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(********* Bounded *****************)
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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.

Theorem Bounded_th1 :
forall (P : Ens -> Prop) (A : Ens) (h : forall X : Ens, P X -> IN X A),
Bounded P.