## Theorem

Let $m \in \Z$ be an integer.

Then addition modulo $m$ has inverses:

For each element $\eqclass x m \in \Z_m$, there exists the element $\eqclass {-x} m \in \Z_m$ with the property:

$\eqclass x m +_m \eqclass {-x} m = \eqclass 0 m = \eqclass {-x} m +_m \eqclass x m$

where $\Z_m$ is the set of integers modulo $m$.

That is:

$\forall a \in \Z: a + \paren {-a} \equiv 0 \equiv \paren {-a} + a \pmod m$

## Proof

 $\ds \eqclass x m +_m \eqclass {-x} m$ $=$ $\ds \eqclass {x + \paren {-x} } m$ Definition of Modulo Addition $\ds$ $=$ $\ds \eqclass 0 m$ $\ds$ $=$ $\ds \eqclass {\paren {-x} + x} m$ $\ds$ $=$ $\ds \eqclass {-x} m +_m \eqclass x m$ Definition of Modulo Addition

As $-x$ is a perfectly good integer, $\eqclass {-x} m \in \Z_m$, whatever $x$ may be.

$\blacksquare$