Section Bounded_th1_proof.
Variable P : Ens -> Prop.
Variable A : Ens.
Hypothesis h : forall X : Ens, P X -> IN X A.
Let step1 : Bounded_By A P. Proof h.
Definition Bounded_th1_proof_Out : EXISTS (fun A : Ens => Bounded_By A P) :=
EXISTS_th1 (fun B : Ens => Bounded_By B P) A step1.
End Bounded_th1_proof.
Proof Bounded_th1_proof_Out.
Proof
fun (P : Ens -> Prop) (A : Ens) (h : Bounded_By A P) => Bounded_th1 P A h.
Definition Bounded_CHOICE (P : Ens -> Prop) := REPLACEMENT P.

Theorem Bounded_CHOICE_pr :
forall (P : Ens -> Prop) (b : Bounded P), Bounded_By (Bounded_CHOICE P) P.