Lemma nat_nat_to_monoid1 {X : gr} (x : X) {n1 n2 m2 : nat} (e : n2 = m2) : nat_nat_to_monoid_fun x (make_dirprod n1 n2) = nat_nat_to_monoid_fun x (make_dirprod n1 m2). Proof. induction e. use idpath. Qed.
Require Import UniMath.MoreFoundations.PartA. Require Import UniMath.Algebra.Monoids. Require Import UniMath.Algebra.Groups.