Modulo Addition has Inverses

Theorem
Let $m \in \R$ be a real number.

Then addition modulo $m$ has inverses:

For each element $\left[\!\left[{x}\right]\!\right]_m \in \R_m$, there exists the element $\left[\!\left[{-x}\right]\!\right]_m \in \R_m$ with the property:


 * $\left[\!\left[{x}\right]\!\right]_m +_m \left[\!\left[{-x}\right]\!\right]_m = \left[\!\left[{0}\right]\!\right]_m = \left[\!\left[{-x}\right]\!\right]_m +_m \left[\!\left[{x}\right]\!\right]_m$

where $\R_m$ is the set of residue classes modulo $m$.

That is:
 * $\forall a \in \R: a + \left({-a}\right) \equiv 0 \equiv \left({-a}\right) + a \pmod m$

Proof
Follows directly from the definition of addition modulo $m$:

As $-x$ is a perfectly good real number, $\left[\!\left[{-x}\right]\!\right]_m \in \R_m$, whatever it may be.