Definition Bounded_th1_proof_Out : EXISTS (fun A : Ens => Bounded_By A P) :=
  EXISTS_th1 (fun B : Ens => Bounded_By B P) A step1.

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