(* prove some properties of naturals by induction *)



Definition in_then_strictsub (N : Ens) :=
  forall X : Ens, IN X N -> StrictSUB X N.Definition not_in_itself (X : Ens) := ~ IN X X.Record naturals_main_ind_hyp (N : Ens) : Prop := 
  {naturals_main_ind1 : in_then_strictsub N;
   naturals_main_ind2 : not_in_itself N}.Record naturals_main_ind_hyp (N : Ens) : Prop := 
  {naturals_main_ind1 : in_then_strictsub N;
   naturals_main_ind2 : not_in_itself N}.Record naturals_main_ind_hyp (N : Ens) : Prop := 
  {naturals_main_ind1 : in_then_strictsub N;
   naturals_main_ind2 : not_in_itself N}.Definition strictsub_next1_Out : False := h step4.Definition strictsub_next_Out : StrictSUB A N :=
  Build_StrictSUB A N step1 step5.

Theorem some_things_strictsub_their_nexts : forall (A : Ens) (h : not_in_itself A), StrictSUB A (Next A).