intros n m p H' H'0.
case H'; intros z1 Hz1.
case H'0; intros z2 Hz2.
exists (z1 * z2)%Z; rewrite Hz1; rewrite Hz2; ring.
Qed.

Theorem ZdividesLessPow :
forall (n : Z) (m p : nat),
m <= p -> Zdivides (Zpower_nat n p) (Zpower_nat n m).