Modulo Addition is Well-Defined/Proof 2

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.

Proof
The equivalence class $\left[\!\left[{a}\right]\!\right]_z$ is defined as:
 * $\left[\!\left[{a}\right]\!\right]_z = \left\{{x \in \R: x = a + k z: k \in \Z}\right\}$

That is, the set of all real numbers which differ from $a$ by an integer multiple of $z$.

Thus the notation for addition of two residue classes modulo $z$ is not usually $\left[\!\left[{a}\right]\!\right]_z +_z \left[\!\left[{b}\right]\!\right]_z$.

What is more normally seen is $a + b \pmod z$.

Using this notation: