Parity Addition is Associative

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

The operation $+$ is associative:


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