intros; generalize (tech_suc_M N n H H0); intro; red in |- *; intro; apply H1;
split with k; auto with arith real.
Qed.

(* Specifications: *)
(**********)
Lemma eq_after_M_N1 :
forall N n : nat, 0 < n -> n < N - first N -> after (M N) n = after N n.