simple induction E; intros A f r.
intros.
simpl in |- *.
elim (CIN_EXType (sup A f) E' X).
intros x e.
cut (EQC (pi2 (sup A f) x) E'); auto with zfc.
intros e1.
cut (CIN E'' (pi2 (sup A f) x)).
intros i1.
elim (CIN_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 CIN_sound_right with E'; auto with zfc.
Qed.

Theorem CIN_CINC_Union : forall E E' : Ens, CIN E' E -> CINC E' (Union E).
unfold CINC in |- *; simple induction E; intros A f r.