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 (order_preserving_flip_nonneg (.*.) x₂);trivial. - apply (order_preserving_nonneg (.*.) y₁); [| trivial]. transitivity x₁;trivial. Qed.
Require Import