simple induction E; unfold Union in |- *; simpl in |- *; intros A f r.
simple induction 1.
simple induction x.
intros a b; simpl in |- *.
intros.
exists (f a).
split.
exists a; auto with zfc.
apply IN_sound_left with (pi2 (f a) b); auto with zfc.
simpl in |- *.
generalize b; elim (f a); simpl in |- *.
intros.
exists b0; auto with zfc.
Qed.
(* extentionality of union  *)

Theorem Union_sound : forall E E' : Ens, EQ E E' -> EQ (Union E) (Union E').
unfold Union in |- *; simple induction E; intros A f r; simple induction E';
intros A' f' r'; simpl in |- *; simple induction 1;
intros e1 e2; split.