(**********)
Definition M (N : nat) := first N + last N.

Lemma tech_suc_N : forall N n : nat, 0 < n -> n < N -> forall k : nat, 0 < k -> k < N -> (frac_part_n_alpha n < frac_part_n_alpha k)%R -> frac_part_n_alpha k <> frac_part_n_alpha (after N n) -> (frac_part_n_alpha (after N n) < frac_part_n_alpha k)%R.