proposition: Lemma cut_minus_plus_toggle1 x y z : x ≤ y → z ≤ y → (y ∸ x) + (x ∸ z) = (y ∸ z) + (z ∸ x).
proof: Proof. intros. destruct (total (≤) x z). rewrite (cut_minus_0 x z), cut_minus_le_trans by easy. ring. rewrite (cut_minus_0 z x), cut_minus_le_trans by easy. ring. Qed.
imports: Coq.setoid_ring.Ring MathClasses.interfaces.abstract_algebra MathClasses.interfaces.additional_operations MathClasses.interfaces.orders MathClasses.orders.minmax