proposition: Lemma hr_plus_lecompat_l : ∏ x y z : hr_commring, hr_le_rel y z <-> hr_le_rel (y + x)%ring (z + x)%ring.
proof: Proof. intros x y z ; split ; intro Hle. - apply hr_notlt_le. apply_pr2_in hr_notlt_le Hle. intro Hlt ; apply Hle. apply_pr2 (hr_plus_ltcompat_l x). exact Hlt. - apply hr_notlt_le. apply_pr2_in hr_notlt_le Hle. intro Hlt ; apply Hle. apply hr_plus_ltcompat_l. exact Hlt. 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