proposition: Lemma mult_lt_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 (strictly_order_preserving_flip_pos (.*.) x₂);trivial. - apply (strictly_order_preserving_pos (.*.) 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