Identity is Unique/Proof 1

Proof
Suppose $e_1$ and $e_2$ are both identity elements of $\left({S, \circ}\right)$.

Then by the definition of identity element:
 * $\forall s \in S: s \circ e_1 = s = e_2 \circ s$

Then:
 * $e_1 = e_2 \circ e_1 = e_2$

So:
 * $e_1 = e_2$

and there is only one identity element after all.