proposition: Lemma plus_lt_le_compat x₁ y₁ x₂ y₂ : x₁ < y₁ → x₂ ≤ y₂ → x₁ + x₂ < y₁ + y₂.
proof: Proof. intros E1 E2. apply lt_le_trans with (y₁ + x₂). now apply (strictly_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