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).