x₁ y₁ x₂ y₂ : 0 < x₁ -> 0 < x₂ -> x₁ < y₁ -> x₂ < y₂ -> x₁ * x₂ < y₁ * y₂. 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.
Require Import