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). - apply (strictly_order_preserving (+ x₂));trivial. - apply (order_preserving (y₁ +));trivial. Qed.
imports: HoTT.Classes.theory.apartness HoTT.Classes.interfaces.abstract_algebra HoTT.Classes.interfaces.orders HoTT.Classes.theory.rings HoTT.Classes.orders.orders HoTT.Classes.orders.maps