Lemma Dcuts_apzero_notempty : ∏ (x : Dcuts), (0%NRat ∈ x) <-> x ≠ 0. Proof. intros x ; split. - now apply Dcuts_notempty_notzero. - apply sumofmaps. + apply hinhuniv ; intros r. apply fromempty. now apply (Dcuts_zero_empty _ (pr2 (pr2 r))). + apply hinhuniv ; intros r. apply is_Dcuts_bot with (1 := pr2 (pr2 r)). now apply isnonnegative_NonnegativeRationals. 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.