
                                                                             λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                                                           ex4_5
                                                                             F
                                                                             C
                                                                             C
                                                                             T
                                                                             T
                                                                             λf:F.λe1:C.λ:C.λu:T.λ:T.eq C e (CHead e1 (Flat f) u)
                                                                             λf:F.λ:C.λe2:C.λ:T.λw:T.drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) w)
                                                                             λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                                                             λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2
                                                                            (h1) 
                                                                               by (refl_equal . .)
eq C (CHead x4 (Flat x3) x6) (CHead x4 (Flat x3) x6)
                                                                            end of h1
                                                                            (h2) 
                                                                               consider H15
                                                                               we proved drop n0 O x1 (CHead x5 (Flat x3) x7)
                                                                               that is equivalent to drop (r (Bind b) n0) O x1 (CHead x5 (Flat x3) x7)
                                                                               by (drop_drop . . . . previous .)
drop (S n0) O (CHead x1 (Bind b) x0) (CHead x5 (Flat x3) x7)
                                                                            end of h2
                                                                            by (ex4_5_intro . . . . . . . . . . . . . . h1 h2 H16 H17)
                                                                            we proved 
                                                                               ex4_5
                                                                                 F
                                                                                 C
                                                                                 C
                                                                                 T
                                                                                 T
                                                                                 λf:F.λe1:C.λ:C.λu:T.λ:T.eq C (CHead x4 (Flat x3) x6) (CHead e1 (Flat f) u)
                                                                                 λf:F.λ:C.λe2:C.λ:T.λw:T.drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) w)
                                                                                 λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                                                                 λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2
                                                                            by (or4_intro3 . . . . previous)
                                                                            we proved 
                                                                               or4
                                                                                 drop (S n0) O (CHead x1 (Bind b) x0) (CHead x4 (Flat x3) x6)
                                                                                 ex3_4
                                                                                   F
                                                                                   C
                                                                                   T
                                                                                   T
                                                                                   λf:F.λe0:C.λu:T.λ:T.eq C (CHead x4 (Flat x3) x6) (CHead e0 (Flat f) u)
                                                                                   λf:F.λe0:C.λ:T.λw:T.drop (S n0) O (CHead x1 (Bind b) x0) (CHead e0 (Flat f) w)
                                                                                   λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                                                                 ex3_4
                                                                                   F
                                                                                   C
                                                                                   C
                                                                                   T
                                                                                   λf:F.λe1:C.λ:C.λu:T.eq C (CHead x4 (Flat x3) x6) (CHead e1 (Flat f) u)
                                                                                   λf:F.λ:C.λe2:C.λu:T.drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) u)
                                                                                   λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                                                                 ex4_5
                                                                                   F
                                                                                   C
                                                                                   C
                                                                                   T
                                                                                   T
                                                                                   λf:F.λe1:C.λ:C.λu:T.λ:T.eq C (CHead x4 (Flat x3) x6) (CHead e1 (Flat f) u)
                                                                                   λf:F.λ:C.λe2:C.λ:T.λw:T.drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) w)
                                                                                   λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                                                                   λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2
                                                                            by (eq_ind_r . . . previous . H14)

                                                                               or4
                                                                                 drop (S n0) O (CHead x1 (Bind b) x0) e
                                                                                 ex3_4
                                                                                   F
                                                                                   C
                                                                                   T
                                                                                   T
                                                                                   λf:F.λe0:C.λu:T.λ:T.eq C e (CHead e0 (Flat f) u)
                                                                                   λf:F.λe0:C.λ:T.λw:T.drop (S n0) O (CHead x1 (Bind b) x0) (CHead e0 (Flat f) w)
                                                                                   λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                                                                 ex3_4
                                                                                   F
                                                                                   C
                                                                                   C
                                                                                   T
                                                                                   λf:F.λe1:C.λ:C.λu:T.eq C e (CHead e1 (Flat f) u)
                                                                                   λf:F.λ:C.λe2:C.λu:T.drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) u)
                                                                                   λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                                                                 ex4_5
                                                                                   F
                                                                                   C
                                                                                   C
                                                                                   T
                                                                                   T
                                                                                   λf:F.λe1:C.λ:C.λu:T.λ:T.eq C e (CHead e1 (Flat f) u)
                                                                                   λf:F.λ:C.λe2:C.λ:T.λw:T.drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) w)
                                                                                   λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                                                                   λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2

                                                                      or4
                                                                        drop (S n0) O (CHead x1 (Bind b) x0) e
                                                                        ex3_4
                                                                          F
                                                                          C
                                                                          T
                                                                          T
                                                                          λf:F.λe0:C.λu:T.λ:T.eq C e (CHead e0 (Flat f) u)
                                                                          λf:F.λe0:C.λ:T.λw:T.drop (S n0) O (CHead x1 (Bind b) x0) (CHead e0 (Flat f) w)
                                                                          λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                                                        ex3_4
                                                                          F
                                                                          C
                                                                          C
                                                                          T
                                                                          λf:F.λe1:C.λ:C.λu:T.eq C e (CHead e1 (Flat f) u)
                                                                          λf:F.λ:C.λe2:C.λu:T.drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) u)
                                                                          λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                                                        ex4_5
                                                                          F
                                                                          C
                                                                          C
                                                                          T
                                                                          T
                                                                          λf:F.λe1:C.λ:C.λu:T.λ:T.eq C e (CHead e1 (Flat f) u)
                                                                          λf:F.λ:C.λe2:C.λ:T.λw:T.drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) w)
                                                                          λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                                                          λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2
                                                          we proved 
                                                             or4
                                                               drop (S n0) O (CHead x1 (Bind b) x0) e
                                                               ex3_4
                                                                 F
                                                                 C
                                                                 T
                                                                 T
                                                                 λf:F.λe0:C.λu:T.λ:T.eq C e (CHead e0 (Flat f) u)
                                                                 λf:F.λe0:C.λ:T.λw:T.drop (S n0) O (CHead x1 (Bind b) x0) (CHead e0 (Flat f) w)
                                                                 λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                                               ex3_4
                                                                 F
                                                                 C
                                                                 C
                                                                 T
                                                                 λf:F.λe1:C.λ:C.λu:T.eq C e (CHead e1 (Flat f) u)
                                                                 λf:F.λ:C.λe2:C.λu:T.drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) u)
                                                                 λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                                               ex4_5
                                                                 F
                                                                 C
                                                                 C
                                                                 T
                                                                 T
                                                                 λf:F.λe1:C.λ:C.λu:T.λ:T.eq C e (CHead e1 (Flat f) u)
                                                                 λf:F.λ:C.λe2:C.λ:T.λw:T.drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) w)
                                                                 λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                                                 λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2

                                                          ∀H7:drop (r (Bind b) n0) O c e
                                                            .∀H8:eq nat (S n0) (s (Bind b) x2)
                                                              .or4
                                                                drop (S n0) O (CHead x1 (Bind b) x0) e
                                                                ex3_4
                                                                  F
                                                                  C
                                                                  T
                                                                  T
                                                                  λf:F.λe0:C.λu:T.λ:T.eq C e (CHead e0 (Flat f) u)
                                                                  λf:F.λe0:C.λ:T.λw:T.drop (S n0) O (CHead x1 (Bind b) x0) (CHead e0 (Flat f) w)
                                                                  λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                                                ex3_4
                                                                  F
                                                                  C
                                                                  C
                                                                  T
                                                                  λf:F.λe1:C.λ:C.λu:T.eq C e (CHead e1 (Flat f) u)
                                                                  λf:F.λ:C.λe2:C.λu:T.drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) u)
                                                                  λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                                                ex4_5
                                                                  F
                                                                  C
                                                                  C
                                                                  T
                                                                  T
                                                                  λf:F.λe1:C.λ:C.λu:T.λ:T.eq C e (CHead e1 (Flat f) u)
                                                                  λf:F.λ:C.λe2:C.λ:T.λw:T.drop (S n0) O (CHead x1 (Bind b) x0) (CHead e2 (Flat f) w)
                                                                  λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                                                  λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2
                                                       assume f: F
                                                        suppose H7: drop (r (Flat f) n0) O c e
                                                        suppose H8: eq nat (S n0) (s (Flat f) x2)
                                                          (H9) 
                                                             by (f_equal . . . . . H8)
                                                             we proved eq nat (S n0) (s (Flat f) x2)
eq nat (λe0:nat.e0 (S n0)) (λe0:nat.e0 (s (Flat f) x2))
                                                          end of H9
                                                          (H10) 
                                                             consider H9
                                                             we proved eq nat (S n0) (s (Flat f) x2)
                                                             that is equivalent to eq nat (S n0) x2
                                                             by (eq_ind_r . . . H6 . previous)
csubst0 (S n0) v c x1
                                                          end of H10
                                                          (H12) 
                                                             consider H7
                                                             we proved drop (r (Flat f) n0) O c e
                                                             that is equivalent to drop (S n0) O c e
                                                             by (H . . H10 . previous)

                                                                or4
                                                                  drop (S n0) O x1 e
                                                                  ex3_4
                                                                    F
                                                                    C
                                                                    T
                                                                    T
                                                                    λf:F.λe0:C.λu:T.λ:T.eq C e (CHead e0 (Flat f) u)
                                                                    λf:F.λe0:C.λ:T.λw:T.drop (S n0) O x1 (CHead e0 (Flat f) w)
                                                                    λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                                                  ex3_4
                                                                    F
                                                                    C
                                                                    C
                                                                    T
                                                                    λf:F.λe1:C.λ:C.λu:T.eq C e (CHead e1 (Flat f) u)
                                                                    λf:F.λ:C.λe2:C.λu:T.drop (S n0) O x1 (CHead e2 (Flat f) u)
                                                                    λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                                                  ex4_5
                                                                    F
                                                                    C
                                                                    C
                                                                    T
                                                                    T
                                                                    λf:F.λe1:C.λ:C.λu:T.λ:T.eq C e (CHead e1 (Flat f) u)
                                                                    λf:F.λ:C.λe2:C.λ:T.λw:T.drop (S n0) O x1 (CHead e2 (Flat f) w)
                                                                    λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                                                    λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2
                                                          end of H12
                                                          we proceed by induction on H12 to prove 
                                                             or4
                                                               drop (S n0) O (CHead x1 (Flat f) x0) e
                                                               ex3_4
                                                                 F
                                                                 C
                                                                 T
                                                                 T
                                                                 λf0:F.λe0:C.λu:T.λ:T.eq C e (CHead e0 (Flat f0) u)
                                                                 λf0:F.λe0:C.λ:T.λw:T.drop (S n0) O (CHead x1 (Flat f) x0) (CHead e0 (Flat f0) w)
                                                                 λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                                               ex3_4
                                                                 F
                                                                 C
                                                                 C
                                                                 T
                                                                 λf0:F.λe1:C.λ:C.λu:T.eq C e (CHead e1 (Flat f0) u)
                                                                 λf0:F.λ:C.λe2:C.λu:T.drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) u)
                                                                 λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                                               ex4_5
                                                                 F
                                                                 C
                                                                 C
                                                                 T
                                                                 T
                                                                 λf0:F.λe1:C.λ:C.λu:T.λ:T.eq C e (CHead e1 (Flat f0) u)
                                                                 λf0:F.λ:C.λe2:C.λ:T.λw:T.drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) w)
                                                                 λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                                                 λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2
                                                             case or4_intro0 : H13:drop (S n0) O x1 e ⇒
                                                                the thesis becomes 
                                                                or4
                                                                  drop (S n0) O (CHead x1 (Flat f) x0) e
                                                                  ex3_4
                                                                    F
                                                                    C
                                                                    T
                                                                    T
                                                                    λf0:F.λe0:C.λu:T.λ:T.eq C e (CHead e0 (Flat f0) u)
                                                                    λf0:F.λe0:C.λ:T.λw:T.drop (S n0) O (CHead x1 (Flat f) x0) (CHead e0 (Flat f0) w)
                                                                    λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                                                  ex3_4
                                                                    F
                                                                    C
                                                                    C
                                                                    T
                                                                    λf0:F.λe1:C.λ:C.λu:T.eq C e (CHead e1 (Flat f0) u)
                                                                    λf0:F.λ:C.λe2:C.λu:T.drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) u)
                                                                    λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                                                  ex4_5
                                                                    F
                                                                    C
                                                                    C
                                                                    T
                                                                    T
                                                                    λf0:F.λe1:C.λ:C.λu:T.λ:T.eq C e (CHead e1 (Flat f0) u)
                                                                    λf0:F.λ:C.λe2:C.λ:T.λw:T.drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) w)
                                                                    λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                                                    λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2
                                                                   consider H13
                                                                   we proved drop (S n0) O x1 e
                                                                   that is equivalent to drop (r (Flat f) n0) O x1 e
                                                                   by (drop_drop . . . . previous .)
                                                                   we proved drop (S n0) O (CHead x1 (Flat f) x0) e
                                                                   by (or4_intro0 . . . . previous)

                                                                      or4
                                                                        drop (S n0) O (CHead x1 (Flat f) x0) e
                                                                        ex3_4
                                                                          F
                                                                          C
                                                                          T
                                                                          T
                                                                          λf0:F.λe0:C.λu:T.λ:T.eq C e (CHead e0 (Flat f0) u)
                                                                          λf0:F.λe0:C.λ:T.λw:T.drop (S n0) O (CHead x1 (Flat f) x0) (CHead e0 (Flat f0) w)
                                                                          λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                                                        ex3_4
                                                                          F
                                                                          C
                                                                          C
                                                                          T
                                                                          λf0:F.λe1:C.λ:C.λu:T.eq C e (CHead e1 (Flat f0) u)
                                                                          λf0:F.λ:C.λe2:C.λu:T.drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) u)
                                                                          λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                                                        ex4_5
                                                                          F
                                                                          C
                                                                          C
                                                                          T
                                                                          T
                                                                          λf0:F.λe1:C.λ:C.λu:T.λ:T.eq C e (CHead e1 (Flat f0) u)
                                                                          λf0:F.λ:C.λe2:C.λ:T.λw:T.drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) w)
                                                                          λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                                                          λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2
                                                             case or4_intro1 : H13:ex3_4 F C T T λf0:F.λe0:C.λu:T.λ:T.eq C e (CHead e0 (Flat f0) u) λf0:F.λe0:C.λ:T.λw:T.drop (S n0) O x1 (CHead e0 (Flat f0) w) λ:F.λ:C.λu:T.λw:T.subst0 O v u w ⇒
                                                                the thesis becomes 
                                                                or4
                                                                  drop (S n0) O (CHead x1 (Flat f) x0) e
                                                                  ex3_4
                                                                    F
                                                                    C
                                                                    T
                                                                    T
                                                                    λf0:F.λe0:C.λu:T.λ:T.eq C e (CHead e0 (Flat f0) u)
                                                                    λf0:F.λe0:C.λ:T.λw:T.drop (S n0) O (CHead x1 (Flat f) x0) (CHead e0 (Flat f0) w)
                                                                    λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                                                  ex3_4
                                                                    F
                                                                    C
                                                                    C
                                                                    T
                                                                    λf0:F.λe1:C.λ:C.λu:T.eq C e (CHead e1 (Flat f0) u)
                                                                    λf0:F.λ:C.λe2:C.λu:T.drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) u)
                                                                    λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                                                  ex4_5
                                                                    F
                                                                    C
                                                                    C
                                                                    T
                                                                    T
                                                                    λf0:F.λe1:C.λ:C.λu:T.λ:T.eq C e (CHead e1 (Flat f0) u)
                                                                    λf0:F.λ:C.λe2:C.λ:T.λw:T.drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) w)
                                                                    λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                                                    λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2
                                                                   we proceed by induction on H13 to prove 
                                                                      or4
                                                                        drop (S n0) O (CHead x1 (Flat f) x0) e
                                                                        ex3_4
                                                                          F
                                                                          C
                                                                          T
                                                                          T
                                                                          λf0:F.λe0:C.λu:T.λ:T.eq C e (CHead e0 (Flat f0) u)
                                                                          λf0:F.λe0:C.λ:T.λw:T.drop (S n0) O (CHead x1 (Flat f) x0) (CHead e0 (Flat f0) w)
                                                                          λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                                                        ex3_4
                                                                          F
                                                                          C
                                                                          C
                                                                          T
                                                                          λf0:F.λe1:C.λ:C.λu:T.eq C e (CHead e1 (Flat f0) u)
                                                                          λf0:F.λ:C.λe2:C.λu:T.drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) u)
                                                                          λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                                                        ex4_5
                                                                          F
                                                                          C
                                                                          C
                                                                          T
                                                                          T
                                                                          λf0:F.λe1:C.λ:C.λu:T.λ:T.eq C e (CHead e1 (Flat f0) u)
                                                                          λf0:F.λ:C.λe2:C.λ:T.λw:T.drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) w)
                                                                          λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                                                          λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2
                                                                      case ex3_4_intro : x3:F x4:C x5:T x6:T H14:eq C e (CHead x4 (Flat x3) x5) H15:drop (S n0) O x1 (CHead x4 (Flat x3) x6) H16:subst0 O v x5 x6 ⇒
                                                                         the thesis becomes 
                                                                         or4
                                                                           drop (S n0) O (CHead x1 (Flat f) x0) e
                                                                           ex3_4
                                                                             F
                                                                             C
                                                                             T
                                                                             T
                                                                             λf0:F.λe0:C.λu:T.λ:T.eq C e (CHead e0 (Flat f0) u)
                                                                             λf0:F.λe0:C.λ:T.λw:T.drop (S n0) O (CHead x1 (Flat f) x0) (CHead e0 (Flat f0) w)
                                                                             λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                                                           ex3_4
                                                                             F
                                                                             C
                                                                             C
                                                                             T
                                                                             λf0:F.λe1:C.λ:C.λu:T.eq C e (CHead e1 (Flat f0) u)
                                                                             λf0:F.λ:C.λe2:C.λu:T.drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) u)
                                                                             λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                                                           ex4_5
                                                                             F
                                                                             C
                                                                             C
                                                                             T
                                                                             T
                                                                             λf0:F.λe1:C.λ:C.λu:T.λ:T.eq C e (CHead e1 (Flat f0) u)
                                                                             λf0:F.λ:C.λe2:C.λ:T.λw:T.drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) w)
                                                                             λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                                                             λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2
                                                                            (h1) 
                                                                               by (refl_equal . .)
eq C (CHead x4 (Flat x3) x5) (CHead x4 (Flat x3) x5)
                                                                            end of h1
                                                                            (h2) 
                                                                               consider H15
                                                                               we proved drop (S n0) O x1 (CHead x4 (Flat x3) x6)
                                                                               that is equivalent to drop (r (Flat f) n0) O x1 (CHead x4 (Flat x3) x6)
                                                                               by (drop_drop . . . . previous .)
drop (S n0) O (CHead x1 (Flat f) x0) (CHead x4 (Flat x3) x6)
                                                                            end of h2
                                                                            by (ex3_4_intro . . . . . . . . . . . h1 h2 H16)
                                                                            we proved 
                                                                               ex3_4
                                                                                 F
                                                                                 C
                                                                                 T
                                                                                 T
                                                                                 λf0:F.λe0:C.λu:T.λ:T.eq C (CHead x4 (Flat x3) x5) (CHead e0 (Flat f0) u)
                                                                                 λf0:F.λe0:C.λ:T.λw:T.drop (S n0) O (CHead x1 (Flat f) x0) (CHead e0 (Flat f0) w)
                                                                                 λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                                                            by (or4_intro1 . . . . previous)
                                                                            we proved 
                                                                               or4
                                                                                 drop (S n0) O (CHead x1 (Flat f) x0) (CHead x4 (Flat x3) x5)
                                                                                 ex3_4
                                                                                   F
                                                                                   C
                                                                                   T
                                                                                   T
                                                                                   λf0:F.λe0:C.λu:T.λ:T.eq C (CHead x4 (Flat x3) x5) (CHead e0 (Flat f0) u)
                                                                                   λf0:F.λe0:C.λ:T.λw:T.drop (S n0) O (CHead x1 (Flat f) x0) (CHead e0 (Flat f0) w)
                                                                                   λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                                                                 ex3_4
                                                                                   F
                                                                                   C
                                                                                   C
                                                                                   T
                                                                                   λf0:F.λe1:C.λ:C.λu:T.eq C (CHead x4 (Flat x3) x5) (CHead e1 (Flat f0) u)
                                                                                   λf0:F.λ:C.λe2:C.λu:T.drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) u)
                                                                                   λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                                                                 ex4_5
                                                                                   F
                                                                                   C
                                                                                   C
                                                                                   T
                                                                                   T
                                                                                   λf0:F.λe1:C.λ:C.λu:T.λ:T.eq C (CHead x4 (Flat x3) x5) (CHead e1 (Flat f0) u)
                                                                                   λf0:F.λ:C.λe2:C.λ:T.λw:T.drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) w)
                                                                                   λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                                                                   λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2
                                                                            by (eq_ind_r . . . previous . H14)

                                                                               or4
                                                                                 drop (S n0) O (CHead x1 (Flat f) x0) e
                                                                                 ex3_4
                                                                                   F
                                                                                   C
                                                                                   T
                                                                                   T
                                                                                   λf0:F.λe0:C.λu:T.λ:T.eq C e (CHead e0 (Flat f0) u)
                                                                                   λf0:F.λe0:C.λ:T.λw:T.drop (S n0) O (CHead x1 (Flat f) x0) (CHead e0 (Flat f0) w)
                                                                                   λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                                                                 ex3_4
                                                                                   F
                                                                                   C
                                                                                   C
                                                                                   T
                                                                                   λf0:F.λe1:C.λ:C.λu:T.eq C e (CHead e1 (Flat f0) u)
                                                                                   λf0:F.λ:C.λe2:C.λu:T.drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) u)
                                                                                   λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                                                                 ex4_5
                                                                                   F
                                                                                   C
                                                                                   C
                                                                                   T
                                                                                   T
                                                                                   λf0:F.λe1:C.λ:C.λu:T.λ:T.eq C e (CHead e1 (Flat f0) u)
                                                                                   λf0:F.λ:C.λe2:C.λ:T.λw:T.drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) w)
                                                                                   λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                                                                   λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2

                                                                      or4
                                                                        drop (S n0) O (CHead x1 (Flat f) x0) e
                                                                        ex3_4
                                                                          F
                                                                          C
                                                                          T
                                                                          T
                                                                          λf0:F.λe0:C.λu:T.λ:T.eq C e (CHead e0 (Flat f0) u)
                                                                          λf0:F.λe0:C.λ:T.λw:T.drop (S n0) O (CHead x1 (Flat f) x0) (CHead e0 (Flat f0) w)
                                                                          λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                                                        ex3_4
                                                                          F
                                                                          C
                                                                          C
                                                                          T
                                                                          λf0:F.λe1:C.λ:C.λu:T.eq C e (CHead e1 (Flat f0) u)
                                                                          λf0:F.λ:C.λe2:C.λu:T.drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) u)
                                                                          λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                                                        ex4_5
                                                                          F
                                                                          C
                                                                          C
                                                                          T
                                                                          T
                                                                          λf0:F.λe1:C.λ:C.λu:T.λ:T.eq C e (CHead e1 (Flat f0) u)
                                                                          λf0:F.λ:C.λe2:C.λ:T.λw:T.drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) w)
                                                                          λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                                                          λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2
                                                             case or4_intro2 : H13:ex3_4 F C C T λf0:F.λe1:C.λ:C.λu:T.eq C e (CHead e1 (Flat f0) u) λf0:F.λ:C.λe2:C.λu:T.drop (S n0) O x1 (CHead e2 (Flat f0) u) λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2 ⇒
                                                                the thesis becomes 
                                                                or4
                                                                  drop (S n0) O (CHead x1 (Flat f) x0) e
                                                                  ex3_4
                                                                    F
                                                                    C
                                                                    T
                                                                    T
                                                                    λf0:F.λe0:C.λu:T.λ:T.eq C e (CHead e0 (Flat f0) u)
                                                                    λf0:F.λe0:C.λ:T.λw:T.drop (S n0) O (CHead x1 (Flat f) x0) (CHead e0 (Flat f0) w)
                                                                    λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                                                  ex3_4
                                                                    F
                                                                    C
                                                                    C
                                                                    T
                                                                    λf0:F.λe1:C.λ:C.λu:T.eq C e (CHead e1 (Flat f0) u)
                                                                    λf0:F.λ:C.λe2:C.λu:T.drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) u)
                                                                    λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                                                  ex4_5
                                                                    F
                                                                    C
                                                                    C
                                                                    T
                                                                    T
                                                                    λf0:F.λe1:C.λ:C.λu:T.λ:T.eq C e (CHead e1 (Flat f0) u)
                                                                    λf0:F.λ:C.λe2:C.λ:T.λw:T.drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) w)
                                                                    λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                                                    λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2
                                                                   we proceed by induction on H13 to prove 
                                                                      or4
                                                                        drop (S n0) O (CHead x1 (Flat f) x0) e
                                                                        ex3_4
                                                                          F
                                                                          C
                                                                          T
                                                                          T
                                                                          λf0:F.λe0:C.λu:T.λ:T.eq C e (CHead e0 (Flat f0) u)
                                                                          λf0:F.λe0:C.λ:T.λw:T.drop (S n0) O (CHead x1 (Flat f) x0) (CHead e0 (Flat f0) w)
                                                                          λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                                                        ex3_4
                                                                          F
                                                                          C
                                                                          C
                                                                          T
                                                                          λf0:F.λe1:C.λ:C.λu:T.eq C e (CHead e1 (Flat f0) u)
                                                                          λf0:F.λ:C.λe2:C.λu:T.drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) u)
                                                                          λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                                                        ex4_5
                                                                          F
                                                                          C
                                                                          C
                                                                          T
                                                                          T
                                                                          λf0:F.λe1:C.λ:C.λu:T.λ:T.eq C e (CHead e1 (Flat f0) u)
                                                                          λf0:F.λ:C.λe2:C.λ:T.λw:T.drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) w)
                                                                          λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                                                          λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2
                                                                      case ex3_4_intro : x3:F x4:C x5:C x6:T H14:eq C e (CHead x4 (Flat x3) x6) H15:drop (S n0) O x1 (CHead x5 (Flat x3) x6) H16:csubst0 O v x4 x5 ⇒
                                                                         the thesis becomes 
                                                                         or4
                                                                           drop (S n0) O (CHead x1 (Flat f) x0) e
                                                                           ex3_4
                                                                             F
                                                                             C
                                                                             T
                                                                             T
                                                                             λf0:F.λe0:C.λu:T.λ:T.eq C e (CHead e0 (Flat f0) u)
                                                                             λf0:F.λe0:C.λ:T.λw:T.drop (S n0) O (CHead x1 (Flat f) x0) (CHead e0 (Flat f0) w)
                                                                             λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                                                           ex3_4
                                                                             F
                                                                             C
                                                                             C
                                                                             T
                                                                             λf0:F.λe1:C.λ:C.λu:T.eq C e (CHead e1 (Flat f0) u)
                                                                             λf0:F.λ:C.λe2:C.λu:T.drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) u)
                                                                             λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                                                           ex4_5
                                                                             F
                                                                             C
                                                                             C
                                                                             T
                                                                             T
                                                                             λf0:F.λe1:C.λ:C.λu:T.λ:T.eq C e (CHead e1 (Flat f0) u)
                                                                             λf0:F.λ:C.λe2:C.λ:T.λw:T.drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) w)
                                                                             λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                                                             λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2
                                                                            (h1) 
                                                                               by (refl_equal . .)
eq C (CHead x4 (Flat x3) x6) (CHead x4 (Flat x3) x6)
                                                                            end of h1
                                                                            (h2) 
                                                                               consider H15
                                                                               we proved drop (S n0) O x1 (CHead x5 (Flat x3) x6)
                                                                               that is equivalent to drop (r (Flat f) n0) O x1 (CHead x5 (Flat x3) x6)
                                                                               by (drop_drop . . . . previous .)
drop (S n0) O (CHead x1 (Flat f) x0) (CHead x5 (Flat x3) x6)
                                                                            end of h2
                                                                            by (ex3_4_intro . . . . . . . . . . . h1 h2 H16)
                                                                            we proved 
                                                                               ex3_4
                                                                                 F
                                                                                 C
                                                                                 C
                                                                                 T
                                                                                 λf0:F.λe1:C.λ:C.λu:T.eq C (CHead x4 (Flat x3) x6) (CHead e1 (Flat f0) u)
                                                                                 λf0:F.λ:C.λe2:C.λu:T.drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) u)
                                                                                 λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                                                            by (or4_intro2 . . . . previous)
                                                                            we proved 
                                                                               or4
                                                                                 drop (S n0) O (CHead x1 (Flat f) x0) (CHead x4 (Flat x3) x6)
                                                                                 ex3_4
                                                                                   F
                                                                                   C
                                                                                   T
                                                                                   T
                                                                                   λf0:F.λe0:C.λu:T.λ:T.eq C (CHead x4 (Flat x3) x6) (CHead e0 (Flat f0) u)
                                                                                   λf0:F.λe0:C.λ:T.λw:T.drop (S n0) O (CHead x1 (Flat f) x0) (CHead e0 (Flat f0) w)
                                                                                   λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                                                                 ex3_4
                                                                                   F
                                                                                   C
                                                                                   C
                                                                                   T
                                                                                   λf0:F.λe1:C.λ:C.λu:T.eq C (CHead x4 (Flat x3) x6) (CHead e1 (Flat f0) u)
                                                                                   λf0:F.λ:C.λe2:C.λu:T.drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) u)
                                                                                   λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                                                                 ex4_5
                                                                                   F
                                                                                   C
                                                                                   C
                                                                                   T
                                                                                   T
                                                                                   λf0:F.λe1:C.λ:C.λu:T.λ:T.eq C (CHead x4 (Flat x3) x6) (CHead e1 (Flat f0) u)
                                                                                   λf0:F.λ:C.λe2:C.λ:T.λw:T.drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) w)
                                                                                   λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                                                                   λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2
                                                                            by (eq_ind_r . . . previous . H14)

                                                                               or4
                                                                                 drop (S n0) O (CHead x1 (Flat f) x0) e
                                                                                 ex3_4
                                                                                   F
                                                                                   C
                                                                                   T
                                                                                   T
                                                                                   λf0:F.λe0:C.λu:T.λ:T.eq C e (CHead e0 (Flat f0) u)
                                                                                   λf0:F.λe0:C.λ:T.λw:T.drop (S n0) O (CHead x1 (Flat f) x0) (CHead e0 (Flat f0) w)
                                                                                   λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                                                                 ex3_4
                                                                                   F
                                                                                   C
                                                                                   C
                                                                                   T
                                                                                   λf0:F.λe1:C.λ:C.λu:T.eq C e (CHead e1 (Flat f0) u)
                                                                                   λf0:F.λ:C.λe2:C.λu:T.drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) u)
                                                                                   λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                                                                 ex4_5
                                                                                   F
                                                                                   C
                                                                                   C
                                                                                   T
                                                                                   T
                                                                                   λf0:F.λe1:C.λ:C.λu:T.λ:T.eq C e (CHead e1 (Flat f0) u)
                                                                                   λf0:F.λ:C.λe2:C.λ:T.λw:T.drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) w)
                                                                                   λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                                                                   λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2

                                                                      or4
                                                                        drop (S n0) O (CHead x1 (Flat f) x0) e
                                                                        ex3_4
                                                                          F
                                                                          C
                                                                          T
                                                                          T
                                                                          λf0:F.λe0:C.λu:T.λ:T.eq C e (CHead e0 (Flat f0) u)
                                                                          λf0:F.λe0:C.λ:T.λw:T.drop (S n0) O (CHead x1 (Flat f) x0) (CHead e0 (Flat f0) w)
                                                                          λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                                                        ex3_4
                                                                          F
                                                                          C
                                                                          C
                                                                          T
                                                                          λf0:F.λe1:C.λ:C.λu:T.eq C e (CHead e1 (Flat f0) u)
                                                                          λf0:F.λ:C.λe2:C.λu:T.drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) u)
                                                                          λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                                                        ex4_5
                                                                          F
                                                                          C
                                                                          C
                                                                          T
                                                                          T
                                                                          λf0:F.λe1:C.λ:C.λu:T.λ:T.eq C e (CHead e1 (Flat f0) u)
                                                                          λf0:F.λ:C.λe2:C.λ:T.λw:T.drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) w)
                                                                          λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                                                          λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2
                                                             case or4_intro3 : H13:ex4_5 F C C T T λf0:F.λe1:C.λ:C.λu:T.λ:T.eq C e (CHead e1 (Flat f0) u) λf0:F.λ:C.λe2:C.λ:T.λw:T.drop (S n0) O x1 (CHead e2 (Flat f0) w) λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2 ⇒
                                                                the thesis becomes 
                                                                or4
                                                                  drop (S n0) O (CHead x1 (Flat f) x0) e
                                                                  ex3_4
                                                                    F
                                                                    C
                                                                    T
                                                                    T
                                                                    λf0:F.λe0:C.λu:T.λ:T.eq C e (CHead e0 (Flat f0) u)
                                                                    λf0:F.λe0:C.λ:T.λw:T.drop (S n0) O (CHead x1 (Flat f) x0) (CHead e0 (Flat f0) w)
                                                                    λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                                                  ex3_4
                                                                    F
                                                                    C
                                                                    C
                                                                    T
                                                                    λf0:F.λe1:C.λ:C.λu:T.eq C e (CHead e1 (Flat f0) u)
                                                                    λf0:F.λ:C.λe2:C.λu:T.drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) u)
                                                                    λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                                                  ex4_5
                                                                    F
                                                                    C
                                                                    C
                                                                    T
                                                                    T
                                                                    λf0:F.λe1:C.λ:C.λu:T.λ:T.eq C e (CHead e1 (Flat f0) u)
                                                                    λf0:F.λ:C.λe2:C.λ:T.λw:T.drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) w)
                                                                    λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                                                    λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2
                                                                   we proceed by induction on H13 to prove 
                                                                      or4
                                                                        drop (S n0) O (CHead x1 (Flat f) x0) e
                                                                        ex3_4
                                                                          F
                                                                          C
                                                                          T
                                                                          T
                                                                          λf0:F.λe0:C.λu:T.λ:T.eq C e (CHead e0 (Flat f0) u)
                                                                          λf0:F.λe0:C.λ:T.λw:T.drop (S n0) O (CHead x1 (Flat f) x0) (CHead e0 (Flat f0) w)
                                                                          λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                                                        ex3_4
                                                                          F
                                                                          C
                                                                          C
                                                                          T
                                                                          λf0:F.λe1:C.λ:C.λu:T.eq C e (CHead e1 (Flat f0) u)
                                                                          λf0:F.λ:C.λe2:C.λu:T.drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) u)
                                                                          λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                                                        ex4_5
                                                                          F
                                                                          C
                                                                          C
                                                                          T
                                                                          T
                                                                          λf0:F.λe1:C.λ:C.λu:T.λ:T.eq C e (CHead e1 (Flat f0) u)
                                                                          λf0:F.λ:C.λe2:C.λ:T.λw:T.drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) w)
                                                                          λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                                                          λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2
                                                                      case ex4_5_intro : x3:F x4:C x5:C x6:T x7:T H14:eq C e (CHead x4 (Flat x3) x6) H15:drop (S n0) O x1 (CHead x5 (Flat x3) x7) H16:subst0 O v x6 x7 H17:csubst0 O v x4 x5 ⇒
                                                                         the thesis becomes 
                                                                         or4
                                                                           drop (S n0) O (CHead x1 (Flat f) x0) e
                                                                           ex3_4
                                                                             F
                                                                             C
                                                                             T
                                                                             T
                                                                             λf0:F.λe0:C.λu:T.λ:T.eq C e (CHead e0 (Flat f0) u)
                                                                             λf0:F.λe0:C.λ:T.λw:T.drop (S n0) O (CHead x1 (Flat f) x0) (CHead e0 (Flat f0) w)
                                                                             λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                                                           ex3_4
                                                                             F
                                                                             C
                                                                             C
                                                                             T
                                                                             λf0:F.λe1:C.λ:C.λu:T.eq C e (CHead e1 (Flat f0) u)
                                                                             λf0:F.λ:C.λe2:C.λu:T.drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) u)
                                                                             λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                                                           ex4_5
                                                                             F
                                                                             C
                                                                             C
                                                                             T
                                                                             T
                                                                             λf0:F.λe1:C.λ:C.λu:T.λ:T.eq C e (CHead e1 (Flat f0) u)
                                                                             λf0:F.λ:C.λe2:C.λ:T.λw:T.drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) w)
                                                                             λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                                                             λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2
                                                                            (h1) 
                                                                               by (refl_equal . .)
eq C (CHead x4 (Flat x3) x6) (CHead x4 (Flat x3) x6)
                                                                            end of h1
                                                                            (h2) 
                                                                               consider H15
                                                                               we proved drop (S n0) O x1 (CHead x5 (Flat x3) x7)
                                                                               that is equivalent to drop (r (Flat f) n0) O x1 (CHead x5 (Flat x3) x7)
                                                                               by (drop_drop . . . . previous .)
drop (S n0) O (CHead x1 (Flat f) x0) (CHead x5 (Flat x3) x7)
                                                                            end of h2
                                                                            by (ex4_5_intro . . . . . . . . . . . . . . h1 h2 H16 H17)
                                                                            we proved 
                                                                               ex4_5
                                                                                 F
                                                                                 C
                                                                                 C
                                                                                 T
                                                                                 T
                                                                                 λf0:F.λe1:C.λ:C.λu:T.λ:T.eq C (CHead x4 (Flat x3) x6) (CHead e1 (Flat f0) u)
                                                                                 λf0:F.λ:C.λe2:C.λ:T.λw:T.drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) w)
                                                                                 λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                                                                 λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2
                                                                            by (or4_intro3 . . . . previous)
                                                                            we proved 
                                                                               or4
                                                                                 drop (S n0) O (CHead x1 (Flat f) x0) (CHead x4 (Flat x3) x6)
                                                                                 ex3_4
                                                                                   F
                                                                                   C
                                                                                   T
                                                                                   T
                                                                                   λf0:F.λe0:C.λu:T.λ:T.eq C (CHead x4 (Flat x3) x6) (CHead e0 (Flat f0) u)
                                                                                   λf0:F.λe0:C.λ:T.λw:T.drop (S n0) O (CHead x1 (Flat f) x0) (CHead e0 (Flat f0) w)
                                                                                   λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                                                                 ex3_4
                                                                                   F
                                                                                   C
                                                                                   C
                                                                                   T
                                                                                   λf0:F.λe1:C.λ:C.λu:T.eq C (CHead x4 (Flat x3) x6) (CHead e1 (Flat f0) u)
                                                                                   λf0:F.λ:C.λe2:C.λu:T.drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) u)
                                                                                   λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                                                                 ex4_5
                                                                                   F
                                                                                   C
                                                                                   C
                                                                                   T
                                                                                   T
                                                                                   λf0:F.λe1:C.λ:C.λu:T.λ:T.eq C (CHead x4 (Flat x3) x6) (CHead e1 (Flat f0) u)
                                                                                   λf0:F.λ:C.λe2:C.λ:T.λw:T.drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) w)
                                                                                   λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                                                                   λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2
                                                                            by (eq_ind_r . . . previous . H14)

                                                                               or4
                                                                                 drop (S n0) O (CHead x1 (Flat f) x0) e
                                                                                 ex3_4
                                                                                   F
                                                                                   C
                                                                                   T
                                                                                   T
                                                                                   λf0:F.λe0:C.λu:T.λ:T.eq C e (CHead e0 (Flat f0) u)
                                                                                   λf0:F.λe0:C.λ:T.λw:T.drop (S n0) O (CHead x1 (Flat f) x0) (CHead e0 (Flat f0) w)
                                                                                   λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                                                                 ex3_4
                                                                                   F
                                                                                   C
                                                                                   C
                                                                                   T
                                                                                   λf0:F.λe1:C.λ:C.λu:T.eq C e (CHead e1 (Flat f0) u)
                                                                                   λf0:F.λ:C.λe2:C.λu:T.drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) u)
                                                                                   λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                                                                 ex4_5
                                                                                   F
                                                                                   C
                                                                                   C
                                                                                   T
                                                                                   T
                                                                                   λf0:F.λe1:C.λ:C.λu:T.λ:T.eq C e (CHead e1 (Flat f0) u)
                                                                                   λf0:F.λ:C.λe2:C.λ:T.λw:T.drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) w)
                                                                                   λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                                                                   λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2

                                                                      or4
                                                                        drop (S n0) O (CHead x1 (Flat f) x0) e
                                                                        ex3_4
                                                                          F
                                                                          C
                                                                          T
                                                                          T
                                                                          λf0:F.λe0:C.λu:T.λ:T.eq C e (CHead e0 (Flat f0) u)
                                                                          λf0:F.λe0:C.λ:T.λw:T.drop (S n0) O (CHead x1 (Flat f) x0) (CHead e0 (Flat f0) w)
                                                                          λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                                                        ex3_4
                                                                          F
                                                                          C
                                                                          C
                                                                          T
                                                                          λf0:F.λe1:C.λ:C.λu:T.eq C e (CHead e1 (Flat f0) u)
                                                                          λf0:F.λ:C.λe2:C.λu:T.drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) u)
                                                                          λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                                                        ex4_5
                                                                          F
                                                                          C
                                                                          C
                                                                          T
                                                                          T
                                                                          λf0:F.λe1:C.λ:C.λu:T.λ:T.eq C e (CHead e1 (Flat f0) u)
                                                                          λf0:F.λ:C.λe2:C.λ:T.λw:T.drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) w)
                                                                          λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                                                          λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2
                                                          we proved 
                                                             or4
                                                               drop (S n0) O (CHead x1 (Flat f) x0) e
                                                               ex3_4
                                                                 F
                                                                 C
                                                                 T
                                                                 T
                                                                 λf0:F.λe0:C.λu:T.λ:T.eq C e (CHead e0 (Flat f0) u)
                                                                 λf0:F.λe0:C.λ:T.λw:T.drop (S n0) O (CHead x1 (Flat f) x0) (CHead e0 (Flat f0) w)
                                                                 λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                                               ex3_4
                                                                 F
                                                                 C
                                                                 C
                                                                 T
                                                                 λf0:F.λe1:C.λ:C.λu:T.eq C e (CHead e1 (Flat f0) u)
                                                                 λf0:F.λ:C.λe2:C.λu:T.drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) u)
                                                                 λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                                               ex4_5
                                                                 F
                                                                 C
                                                                 C
                                                                 T
                                                                 T
                                                                 λf0:F.λe1:C.λ:C.λu:T.λ:T.eq C e (CHead e1 (Flat f0) u)
                                                                 λf0:F.λ:C.λe2:C.λ:T.λw:T.drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) w)
                                                                 λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                                                 λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2

                                                          ∀H7:drop (r (Flat f) n0) O c e
                                                            .∀H8:eq nat (S n0) (s (Flat f) x2)
                                                              .or4
                                                                drop (S n0) O (CHead x1 (Flat f) x0) e
                                                                ex3_4
                                                                  F
                                                                  C
                                                                  T
                                                                  T
                                                                  λf0:F.λe0:C.λu:T.λ:T.eq C e (CHead e0 (Flat f0) u)
                                                                  λf0:F.λe0:C.λ:T.λw:T.drop (S n0) O (CHead x1 (Flat f) x0) (CHead e0 (Flat f0) w)
                                                                  λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                                                ex3_4
                                                                  F
                                                                  C
                                                                  C
                                                                  T
                                                                  λf0:F.λe1:C.λ:C.λu:T.eq C e (CHead e1 (Flat f0) u)
                                                                  λf0:F.λ:C.λe2:C.λu:T.drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) u)
                                                                  λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                                                ex4_5
                                                                  F
                                                                  C
                                                                  C
                                                                  T
                                                                  T
                                                                  λf0:F.λe1:C.λ:C.λu:T.λ:T.eq C e (CHead e1 (Flat f0) u)
                                                                  λf0:F.λ:C.λe2:C.λ:T.λw:T.drop (S n0) O (CHead x1 (Flat f) x0) (CHead e2 (Flat f0) w)
                                                                  λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                                                  λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2
                                                    by (previous . previous H3)
                                                    we proved 
                                                       or4
                                                         drop (S n0) O (CHead x1 k x0) e
                                                         ex3_4
                                                           F
                                                           C
                                                           T
                                                           T
                                                           λf:F.λe0:C.λu:T.λ:T.eq C e (CHead e0 (Flat f) u)
                                                           λf:F.λe0:C.λ:T.λw:T.drop (S n0) O (CHead x1 k x0) (CHead e0 (Flat f) w)
                                                           λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                                         ex3_4
                                                           F
                                                           C
                                                           C
                                                           T
                                                           λf:F.λe1:C.λ:C.λu:T.eq C e (CHead e1 (Flat f) u)
                                                           λf:F.λ:C.λe2:C.λu:T.drop (S n0) O (CHead x1 k x0) (CHead e2 (Flat f) u)
                                                           λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                                         ex4_5
                                                           F
                                                           C
                                                           C
                                                           T
                                                           T
                                                           λf:F.λe1:C.λ:C.λu:T.λ:T.eq C e (CHead e1 (Flat f) u)
                                                           λf:F.λ:C.λe2:C.λ:T.λw:T.drop (S n0) O (CHead x1 k x0) (CHead e2 (Flat f) w)
                                                           λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                                           λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2
                                                    by (eq_ind_r . . . previous . H4)

                                                       or4
                                                         drop (S n0) O c2 e
                                                         ex3_4
                                                           F
                                                           C
                                                           T
                                                           T
                                                           λf:F.λe0:C.λu:T.λ:T.eq C e (CHead e0 (Flat f) u)
                                                           λf:F.λe0:C.λ:T.λw:T.drop (S n0) O c2 (CHead e0 (Flat f) w)
                                                           λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                                         ex3_4
                                                           F
                                                           C
                                                           C
                                                           T
                                                           λf:F.λe1:C.λ:C.λu:T.eq C e (CHead e1 (Flat f) u)
                                                           λf:F.λ:C.λe2:C.λu:T.drop (S n0) O c2 (CHead e2 (Flat f) u)
                                                           λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                                         ex4_5
                                                           F
                                                           C
                                                           C
                                                           T
                                                           T
                                                           λf:F.λe1:C.λ:C.λu:T.λ:T.eq C e (CHead e1 (Flat f) u)
                                                           λf:F.λ:C.λe2:C.λ:T.λw:T.drop (S n0) O c2 (CHead e2 (Flat f) w)
                                                           λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                                           λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2

                                              or4
                                                drop (S n0) O c2 e
                                                ex3_4
                                                  F
                                                  C
                                                  T
                                                  T
                                                  λf:F.λe0:C.λu:T.λ:T.eq C e (CHead e0 (Flat f) u)
                                                  λf:F.λe0:C.λ:T.λw:T.drop (S n0) O c2 (CHead e0 (Flat f) w)
                                                  λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                                ex3_4
                                                  F
                                                  C
                                                  C
                                                  T
                                                  λf:F.λe1:C.λ:C.λu:T.eq C e (CHead e1 (Flat f) u)
                                                  λf:F.λ:C.λe2:C.λu:T.drop (S n0) O c2 (CHead e2 (Flat f) u)
                                                  λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                                ex4_5
                                                  F
                                                  C
                                                  C
                                                  T
                                                  T
                                                  λf:F.λe1:C.λ:C.λu:T.λ:T.eq C e (CHead e1 (Flat f) u)
                                                  λf:F.λ:C.λe2:C.λ:T.λw:T.drop (S n0) O c2 (CHead e2 (Flat f) w)
                                                  λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                                  λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2
                                  we proved 
                                     or4
                                       drop (S n0) O c2 e
                                       ex3_4
                                         F
                                         C
                                         T
                                         T
                                         λf:F.λe0:C.λu:T.λ:T.eq C e (CHead e0 (Flat f) u)
                                         λf:F.λe0:C.λ:T.λw:T.drop (S n0) O c2 (CHead e0 (Flat f) w)
                                         λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                       ex3_4
                                         F
                                         C
                                         C
                                         T
                                         λf:F.λe1:C.λ:C.λu:T.eq C e (CHead e1 (Flat f) u)
                                         λf:F.λ:C.λe2:C.λu:T.drop (S n0) O c2 (CHead e2 (Flat f) u)
                                         λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                       ex4_5
                                         F
                                         C
                                         C
                                         T
                                         T
                                         λf:F.λe1:C.λ:C.λu:T.λ:T.eq C e (CHead e1 (Flat f) u)
                                         λf:F.λ:C.λe2:C.λ:T.λw:T.drop (S n0) O c2 (CHead e2 (Flat f) w)
                                         λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                         λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2

                                  ∀c2:C
                                    .∀v:T
                                      .∀H0:csubst0 (S n0) v (CHead c k t) c2
                                        .∀e:C
                                          .∀H1:drop (S n0) O (CHead c k t) e
                                            .or4
                                              drop (S n0) O c2 e
                                              ex3_4
                                                F
                                                C
                                                T
                                                T
                                                λf:F.λe0:C.λu:T.λ:T.eq C e (CHead e0 (Flat f) u)
                                                λf:F.λe0:C.λ:T.λw:T.drop (S n0) O c2 (CHead e0 (Flat f) w)
                                                λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                              ex3_4
                                                F
                                                C
                                                C
                                                T
                                                λf:F.λe1:C.λ:C.λu:T.eq C e (CHead e1 (Flat f) u)
                                                λf:F.λ:C.λe2:C.λu:T.drop (S n0) O c2 (CHead e2 (Flat f) u)
                                                λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                              ex4_5
                                                F
                                                C
                                                C
                                                T
                                                T
                                                λf:F.λe1:C.λ:C.λu:T.λ:T.eq C e (CHead e1 (Flat f) u)
                                                λf:F.λ:C.λe2:C.λ:T.λw:T.drop (S n0) O c2 (CHead e2 (Flat f) w)
                                                λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                                λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2
                      we proved 
                         ∀c2:C
                           .∀v:T
                             .csubst0 (S n0) v c1 c2
                               →∀e:C
                                    .drop (S n0) O c1 e
                                      →(or4
                                           drop (S n0) O c2 e
                                           ex3_4
                                             F
                                             C
                                             T
                                             T
                                             λf:F.λe0:C.λu:T.λ:T.eq C e (CHead e0 (Flat f) u)
                                             λf:F.λe0:C.λ:T.λw:T.drop (S n0) O c2 (CHead e0 (Flat f) w)
                                             λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                           ex3_4
                                             F
                                             C
                                             C
                                             T
                                             λf:F.λe1:C.λ:C.λu:T.eq C e (CHead e1 (Flat f) u)
                                             λf:F.λ:C.λe2:C.λu:T.drop (S n0) O c2 (CHead e2 (Flat f) u)
                                             λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                           ex4_5
                                             F
                                             C
                                             C
                                             T
                                             T
                                             λf:F.λe1:C.λ:C.λu:T.λ:T.eq C e (CHead e1 (Flat f) u)
                                             λf:F.λ:C.λe2:C.λ:T.λw:T.drop (S n0) O c2 (CHead e2 (Flat f) w)
                                             λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                             λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2)

                      ∀c1:C
                        .∀c2:C
                          .∀v:T
                            .csubst0 (S n0) v c1 c2
                              →∀e:C
                                   .drop (S n0) O c1 e
                                     →(or4
                                          drop (S n0) O c2 e
                                          ex3_4
                                            F
                                            C
                                            T
                                            T
                                            λf:F.λe0:C.λu:T.λ:T.eq C e (CHead e0 (Flat f) u)
                                            λf:F.λe0:C.λ:T.λw:T.drop (S n0) O c2 (CHead e0 (Flat f) w)
                                            λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                          ex3_4
                                            F
                                            C
                                            C
                                            T
                                            λf:F.λe1:C.λ:C.λu:T.eq C e (CHead e1 (Flat f) u)
                                            λf:F.λ:C.λe2:C.λu:T.drop (S n0) O c2 (CHead e2 (Flat f) u)
                                            λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                          ex4_5
                                            F
                                            C
                                            C
                                            T
                                            T
                                            λf:F.λe1:C.λ:C.λu:T.λ:T.eq C e (CHead e1 (Flat f) u)
                                            λf:F.λ:C.λe2:C.λ:T.λw:T.drop (S n0) O c2 (CHead e2 (Flat f) w)
                                            λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                            λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2)
          we proved 
             ∀c1:C
               .∀c2:C
                 .∀v:T
                   .csubst0 n v c1 c2
                     →∀e:C
                          .drop n O c1 e
                            →(or4
                                 drop n O c2 e
                                 ex3_4
                                   F
                                   C
                                   T
                                   T
                                   λf:F.λe0:C.λu:T.λ:T.eq C e (CHead e0 (Flat f) u)
                                   λf:F.λe0:C.λ:T.λw:T.drop n O c2 (CHead e0 (Flat f) w)
                                   λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                 ex3_4
                                   F
                                   C
                                   C
                                   T
                                   λf:F.λe1:C.λ:C.λu:T.eq C e (CHead e1 (Flat f) u)
                                   λf:F.λ:C.λe2:C.λu:T.drop n O c2 (CHead e2 (Flat f) u)
                                   λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                 ex4_5
                                   F
                                   C
                                   C
                                   T
                                   T
                                   λf:F.λe1:C.λ:C.λu:T.λ:T.eq C e (CHead e1 (Flat f) u)
                                   λf:F.λ:C.λe2:C.λ:T.λw:T.drop n O c2 (CHead e2 (Flat f) w)
                                   λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                   λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2)
       we proved 
          ∀n:nat
            .∀c1:C
              .∀c2:C
                .∀v:T
                  .csubst0 n v c1 c2
                    →∀e:C
                         .drop n O c1 e
                           →(or4
                                drop n O c2 e
                                ex3_4
                                  F
                                  C
                                  T
                                  T
                                  λf:F.λe0:C.λu:T.λ:T.eq C e (CHead e0 (Flat f) u)
                                  λf:F.λe0:C.λ:T.λw:T.drop n O c2 (CHead e0 (Flat f) w)
                                  λ:F.λ:C.λu:T.λw:T.subst0 O v u w
                                ex3_4
                                  F
                                  C
                                  C
                                  T
                                  λf:F.λe1:C.λ:C.λu:T.eq C e (CHead e1 (Flat f) u)
                                  λf:F.λ:C.λe2:C.λu:T.drop n O c2 (CHead e2 (Flat f) u)
                                  λ:F.λe1:C.λe2:C.λ:T.csubst0 O v e1 e2
                                ex4_5
                                  F
                                  C
                                  C
                                  T
                                  T
                                  λf:F.λe1:C.λ:C.λu:T.λ:T.eq C e (CHead e1 (Flat f) u)
                                  λf:F.λ:C.λe2:C.λ:T.λw:T.drop n O c2 (CHead e2 (Flat f) w)
                                  λ:F.λ:C.λ:C.λu:T.λw:T.subst0 O v u w
                                  λ:F.λe1:C.λe2:C.λ:T.λ:T.csubst0 O v e1 e2)