Lemma istrans_Rle_lt : ∏ x y z : Reals, x <= y -> y < z -> x < z. Proof. intros x y z Hxy Hyz. generalize (iscotrans_Rlt _ x _ Hyz). apply hinhuniv. apply sumofmaps ; intros H. - apply fromempty. revert H. apply_pr2 notRlt_Rle. exact Hxy. - exact H. 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.