User:Jshflynn/Language Product is Associative

Theorem
Let $\Sigma$ be an alphabet.

Let $U$, $V$ and $W$ be languages over $\Sigma$.

Then:


 * $U \circ_L \left({V \circ_L W}\right) = \left({U \circ_L V}\right) \circ_L W$

That is, language product is associative.

Proof
Let $U$, $V$ and $W$ be languages over $\Sigma$.

Then:

Hence the result