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_increasing_chain :
forall (A : Set) (T : mEnsemble A) (r : mRelation A)
(f : Map A -> Map A) (x : Map A) (n : nat),
T x ->
def_ok_app A T f ->
increasing_app A r f -> r x (f x) -> chain A T r (iteres A f x n).