Lemma natlehsnminussmsn ( n m : nat ) ( p : natleh m n ) : natleh (sub ( S n ) ( S m ) ) ( S n ). Proof. revert m p. induction n. - intros m p. apply negnatgthtoleh. intro X. apply nopathsfalsetotrue. assumption. - intros m p. destruct m. + apply natlthtoleh. apply natlthnsn. + apply ( istransnatleh( m := S n ) ). * apply IHn. assumption. * apply natlthtoleh. apply natlthnsn. Defined.
Require Import UniMath.Foundations.PartA. Require Import UniMath.Foundations.Propositions. Require Import UniMath.Foundations.NaturalNumbers. Require Import UniMath.Algebra.RigsAndRings. Require Import UniMath.Algebra.Domains_and_Fields. Require Import UniMath.NumberSystems.Integers. Require Import UniMath.Algebra.Monoids.