Definition iteres_ult_const_def_0 (A : Set) (p : prechain A) : Prop :=
  forall (f : Map A -> Map A) (x : Map A) (n : nat),
  p = iteres A f x n ->
  non_dist_chain A p ->
  exists q : nat, S q <= n /\ power (Map A) f x q = power (Map A) f x (S q).

Lemma iteres_lower_fix_point : forall (A : Set) (T : mEnsemble A) (r : mRelation A) (f : Map A -> Map A) (e : Map A) (n m : nat), mini A r T e -> def_ok_app A T f -> increasing_app A r f -> bounded_sas_chain A T r m -> m <= n -> lower_fix_point A T r f (power (Map A) f e n).