Parity Addition is Commutative/Proof 1

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

the ring of integers modulo $2$.

The result follows from:
 * Modulo Addition is Commutative

and:
 * Isomorphism Preserves Commutativity.