∀ {G : Type u_1} {G' : Type u_2} [inst : AddGroup G] [inst_1 : AddGroup G'] {g : G}
  (hg : ∀ (x : G), x ∈ AddSubgroup.zmultiples g) {g' : G'} (hg' : ∀ (x : G'), x ∈ AddSubgroup.zmultiples g')
  (h : addOrderOf g = addOrderOf g'),
  (addMonoidHomOfForallMemZMultiples hg' ⋯).comp (addMonoidHomOfForallMemZMultiples hg ⋯) = AddMonoidHom.id G