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 NDivides_minus :
forall n m p : Z, Zdivides n p -> Zdivides m p -> Zdivides (n - m) p.