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

(**********)
Lemma last_0 : forall N : nat, N >= 2 -> 0 < last N.
intros; unfold last in |- *; case (N_classic N); intro.