Parity Multiplication is Associative

Theorem
Let $R := \struct {\set {\text{even}, \text{odd} }, +, \times}$ be the parity ring.

The operation $\times$ is associative:


 * $\forall a, b, c \in R: \paren {a \times b} \times c = a \times \paren {b \times c}$