Modulo Addition is Well-Defined/Proof 2

Theorem
Let $m \in \Z$ be an integer.

Let $\Z_m$ be the set of integers modulo $m$.

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

is a well-defined operation.

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

That is, the set of all integers which differ from $a$ by an integer multiple of $m$.

Thus the notation for addition of two set of integers modulo $m$ is not usually $\left[\!\left[{a}\right]\!\right]_m +_m \left[\!\left[{b}\right]\!\right]_m$.

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

Using this notation: