simple induction 1.
cut (IN E Vide).
simpl in |- *; simple induction 1; intros xx; elim xx; simple induction 1.
apply IN_sound_right with (Sing E); auto with zfc.
Qed.
(* The cartesian product and its properties *)
(* This definition of the ordered pair is slightly different from *)
(* the usual one, since we want it to work in an intuisionistic   *)
(* setting. Works the same, neitherless. The soundness proofs are *)
(* unpleasant.                                                    *)
Definition Couple (E E' : Ens) := Paire (Sing E) (Paire Vide (Sing E')).

Theorem Couple_inj_left :
forall A A' B B' : Ens, EQ (Couple A A') (Couple B B') -> EQ A B.