simple induction E; intros A f r.
intros E' E'' i H.
elim (IN_EXType (sup A f) E'' i).
intros a e; simpl in a.
simpl in e.
cut (IN E' E''); auto with zfc.
intros i'.
cut (IN E' (f a)); auto with zfc.
intros i0.
elim (IN_EXType (f a) E' i0).
intros b e'.
simpl in |- *.
cut (forall x : A, IN (pi2 (f a) b) (f x)).
intros.
exists
(dep_i A
(fun a : A =>
depprod (pi1 (f a))
(fun b : pi1 (f a) => forall x : A, IN (pi2 (f a) b) (f x))) a
(dep_i (pi1 (f a))
(fun b : pi1 (f a) => forall x : A, IN (pi2 (f a) b) (f x)) b H0)).
simpl in |- *.
auto with zfc.
auto with zfc.
intros.
apply IN_sound_left with E'.
auto with zfc.
apply H.
auto with zfc.
simpl in |- *.
exists x; auto with zfc.
apply IN_sound_right with E''; auto with zfc.
Qed.
Definition Inter' (E : Ens) : Ens :=
Comp (Union E) (fun e : Ens => forall a : Ens, IN a E -> IN e a).

Theorem IN_Inter'_all :
forall E E' : Ens,
IN E' (Inter' E) -> forall E'' : Ens, IN E'' E -> IN E' E''.