Lemma NR_to_hr_opp : ∏ x : NonnegativeReals × NonnegativeReals, (- NR_to_hr x)%ring = NR_to_hr (pr2 x ,, pr1 x). Proof. intros x. unfold ringinv1, grinv_is ; simpl. unfold abgrdiffinv. unfold NR_to_hr. apply (setquotfuncomm (binopeqrelabgrdiff (rigaddabmonoid NonnegativeReals)) (binopeqrelabgrdiff (rigaddabmonoid NonnegativeReals))). Qed.
Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.OrderTheory.Lattice.Lattice. Require Import UniMath.RealNumbers.Prelim. Require Import UniMath.RealNumbers.Sets. Require Import UniMath.RealNumbers.NonnegativeRationals.