Lemma maxNonnegativeReals_lt' : ∏ x y z : NonnegativeReals, z < maxNonnegativeReals x y -> z < x ∨ z < y. 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.
Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.OrderTheory.Lattice.Lattice. Require Import UniMath.RealNumbers.Prelim. Require Import UniMath.RealNumbers.Sets. Require Import UniMath.RealNumbers.NonnegativeRationals.