intros m; case m; simpl in |- *; auto.
intros H'; inversion H'; auto.
intros n0 H' m; case m; simpl in |- *; auto.
intros H'0; inversion H'0; auto.
intros n1 H'0.
rewrite H'; auto with arith.
rewrite H'; auto with arith.
Qed.

Lemma binomial_def4 :
forall n k : nat,
k < n -> binomial (S n) (S k) = binomial n (S k) + binomial n k.