Modulo Subtraction is Well-Defined

Corollary to Modulo Addition is Well-Defined
Let $m \in \Z$ be an integer.

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

The modulo subtraction 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 $a - x \equiv b - y \pmod m$.

Proof
We have:

The result follows from the fact that Modulo Addition is Well-Defined for all integers.