Proof Bounded_sub2_Out.
(************************************************)
(************ Restriction ***********************)
(***** a more practical form of replacement *****)
(************************************************)
Record Restriction (P : Ens -> Prop) (A X : Ens) : Prop :=
{remaining_property : P X; the_restriction : IN X A}.
Section restriction_bounded_proof.
Variable P : Ens -> Prop.
Variable A : Ens.
Let R := Restriction P A.
Let t : Bounded_By A R. Proof fun (X : Ens) (h : R X) => the_restriction P A X h.
Definition restriction_bounded_Out : Bounded R :=
EXISTS_th1 (fun Y : Ens => Bounded_By Y R) A t.
End restriction_bounded_proof.

Theorem Restriction_bounded :
forall (P : Ens -> Prop) (A : Ens), Bounded (Restriction P A).