apply Ord_intro.
intros E'; simple induction 1.
simple induction x.
intros a b.
simpl in |- *.
intros e.
apply IN_Ord_Ord with (f a); auto with zfc.
apply IN_Ord_Ord with (sup A f); try exists a; auto with zfc.
apply IN_sound_left with (pi2 (f a) b); auto with zfc.
generalize b; elim (f a).
simpl in |- *.
intros.
exists b0; auto with zfc.
intros E1 E2 o1 o2; simple induction 1.
simple induction x.
intros a b e inc.
simpl in e.
apply IN_Union with (f a).
exists a; try trivial with zfc.
apply plump with E1; auto with zfc.
apply IN_Ord_Ord with (sup A f); try exists a; auto with zfc.
apply IN_sound_left with (pi2 (f a) b); auto with zfc.
generalize b; elim (f a); simpl in |- *; intros.
exists b0; auto with zfc.
Qed.

Lemma Inter_Ord :
forall E : Ens, (forall E' : Ens, IN E' E -> Ord E') -> Ord (Inter E).