simple induction E; intros A f HR X.
intros i.
change (IN X (Union (sup A (fun a : A => Power (Vee (f a)))))) in i.
elim (Union_IN _ _ i).
intros T c.
elim c; intros i1 i2.
elim i1; intros a e.
exists (f a); split.
exists a; auto with zfc.
apply IN_Power_INC.
apply IN_sound_right with T; auto with zfc.
Qed.

(* Specifications: *)
Lemma mon_Vee : forall E1 E2 : Ens, INC E1 E2 -> INC (Vee E1) (Vee E2).
intros E1 E2 inc; unfold INC in |- *; intros X i.