Lemma tech_after : forall N n m : nat, 0 < n -> n < N -> 0 <= m -> m < N -> (frac_part_n_alpha n < frac_part_n_alpha m)%R -> ((exists 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 m)%R) -> False) -> m = after N n.