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.
Proof restriction_bounded_Out.
Definition Set_Of (A : Ens) (P : Ens -> Prop) :=
REPLACEMENT (Restriction P A).

Theorem Set_Of_pr1 :
forall (A : Ens) (P : Ens -> Prop) (X : Ens),
Restriction P A X -> IN X (Set_Of A P).