intros; generalize (tech_fp_alp_irr n m); tauto.
Qed.

(**********)
Lemma contradiction1 :
forall N k : nat,
N <= k ->
k < first N + last N ->
(frac_part_n_alpha k < frac_part_n_alpha (first N))%R \/
(frac_part_n_alpha k > frac_part_n_alpha (last N))%R -> False.