left; apply (Even_intro 0 0); auto with arith.
intros y hrec.
elim hrec.
intro eveny.
right; elim eveny.
intros q hy; apply (Odd_intro (S y) q).
elim hy.
elim plus_n_Sm.
auto with arith.
intro oddy; left.
elim oddy.
intros q hy; apply (Even_intro (S y) (S q)).
rewrite hy.
rewrite plus_n_Sm.
elim (mult_comm (S q) 2).
elim (mult_comm q 2).
elim plus_comm.
auto with arith.
Qed.

Lemma Odd_odd : forall n : nat, Odd n -> odd n.
intros.