Lemma hzgthsntogeh ( n m : hz ) ( p : hzgth ( n + 1 ) m ) : hzgeh n m. Proof. intros. set ( c := hzgthchoice2 ( n + 1 ) m ). destruct c as [ h | k ]. - exact p. - assert ( hzgth n m ) as a by exact ( hzgthandplusrinv n m 1 h ). apply hzgthtogeh. exact a. - rewrite ( hzplusrcan n m 1 k ). apply isreflhzgeh. 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.