Parity Multiplication is Associative

Theorem
Let $R := \left({\left\{ {\text{even}, \text{odd} }\right\}, +, \times}\right)$ be the parity ring.

The operation $\times$ is associative:


 * $\forall a, b, c \in R: \left({a \times b}\right) \times c = a \times \left({b \times c}\right)$