intros; rewrite H in H2; apply (absurd_last N b H1 H2).
Qed.

(**********)
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.