Section Bounded_NN_proof2.
Variable Z : Ens.
Hypothesis w : Element_of_NN Z.
Definition Bounded_NN_proof2_Out : IN Z B := w B v.
End Bounded_NN_proof2.
Let step2 : Bounded_By B Element_of_NN := Bounded_NN_proof2_Out.
Definition Bounded_NN_proof_Out : False := step1 B step2.
End Bounded_NN_proof.
Proof Bounded_NN_proof_Out.
Proof Bounded_NN_pre.
Definition Naturals := REPLACEMENT Element_of_NN.
Proof REPLACEMENT_pr1 Element_of_NN Bounded_NN.
Proof REPLACEMENT_pr2 Element_of_NN Bounded_NN.
(* Now prove Contains_NN Naturals *)
(* prove the two inductive properties separately *)
Proof Naturals_pr1 Zero Zero_element_of_NN.

Theorem Nexts_in_Naturals :
forall (A : Ens) (h : IN A Naturals), IN (Next A) Naturals.