## Theorem

$\forall x, y, z \in \Z: x + y \pmod m = y + x \pmod m$

## Proof

From the definition of modulo addition, this is also written:

$\forall m \in \Z: \forall \eqclass x m, \eqclass y m \in \Z_m: \eqclass x m +_m \eqclass y m = \eqclass y m +_m \eqclass x m$

Hence:

 $\displaystyle \eqclass x m +_m \eqclass y m$ $=$ $\displaystyle \eqclass {x + y} m$ Definition of Modulo Addition $\displaystyle$ $=$ $\displaystyle \eqclass {y + x} m$ Commutative Law of Addition $\displaystyle$ $=$ $\displaystyle \eqclass y m +_m \eqclass x m$ Definition of Modulo Addition

$\blacksquare$