proposition: Lemma hr_to_NRnegpos_zero : ∏ x : hr_commring, 0 < hr_to_NRneg x -> hr_to_NRpos x = 0.
proof: Proof. intros x. rewrite <- (hr_to_NR_bij x). generalize (hr_to_NR x) ; clear x ; intros x. rewrite hr_to_NRpos_NR_to_hr, hr_to_NRneg_NR_to_hr. intros Hx. apply minusNonnegativeReals_eq_zero. apply lt_leNonnegativeReals. apply_pr2 ispositive_minusNonnegativeReals. exact Hx. Qed.
imports: UniMath.Algebra.Groups UniMath.Foundations.Preamble UniMath.MoreFoundations.Orders UniMath.MoreFoundations.Tactics UniMath.OrderTheory.Lattice.Lattice UniMath.RealNumbers.Prelim UniMath.RealNumbers.Sets UniMath.RealNumbers.NonnegativeRationals UniMath.RealNumbers.NonnegativeReals