intros; elim inser_trans_lt with 0 k (first N + last N) N.
intro y; elim y; intros; apply (first_n N k (prop_N N) H1 H2 prop_alpha).
intro y; elim y; intros.
elim (absurd1 N k H1 H2); intros.
unfold Rge in H3; unfold Rle in |- *; unfold Rgt in H3; elim H3; intro.
left; assumption.
right; auto.
auto.
Qed.

(**********)
Lemma absurd_last :
forall N k : nat,
0 < k ->
k < first N + last N ->
(frac_part_n_alpha k <= frac_part_n_alpha (last N))%R.