proposition: Lemma maxNonnegativeReals_lt' : ∏ x y z : NonnegativeReals, z < maxNonnegativeReals x y -> z < x ∨ z < y.
proof: Proof. intros x y z. intros H. generalize (iscotrans_ltNonnegativeReals _ x _ H). apply hinhfun. apply sumofmaps ; intros Hx. - now left. - right. rewrite <- (maxNonnegativeReals_carac_r x y). + apply H. + apply notlt_leNonnegativeReals ; intros Hy. apply (isirrefl_ltNonnegativeReals (maxNonnegativeReals x y)). apply maxNonnegativeReals_lt. * exact Hx. * now apply istrans_ltNonnegativeReals with x. 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