Lemma Dcuts_finite : ∏ X : Dcuts_set, ∏ r : NonnegativeRationals, neg (r ∈ X) -> ∏ n : NonnegativeRationals, n ∈ X -> n < r. Proof. intros X r Hr n Hn. apply notge_ltNonnegativeRationals ; intro Hn'. apply Hr. apply is_Dcuts_bot with n. - exact Hn. - exact Hn'. 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.