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).
Proof
fun (A : Ens) (P : Ens -> Prop) =>
REPLACEMENT_pr1 (Restriction P A) (Restriction_bounded P A).
Proof
fun (A : Ens) (P : Ens -> Prop) =>
REPLACEMENT_pr2 (Restriction P A) (Restriction_bounded P A).
Proof
fun (A : Ens) (P : Ens -> Prop) (X : Ens) (h : IN X A) (k : P X) =>
Set_Of_pr1 A P X (Build_Restriction P A X k h).
Proof
fun (A : Ens) (P : Ens -> Prop) (X : Ens) (h : IN X (Set_Of A P)) =>
remaining_property P A X (Set_Of_pr2 A P X h).
Proof
fun (A : Ens) (P : Ens -> Prop) (X : Ens) (h : IN X (Set_Of A P)) =>
the_restriction P A X (Set_Of_pr2 A P X h).

Theorem Set_Of_th4 : forall (A : Ens) (P : Ens -> Prop), SUB (Set_Of A P) A.