Null String is Identity Element for Concatenation Operator

Theorem
Let $\AA$ be an alphabet of symbols.

Let $\WW$ denote the set of words in $\AA$.

Let $\epsilon$ denote the null string.

Let $C: \WW \times \WW \to \WW$ denote the concatenation operator on $\WW$:
 * $\forall A, B \in \WW: \map C {A, B} := A B$

Then $\epsilon$ is the identity element for $C$.

Proof
As $\epsilon$ is the null string:


 * $\map C {\epsilon, A}$

Hence the result by definition of identity element.