Modulo Addition is Well-Defined

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.

That is:
 * If $a \equiv b \pmod m$ and $x \equiv y \pmod m$, then $a + x \equiv b + y \pmod m$.

Corollary
It follows that: