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 fraction_lt_trans :
forall a b c d e f : Z,
(0 < b)%Z ->
(0 < d)%Z ->
(0 < f)%Z -> (a * d < c * b)%Z -> (c * f < e * d)%Z -> (a * f < e * b)%Z.