Lemma Dcuts_minus_open : Dcuts_def_open Dcuts_minus_val. Proof. intros r. apply hinhuniv ; intros x. generalize (X_open _ (pr1 (pr2 x))). apply hinhfun ; intros x'. exists (r + (pr1 x' - pr1 x)) ; split. - apply hinhpr ; exists (pr1 x') ; split. + exact (pr1 (pr2 x')). + intros y Yy. pattern (pr1 x') at 2 ; rewrite <- (minusNonnegativeRationals_plus_r _ _ (lt_leNonnegativeRationals _ _ (pr2 (pr2 x')))). rewrite (iscomm_plusNonnegativeRationals r), isassoc_plusNonnegativeRationals. apply plusNonnegativeRationals_ltcompat_l. now apply (pr2 (pr2 x)). - apply plusNonnegativeRationals_lt_r. apply ispositive_minusNonnegativeRationals. exact (pr2 (pr2 x')). 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.