+(* derived *)
+lemma q_lt_canc_plus_r:
+ ∀x,y,z:Q.x + z < y + z → x < y.
+intros; rewrite < (q_plus_OQ y); rewrite < (q_plus_minus z);
+rewrite > q_elim_minus; rewrite > q_plus_assoc;
+apply q_lt_plus; rewrite > q_elim_opp; assumption;
+qed.
+
+lemma q_lt_inj_plus_r:
+ ∀x,y,z:Q.x < y → x + z < y + z.
+intros; apply (q_lt_canc_plus_r ?? (Qopp z));
+do 2 (rewrite < q_plus_assoc;rewrite < q_elim_minus);
+rewrite > q_plus_minus;
+do 2 rewrite > q_plus_OQ; assumption;
+qed.
+
+lemma q_le_inj_plus_r:
+ ∀x,y,z:Q.x ≤ y → x + z ≤ y + z.
+intros;cases (q_le_cases ?? H);
+[1: rewrite > H1; apply q_eq_to_le; reflexivity;
+|2: apply q_lt_to_le; apply q_lt_inj_plus_r; assumption;]
+qed.
+
+lemma q_le_canc_plus_r:
+ ∀x,y,z:Q.x + z ≤ y + z → x ≤ y.
+intros; lapply (q_le_inj_plus_r ?? (Qopp z) H) as H1;
+do 2 rewrite < q_plus_assoc in H1;
+rewrite < q_elim_minus in H1; rewrite > q_plus_minus in H1;
+do 2 rewrite > q_plus_OQ in H1; assumption;
+qed.