elim (Union_IN E E'').
intros E'''; simple induction 1; intros I1 I2.
apply IN_Union with E'''; auto with zfc.
auto with zfc.
Qed.
(* The  Intersection set   *)
Definition Inter (E : Ens) : Ens :=
match E with
| sup A f =>
sup _
(fun
c : depprod _
(fun a : A =>
depprod _
(fun b : pi1 (f a) =>
forall x : A, IN (pi2 (f a) b) (f x))) =>
match c with
| dep_i a (dep_i b p) => pi2 (f a) b
end)
end.
(* the axioms *)

(* Specifications: *)
Theorem IN_Inter_all :
forall E E' : Ens,
IN E' (Inter E) -> forall E'' : Ens, IN E'' E -> IN E' E''.
Theorem IN_Inter_all :
forall E E' : Ens,
IN E' (Inter E) -> forall E'' : Ens, IN E'' E -> IN E' E''.