cut False.
simple induction 1.
elim (IN_EXType (sup A f) (sup A f) i); intros a e.
simpl in a.
change (EQ (sup A f) (f a)) in e.
elim (r a).
apply IN_sound_right with (sup A f); auto with zfc.
exists a; auto with zfc.
Qed.

(* Specifications: *)
Theorem Nat_IN_Omega : forall n : nat, IN (Nat n) Omega.
intros; simpl in |- *; exists n; auto with zfc.
Theorem Nat_IN_Omega : forall n : nat, IN (Nat n) Omega.
intros; simpl in |- *; exists n; auto with zfc.