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.

(* Specifications: *)
Theorem naturals_main_induction_next : forall (A : Ens) (h : naturals_main_ind_hyp A), naturals_main_ind_hyp (Next A).
Theorem naturals_main_induction_next_1 : naturals_induction_nexts naturals_main_ind_hyp.
Theorem naturals_main_th1 : forall (B : Ens) (i : IN B Naturals), naturals_main_ind_hyp B.
Theorem naturals_main_th2 : forall (B : Ens) (i : IN B Naturals), in_then_strictsub B.
Theorem naturals_main_th2_rewrite : forall (B : Ens) (i : IN B Naturals) (A : Ens) (j : IN A B), StrictSUB A B.
Theorem naturals_main_th3 : forall (B : Ens) (i : IN B Naturals), not_in_itself B.
Theorem naturals_main_th3_rewrite : forall (B : Ens) (i : IN B Naturals), ~ IN B B.