DEFINITION getl_gen_O()
TYPE =
∀e:C.∀x:C.(getl O e x)→(clear e x)
BODY =
assume e: C
assume x: C
suppose H: getl O e x
(H0)
by (getl_gen_all . . . H)
ex2 C λe:C.drop O O e e λe:C.clear e x
end of H0
we proceed by induction on H0 to prove clear e x
case ex_intro2 : x0:C H1:drop O O e x0 H2:clear x0 x ⇒
the thesis becomes clear e x
(H3)
by (drop_gen_refl . . H1)
we proved eq C e x0
by (eq_ind_r . . . H2 . previous)
clear e x
end of H3
consider H3
clear e x
we proved clear e x
we proved ∀e:C.∀x:C.(getl O e x)→(clear e x)