Section Nexts_element_of_NN_proof.
Variable A : Ens.
Hypothesis h : Element_of_NN A.
Variable X : Ens.
Hypothesis k : Contains_NN X.
Let N := Next A.
Let j : IN A X := h X k.
Definition Nexts_element_of_NN_proof_Out : IN N X :=
Contains_NN_nexts X k A j.
End Nexts_element_of_NN_proof.
Proof Nexts_element_of_NN_proof_Out.
Definition Set_containing_NN := CHOICE Contains_NN.

Theorem Set_containing_NN_pr : Contains_NN Set_containing_NN.