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 Zle_reg_mult_l :
forall x y a : Z, (0 < a)%Z -> (x <= y)%Z -> (a * x <= a * y)%Z.