intros; elim inser_trans_lt with 0 k (first N + last N) N.
intro y; elim y; intros; apply (last_n N k (prop_N N) H1 H2 prop_alpha).
intro y; elim y; intros.
elim (absurd1 N k H1 H2); intros; auto.
auto.
Qed.

(**********)
Lemma tech_first_aux :
forall N a : nat,
0 < a ->
a < N ->
((forall b : nat,
0 < b -> b < N -> (frac_part_n_alpha a <= frac_part_n_alpha b)%R) <->
a = first N).