Parity Multiplication is Commutative/Proof 1

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

The operation $\times$ is commutative:


 * $\forall a, b \in R: a \times b = b \times a$

Proof
From Isomorphism between Ring of Integers Modulo 2 and Parity Ring:
 * $\left({\left\{ {\text{even}, \text{odd} }\right\}, +, \times}\right)$ is isomorphic with $\left({\Z_2, +_2, \times_2}\right)$

the ring of integers modulo $2$.

The result follows from:
 * Modulo Multiplication is Commutative

and:
 * Isomorphism Preserves Associativity.