+lemma minus_minus_comm: ∀a,b,c. a - b - c = a - c - b.
+/3 by monotonic_le_minus_l, le_to_le_to_eq/ qed.
+
+lemma minus_le_minus_minus_comm: ∀b,c,a. c ≤ b → a - (b - c) = a + c - b.
+#b #c #a #H >(plus_minus_m_m b c) in ⊢ (? ? ?%); //
+qed.
+
+lemma minus_minus_m_m: ∀m,n. n ≤ m → m - (m - n) = n.
+/2 width=1/ qed.
+
+(* Stilll more atomic conclusion ********************************************)
+
+(* le *)
+
+lemma le_fwd_plus_plus_ge: ∀m1,m2. m2 ≤ m1 → ∀n1,n2. m1 + n1 ≤ m2 + n2 → n1 ≤ n2.
+#m1 #m2 #H #n1 #n2 >commutative_plus
+#H elim (le_inv_plus_l … H) -H >commutative_plus <minus_le_minus_minus_comm //
+#H #_ @(transitive_le … H) /2 width=1/
+qed-.
+