intro; elim s; intros x y; elim y; intros; elim H0; intros; elim H2; intros;
clear s y H0 H2 H4; generalize (last_n N x (prop_N N) H H1 prop_alpha);
intro;
generalize
(Rgt_not_le (frac_part_n_alpha x) (frac_part_n_alpha (last N)) H3);
intro; elimtype False; auto.
Qed.

(**********)
Lemma prop_M :
forall N M : nat,
M = first N + last N ->
(frac_part_n_alpha M < frac_part_n_alpha (first N))%R \/
(frac_part_n_alpha M > frac_part_n_alpha (last N))%R.