Lemma Dcuts_of_Dcuts_bot : ∏ (x : NonnegativeRationals), Dcuts_of_Dcuts_val x -> ∏ y : NonnegativeRationals, (y <= x)%NRat -> Dcuts_of_Dcuts_val y. Proof. intros r Xr n Xn. revert Xr ; apply hinhfun ; intros X. exists (pr1 X) ; split. - exact (pr1 (pr2 X)). - apply is_Dcuts_bot with r. + exact (pr2 (pr2 X)). + exact Xn. 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.