generalize noetherian_course_of_values.
unfold adjoint in |- *; unfold universal in |- *; unfold sub in |- *; intros.
unfold explicit_noetherian in |- *; unfold universal in |- *;
unfold sub in |- *; intros.
apply (H A R H0 A0 H1 x x).
auto.
Qed.

Lemma noetherian_induction1 :
forall (A : Set) (R : A -> A -> Prop),
explicit_noetherian _ R ->
forall (x : A) (P : A -> Prop),
(forall y : A, (forall z : A, R y z -> P z) -> P y) -> P x.