proposition: Lemma NonnegativeRationals_to_NonnegativeReals_le : ∏ x y : NonnegativeRationals, (x <= y)%NRat <-> NonnegativeRationals_to_NonnegativeReals x <= NonnegativeRationals_to_NonnegativeReals y.
proof: Proof. intros x y ; split. - intros H. apply Dcuts_nlt_ge. intro H0. revert H. apply_pr2 notge_ltNonnegativeRationals. apply_pr2 NonnegativeRationals_to_NonnegativeReals_lt. exact H0. - intros H. apply notlt_geNonnegativeRationals. intros H0. revert H. apply Dcuts_gt_nle. apply NonnegativeRationals_to_NonnegativeReals_lt. exact H0. Qed.
imports: UniMath.MoreFoundations.Orders UniMath.MoreFoundations.Tactics UniMath.MoreFoundations.Notations UniMath.RealNumbers.Sets UniMath.Algebra.ConstructiveStructures UniMath.OrderTheory.Lattice.Lattice UniMath.RealNumbers.Prelim UniMath.RealNumbers.NonnegativeRationals