Lemma between_ltNonnegativeRationals : ∏ x y : NonnegativeRationals, x < y -> ∑ t : NonnegativeRationals, x < t × t < y. Proof. intros x y H. set (z := hqlth_between (pr1 x) (pr1 y) H). assert (Hz : hqleh 0%hq (pr1 z)). { apply istranshqleh with (pr1 x). { now apply (pr2 x). } apply (hqlthtoleh (pr1 x) (pr1 z)), (pr1 (pr2 z)). } exists (hq_to_hnnq_set _ Hz). exact (pr2 z). 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.