simple induction E; intros A f r.
intros.
simpl in |- *.
elim (IN_EXType (sup A f) E' H).
intros x e.
cut (EQ (pi2 (sup A f) x) E'); auto with zfc.
intros e1.
cut (IN E'' (pi2 (sup A f) x)).
intros i1.
elim (IN_EXType _ _ i1).
intros x0 e2.
simpl in x0.
exists (dep_i A (fun x : A => pi1 (f x)) x x0).
simpl in |- *.
exact e2.
apply IN_sound_right with E'; auto with zfc.
Qed.

(* Specifications: *)
Theorem IN_INC_Union : forall E E' : Ens, IN E' E -> INC E' (Union E).
unfold INC in |- *; simple induction E; intros A f r; unfold Union in |- *.
Theorem IN_INC_Union : forall E E' : Ens, IN E' E -> INC E' (Union E).
unfold INC in |- *; simple induction E; intros A f r; unfold Union in |- *.