simple induction E; intros A f HR H.
apply Ord_intro.
intros E' i.
elim i.
simple induction x.
intro a; simple induction p.
intros b h.
intros e.
apply IN_Ord_Ord with (f a).
apply H; exists a; auto with zfc.
apply IN_sound_left with (pi2 (f a) b); auto with zfc.
intros E1 E2 o1 o2 i inc.
elim i.
simple induction x.
intro a; simple induction p.
intros b h e.
apply all_IN_Inter with (f a).
exists a; auto with zfc.
intros.
apply plump with E1; auto with zfc.
apply IN_Inter_all with (sup A f); auto with zfc.
Qed.

(* Specifications: *)
Lemma Union_Ord :
forall E : Ens, (forall E' : Ens, IN E' E -> Ord E') -> Ord (Union E).