simple induction E'; simple induction E''; unfold INC in |- *; simpl in |- *;
intros.
elim (H2 E0); try assumption; intros.
elim H1; intros HA HB; elim (HA x); intros.
exists x0; apply EQ_tran with (e x); auto with zfc.
Qed.
(* Useful types (actually top and bottom)   *)
Inductive Un : Set :=
void : Un.
Inductive F : Set :=.
(* The empty set  (vide = french for empty)   *)
Definition Vide : Ens := sup F (fun f : F => match f return Ens with
end).
(* The axioms of the empty set *)

Theorem Vide_est_vide : forall E : Ens, IN E Vide -> F.
unfold Vide in |- *; simpl in |- *; intros E H; cut False.