Lemma NonnegativeReals_dense : ∏ x y : NonnegativeReals, x < y -> ∃ r : NonnegativeRationals, x < NonnegativeRationals_to_NonnegativeReals r × NonnegativeRationals_to_NonnegativeReals r < y. Proof. intros x y. apply hinhuniv ; intros q. generalize (is_Dcuts_open y (pr1 q) (pr2 (pr2 q))). apply hinhfun ; intros r. exists (pr1 r) ; split ; apply hinhpr. - exists (pr1 q) ; split. + exact (pr1 (pr2 q)). + exact (pr2 (pr2 r)). - exists (pr1 r) ; split. + exact (isirrefl_ltNonnegativeRationals _). + exact (pr1 (pr2 r)). Qed.
Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.Notations. Require Import UniMath.RealNumbers.Sets. Require Import UniMath.OrderTheory.Lattice.Lattice. Require Import UniMath.RealNumbers.Prelim. Require Import UniMath.RealNumbers.NonnegativeRationals.