proposition: Lemma hzabsvalandlth ( n m : hz ) ( p : hzleh 0 n ) ( p' : hzlth n m ) : natlth ( hzabsval n ) ( hzabsval m ).
proof: Proof. intros. destruct ( natlthorgeh ( hzabsval n ) ( hzabsval m ) ) as [ h | k ]. - assumption. - apply fromempty. apply ( isirreflhzlth m ). apply ( hzlehlthtrans _ n _ ). + rewrite <- ( hzabsvalgeh0 ). * rewrite <- ( hzabsvalgeh0 p ). apply nattohzandleh. apply k. * apply hzgthtogeh. apply ( hzgthgehtrans _ n _ ); assumption. + assumption. Defined.
imports: UniMath.Tactics.EnsureStructuredProofs UniMath.Foundations.PartA UniMath.Foundations.Propositions UniMath.Foundations.NaturalNumbers UniMath.Algebra.RigsAndRings UniMath.Algebra.Domains_and_Fields UniMath.NumberSystems.Integers UniMath.Algebra.Monoids