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.