Proof in_iff_strictsub_Zero_Out.
Section not_in_itself_Zero.
Hypothesis h : IN Zero Zero.
Definition not_in_itself_Zero_Out : False := EmptySet_th1 Zero h.
End not_in_itself_Zero.

(* Specifications: *)
Theorem zero_not_in_itself : not_in_itself Zero.
Proof not_in_itself_Zero_Out.
Theorem naturals_main_ind_for_Zero : naturals_main_ind_hyp Zero.
Proof
Build_naturals_main_ind_hyp Zero in_then_strictsub_Zero zero_not_in_itself.