simple induction E; intros A f r; simple induction E'; intros A' f' r'.
intros; elim (IN_EXType (Prod (sup A f) (sup A' f')) E'').
simple induction x.
intros; exists (f a); exists (f' b); auto with zfc.
auto with zfc.
Qed.
(* Ordinals *)
Definition Succ (E : Ens) := Union (Paire E (Sing E)).
Inductive Ord : Ens -> Prop :=
| Oo : Ord Vide
| So : forall E : Ens, Ord E -> Ord (Succ E)
| Lo : forall E : Ens, (forall e : Ens, IN e E -> Ord e) -> Ord (Union E)
| Eo : forall E E' : Ens, Ord E -> EQ E E' -> Ord E'.
Hint Resolve Oo So Lo: zfc.
Definition Nat : nat -> Ens.
simple induction 1; intros.
exact Vide.
exact (Succ X).
Defined.

(* Specifications: *)
Theorem Nat_Ord : forall n : nat, Ord (Nat n).
simple induction n; simpl in |- *; auto with zfc.