User:Jshflynn/Empty Word is Two-sided Identity

Theorem
Let $\Sigma$ be an alphabet, $x$ be a word over $\Sigma$, $\lambda$ be the empty word over $\Sigma$ and $\circ$ denote concatenation. Then $x \circ \lambda = x$ and $\lambda \circ x = x$. That is, $\lambda$ is a two-sided identity element of concatenation.

Proof
Follows immediately from the definition of concatenation with the empty word.