absurd (N >= 2).
elim s; intro y; rewrite y; unfold ge in |- *; auto with arith real.
exact H.
case (exist_last N g); intros; elim a; intros; elim H1; intros;
auto with arith real.
Qed.

(**********)
Lemma inter1 : forall N : nat, N >= 2 -> max (first N) (last N) < N.
intros; apply max_case2.