Modulo Addition is Commutative

Theorem
Modulo addition is commutative:


 * $\forall x, y, z \in \R: x + y \left({\bmod\,z}\right) = y + x \left({\bmod\,z}\right)$

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


 * $\forall z \in \R: \forall \left[\!\left[{x}\right]\!\right]_z, \left[\!\left[{y}\right]\!\right]_z \in \R_z: \left[\!\left[{x}\right]\!\right]_z +_z \left[\!\left[{y}\right]\!\right]_z = \left[\!\left[{y}\right]\!\right]_z +_z \left[\!\left[{x}\right]\!\right]_z$

Hence: