proposition: Lemma mult_le_compat x₁ y₁ x₂ y₂ : 0 ≤ x₁ -> 0 ≤ x₂ -> x₁ ≤ y₁ -> x₂ ≤ y₂ -> x₁ * x₂ ≤ y₁ * y₂.
proof: Proof. intros Ex₁ Ey₁ E1 E2. transitivity (y₁ * x₂). - apply (order_preserving_flip_nonneg (.*.) x₂);trivial. - apply (order_preserving_nonneg (.*.) y₁); [| trivial]. transitivity x₁;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