Modulo Addition is Well-Defined

Theorem
Let $z \in \R$.

Let $\R_z$ be the set of all residue classes modulo $z$ of $\R$.

The modulo addition operation on $\R_z$, defined by the rule:
 * $\left[\!\left[{a}\right]\!\right]_z +_z \left[\!\left[{b}\right]\!\right]_z = \left[\!\left[{a + b}\right]\!\right]_z$

is a well-defined operation.

Corollary
It follows that: