Modulo Multiplication is Commutative

Theorem
Multiplication modulo $m$ is commutative:


 * $$\forall \left[\!\left[{x}\right]\!\right]_m, \left[\!\left[{y}\right]\!\right]_m \in \Z_m: \left[\!\left[{x}\right]\!\right]_m \times_m \left[\!\left[{y}\right]\!\right]_m = \left[\!\left[{y}\right]\!\right]_m \times_m \left[\!\left[{x}\right]\!\right]_m$$.

Proof
Follows directly from the definition of multiplication modulo $m$:

$$ $$ $$