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.

(* Specifications: *)
Lemma binomial_def3 : forall n : nat, binomial n n = 1.
Lemma binomial_def4 :
forall n k : nat,
k < n -> binomial (S n) (S k) = binomial n (S k) + binomial n k.
Lemma binomial_fact :
forall m n : nat,
binomial (n + m) n * (factorial n * factorial m) = factorial (n + m).
Theorem exp_Pascal :
forall a b n : nat,
power (a + b) n =
sum_nm n 0 (fun k : nat => binomial n k * (power a (n - k) * power b k)).