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

Lemma tech_suc_M : forall N n : nat, N - first N <= n -> n < last N -> (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 (n + first N - last N))%R) -> False.