Proof power5_Out.
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(**********     Naturals  part 1  ************)
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Definition Element_of_NN (Y : Ens) :=
forall (X : Ens) (k : Contains_NN X), IN Y X.
Section Zero_element_of_NN_proof.
Variable X : Ens.
Hypothesis k : Contains_NN X.
Definition Zero_element_of_NN_proof_Out : IN Zero X := Contains_NN_zero X k.
End Zero_element_of_NN_proof.

Theorem Zero_element_of_NN : Element_of_NN Zero.
Proof Zero_element_of_NN_proof_Out.