intros.
apply Zgt_trans with (m := (n + p)%Z).
rewrite Zplus_comm.
rewrite Zplus_comm with (n := n).
apply Zplus_gt_compat_l.
assumption.
apply Zplus_gt_compat_l.
assumption.
Qed.

Lemma Zlt_mult_mult :
forall a b c d : Z,
(0 < a)%Z -> (0 < d)%Z -> (a < b)%Z -> (c < d)%Z -> (a * c < b * d)%Z.