Definition naturals_inductionA1_Out : IN N S := step5 N step1_2 step1_1.Definition naturals_inductionA_Out : P B := step3 B step9.Definition naturals_induction_nexts (P : Ens -> Prop) :=
  forall (A : Ens) (h : IN A Naturals) (k : P A), P (Next A).

Theorem naturals_inductionA : forall (P : Ens -> Prop) (IH_zero : P Zero) (IH_nexts : naturals_induction_nexts P) (B : Ens) (p : IN B Naturals), P B.