Suppose that * is a binary operation on a non-empty set S. Show that if (*,S) contains an identity e then e is unique.

Is this right?

e1 * x = x * e1 = x

x = e2

e1 * e2 = e2 * e1 = e2

then do for e2

e2 * x = x * e2 = x

x = e1

e2 * e1 = e1 * e2 = e1