unfold Inter' in |- *.
intros.
apply IN_P_Comp.
intros; apply IN_sound_left with w1; auto with zfc.
apply IN_Union with (E' := E''); auto with zfc.
auto with zfc.
Qed.
(*  The powerset and its axioms   *)
Definition Power (E : Ens) : Ens :=
match E with
| sup A f =>
sup _
(fun P : A -> Prop =>
sup _
(fun c : depprod A (fun a : A => P a) =>
match c with
| dep_i a p => f a
end))
end.

(* Specifications: *)
Theorem IN_Power_INC : forall E E' : Ens, IN E' (Power E) -> INC E' E.
simple induction E.