Unicity of the identity element

Suppose that $e'$ is another identity element. In particular, this means that, for $x:=e$:
$$e*e'=x*e'=x=e .$$But since $e$ is an identity element, by definition, for $y:=e'$:
$$e*e'=e*y=y=e'.$$So we can conclude that $e=e'$.