proposition: Lemma NR_to_hr_lt : ∏ x y : NonnegativeReals × NonnegativeReals, pr1 x + pr2 y < pr1 y + pr2 x <-> hr_lt_rel (NR_to_hr x) (NR_to_hr y).
proof: Proof. intros x y. split. - intros H. apply hinhpr ; exists 0 ; simpl. apply plusNonnegativeReals_ltcompat_l, H. - apply hinhuniv ; intros H. apply_pr2 (plusNonnegativeReals_ltcompat_l (pr1 H)). exact (pr2 H). 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