intros m n p Z1; case (Z_eq_dec n 0); intros Z2.
rewrite Z2; unfold Zquotient in |- *; case (m * p)%Z; case m; simpl in |- *;
auto.
case (ZquotientProp m n); auto; intros r (H1, (H2, H3)).
apply sym_equal; apply ZquotientUnique with (r := (r * p)%Z);
auto with zarith.
pattern m at 1 in |- *; rewrite H1; ring.
rewrite Zmult_assoc.
repeat rewrite (fun x => Zabs_Zmult x p); auto with zarith.
repeat rewrite Zabs_Zmult; auto with zarith.
apply Zmult_gt_0_lt_compat_r; auto with zarith.
apply Zlt_gt; generalize Z1; case p; simpl in |- *;
try (intros H4; case H4; auto; fail); unfold Zlt in |- *;
simpl in |- *; auto; intros; red in |- *; intros;
discriminate.
Qed.

Theorem ZDivides_add :
forall n m p : Z, Zdivides n p -> Zdivides m p -> Zdivides (n + m) p.