induction x; intuition; simpl in *.
subst.
trivial.
assert (x * p + (y + p) = z).
omega.
apply IHx in H0.
rewrite H0.
rewrite plus_comm.
rewrite modNat_plus.
rewrite modNat_arg_eq.
rewrite plus_0_l.
trivial.
Qed.

(* Specifications: *)
Lemma modNat_mult : forall x (p : posnat),
modNat (x * p) p = 0.