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.

Theorem Nexts_element_of_NN :
forall (A : Ens) (h : Element_of_NN A), Element_of_NN (Next A).