半群满足结合律，又因为V满足交换律，则 fa(a ◦ x)= a ◦ (a ◦ x)=(a ◦ a) ◦ x (因为满足结合律)=a ◦ x (因为a是幂等元) ①fa(a)= a ◦ a =a (因为a是幂等元)并且fa(x)=a ◦ x则fa(a) ◦ fa(x) = a ◦(a ◦ x) =a ◦ x （由 ①式）则 fa(a ◦ x) = fa(a) ◦ fa(x)即f是V上的自同态。
