# Identity of Group is Unique/Proof 2

## Theorem

Let $\struct {G, \circ}$ be a group which has an identity element $e \in G$.

Then $e$ is unique.

## Proof

Let $e$ and $f$ both be identity elements of a group $\struct {G, \circ}$.

Then:

 $\displaystyle e$ $=$ $\displaystyle e \circ f$ $f$ is an identity $\displaystyle$ $=$ $\displaystyle f$ $e$ is an identity

So $e = f$ and there is only one identity after all.

$\blacksquare$