proposition: Lemma plus_le_compat x₁ y₁ x₂ y₂ : x₁ ≤ y₁ → x₂ ≤ y₂ → x₁ + x₂ ≤ y₁ + y₂.
proof: Proof. intros E1 E2. transitivity (y₁ + x₂). now apply (order_preserving (+ x₂)). now apply (order_preserving (y₁ +)). Qed.
imports: Coq.setoid_ring.Ring MathClasses.interfaces.abstract_algebra MathClasses.interfaces.orders MathClasses.theory.rings MathClasses.orders.orders MathClasses.orders.maps