intros E'.
simple induction 1.
simple induction x; intro a.
simple induction p.
intros b.
intros h e.
apply Ord_sound with (pi2 (f a) b); auto with zfc.
apply IN_Ord_Ord with (f a); auto with zfc.
apply IN_Ord_Ord with (sup A f).
assumption.
exists a; auto with zfc.
intros.
elim H1; 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_Ord_Ord with (sup A f); auto with zfc.
auto with zfc.
auto with zfc.
apply IN_Inter_all with (sup A f); auto with zfc.
auto with zfc.
Qed.

Lemma union_Ord : forall E : Ens, Ord E -> Ord (Union E).
simple induction E; intros A f HR o.