Identity is Unique/Proof 1

Theorem

Let $\left({S, \circ}\right)$ be an algebraic structure that has an identity element $e \in S$.

Then $e$ is unique.

Proof

Suppose $e_1$ and $e_2$ are both identity elements of $\struct {S, \circ}$.

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.

$\blacksquare$