Lemma invNonnegativeRationals_lecompat : ∏ x y : NonnegativeRationals, 0 < x -> x <= y -> / y <= / x. Proof. intros x y Hx0 Hxy. assert (Hy0 : 0 < y). { apply istrans_lt_le_ltNonnegativeRationals with x. { exact Hx0. } exact Hxy. } apply (multNonnegativeRationals_lecompat_l' x). - exact Hx0. - rewrite isrinv_NonnegativeRationals. + apply (multNonnegativeRationals_lecompat_r' y). * exact Hy0. * rewrite isassoc_multNonnegativeRationals, islinv_NonnegativeRationals, islunit_oneNonnegativeRationals, isrunit_oneNonnegativeRationals. ** exact Hxy. ** exact Hy0. + exact Hx0. Qed.
Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.PartA. Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.RealNumbers.Sets. Require Import UniMath.RealNumbers.Fields. Require Import UniMath.OrderTheory.Lattice.Lattice. Require Import UniMath.RealNumbers.Prelim.