Proof fun (P : Ens -> Prop) (b : Bounded P) => REPLACEMENT_pr1 P b.
Definition SUBPROP (P Q : Ens -> Prop) := forall X : Ens, P X -> Q X.
Section Bounded_subprops1.
Variable P : Ens -> Prop.
Variable Q : Ens -> Prop.
Hypothesis s : SUBPROP P Q.
Variable A : Ens.
Hypothesis h : Bounded_By A Q.
Section boundedsubprops1.
Variable X : Ens.
Hypothesis k : P X.
Let step1 : Q X. Proof s X k.
Definition boundedsubprops1_Out : IN X A := h X step1.
End boundedsubprops1.
Definition Bounded_subprops1_Out : Bounded_By A P := boundedsubprops1_Out.
End Bounded_subprops1.

Theorem Bounded_subprops_th1 :
forall (P Q : Ens -> Prop) (s : SUBPROP P Q) (A : Ens) (h : Bounded_By A Q),
Bounded_By A P.