+lemma minus_le_trans_sn: ∀x1,x2. x1 ≤ x2 → ∀x. x1-x ≤ x2.
+/2 width=3 by transitive_le/ qed.
+
+lemma le_plus_to_minus_l: ∀a,b,c. a + b ≤ c → b ≤ c-a.
+/2 width=1 by le_plus_to_minus_r/
+qed-.
+
+lemma le_plus_to_minus_comm: ∀n,m,p. n ≤ p+m → n-p ≤ m.
+/2 width=1 by le_plus_to_minus/ qed-.
+