Theorem 3.13

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Statement:

(Uniqueness of Identity Element)

A binary structure $\langle S,* \rangle$ has at most one identity element. That is, if there is an identity element, it is unique.

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Proof:
Proceeding in the standard way to show uniqueness, suppose that both $e$ and $\bar{e}$ are elements of $S$ serving as identity elements. We let them compete with each other. Regarding $e$ as an identity element, we must have $e*\bar{e}=\bar{e}$. However, regarding $\bar{e}$ as an identity element, we must have $e*\bar{e}=e$. We thus obtain $e=\bar{e}$, showing that an identity element must be unique.