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 Zgt_ge_plus_plus :
forall m n p q : Z, (m > n)%Z -> (p >= q)%Z -> (m + p > n + q)%Z.