# Identity is only Idempotent Cancellable Element

## Theorem

Let $e_S$ is the identity of an algebraic structure $\struct {S, \circ}$.

Then $e_S$ is the only cancellable element of $\struct {S, \circ}$ that is idempotent.

## Proof

By Identity Element is Idempotent, $e_S$ is idempotent.

Let $x$ be a cancellable idempotent element of $\struct {S, \circ}$.

 $\ds x \circ x$ $=$ $\ds x$ $x$ is idempotent $\ds$ $=$ $\ds x \circ e_S$ Definition of Identity Element

So $x \circ x = x \circ e_S$.

But because $x$ is also by hypothesis cancellable, it follows that $x = e_S$.

$\blacksquare$