Definition naturals_main_induction2a_Out : False := h2 step2a3.Definition naturals_main_induction2b_Out : False := step2_1 step2b1.Definition naturals_main_induction2c_Out : False := h2 step2c4.Definition naturals_main_induction2_Out : False := step2_4 k.Definition naturals_main_induction3a_Out : False := contra step3a2.Definition naturals_main_induction3bI_Out : False := h2 step3bI3.Definition naturals_main_induction3b_Out : False := contra step3b4.Definition naturals_main_induction3em_Out : False := step3_3 p.Definition naturals_main_induction3_Out : StrictSUB X N :=
  excluded_middle (StrictSUB X N) sublemma2.Definition naturals_main_induction_step_Out : naturals_main_ind_hyp N :=
  Build_naturals_main_ind_hyp N sublemma3 sublemma1.

Theorem naturals_main_induction_next : forall (A : Ens) (h : naturals_main_ind_hyp A), naturals_main_ind_hyp (Next A).