intros; split; intro.
elim (tech_first_aux M (first N) (first_0 N (prop_N N))).
intros; clear H2; auto.
rewrite H; apply (lt_plus (first N) (last N) (last_0 N (prop_N N))).
elim
(tech_first_aux M (first M) (first_0 M (prop_N M)) (first_N M (prop_N M))).
intros; clear H1; rewrite H0; auto.
Qed.

(**********)
Lemma first_eq_M_N :
forall N M : nat, M = first N + last N -> first N = first M.