intros; elim (eq_first_M_N N M H); intros; clear H1; apply H0.
intros; rewrite H in H2; apply (absurd_first N b H1 H2).
Qed.

(**********)
Lemma tech_last_aux :
forall N a : nat,
0 < a ->
a < N ->
((forall b : nat,
0 < b -> b < N -> (frac_part_n_alpha b <= frac_part_n_alpha a)%R) <->
a = last N).