Modulo Subtraction is Well-Defined

Corollary to Modulo Addition is Well-Defined
Let $z \in \R$.

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

The modulo subtraction 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
We have:

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