unfold Inter' in |- *.
intros E E' i.
change ((fun e : Ens => forall a : Ens, IN a E -> IN e a) E') in |- *.
apply (IN_Comp_P (Union E) E').
intros.
apply IN_sound_left with w1; auto with zfc.
assumption.
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).