proposition: Lemma minusNonnegativeRationals_ltcompat_l' : ∏ x y z : NonnegativeRationals, x - z < y - z -> x < y.
proof: Proof. intros x y z Hlt. assert (Hyz : (z < y)%NRat). { apply_pr2 ispositive_minusNonnegativeRationals. apply istrans_le_lt_ltNonnegativeRationals with (x - z). { now apply isnonnegative_NonnegativeRationals. } exact Hlt. } generalize (isdecrel_leNonnegativeRationals x z) ; apply sumofmaps ; intro Hxz. - apply istrans_le_lt_ltNonnegativeRationals with (1 := Hxz). exact Hyz. - apply notge_ltNonnegativeRationals in Hxz ; apply lt_leNonnegativeRationals in Hxz. apply lt_leNonnegativeRationals in Hyz. rewrite <- (minusNonnegativeRationals_plus_r _ _ Hxz), <- (minusNonnegativeRationals_plus_r _ _ Hyz). apply plusNonnegativeRationals_ltcompat_r. exact Hlt. Qed.
imports: UniMath.MoreFoundations.Tactics UniMath.MoreFoundations.PartA UniMath.MoreFoundations.Sets UniMath.MoreFoundations.Orders UniMath.RealNumbers.Sets UniMath.RealNumbers.Fields UniMath.Algebra.DivisionRig UniMath.OrderTheory.Lattice.Lattice UniMath.RealNumbers.Prelim