proposition: 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.
proof: Proof. intros. apply Z.gt_lt. apply Zgt_mult_reg_absorb_l with d. Flip. apply Zgt_trans with (c * b * f)%Z. replace (d * (e * b))%Z with (b * (e * d))%Z. replace (c * b * f)%Z with (b * (c * f))%Z. apply Z.lt_gt. apply Zlt_reg_mult_l. Flip. assumption. ring. ring. replace (c * b * f)%Z with (f * (c * b))%Z. replace (d * (a * f))%Z with (f * (a * d))%Z. apply Z.lt_gt. apply Zlt_reg_mult_l. Flip. assumption. ring. ring. Qed.
imports: Coq(Lia)