simple induction E; intros A f HR h.
apply Ord_intro.
intros E' i.
elim i.
simple induction x; intros a.
intros b e.
simpl in e.
apply Ord_sound with (pi2 (f a) b).
auto with zfc.
apply IN_Ord_Ord with (f a); auto with zfc.
apply h; exists a; auto with zfc.
generalize b; elim (f a); simpl in |- *.
intros.
exists b0; auto with zfc.
intros.
elim (Union_IN (sup A f) E1); auto with zfc.
intros E3.
simple induction 1.
intros i1 i2.
apply IN_Union with E3.
auto with zfc.
apply plump with E1; auto with zfc.
Qed.
Definition Succ (E : Ens) := Comp (Power E) Ord.

(* Specifications: *)
Lemma Ord_Succ : forall E : Ens, Ord E -> Ord (Succ E).
unfold Succ in |- *; intros E o.