Lemma hzqrandplusineq ( p : hz ) ( x : hzneq 0 p ) ( a c : hz ) : hzleh 0 ( hzremaindermod p x ( hzremaindermod p x a + hzremaindermod p x c ) ) × hzlth ( hzremaindermod p x ( hzremaindermod p x a + hzremaindermod p x c ) ) ( nattohz ( hzabsval p ) ). Proof. intros. split. - apply hzleh0remaindermod. - apply hzlthremaindermodmod. Defined.
Require Import UniMath.PAdics.lemmas. Require Import UniMath.NumberSystems.Integers. Require Import UniMath.Foundations.Preamble.