simple induction E; intros A f r; simpl in |- *; intros E'.
simple induction 1; intros c; elim c; intros a ca; elim ca; intros aa paa;
simpl in |- *.
intros e E'' e''.
elim e''; intros a1 ea1.
apply IN_sound_right with (f a1); auto with zfc.
apply IN_sound_left with (pi2 (f a) aa); auto with zfc.
Qed.

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