## Theorem

Each element $x$ of the set of real numbers $\R$ has an inverse element $-x$ under the operation of real number addition:

$\forall x \in \R: \exists -x \in \R: x + \left({-x}\right) = 0 = \left({-x}\right) + x$

## Proof

We have:

 $\displaystyle \left[\!\left[{\left \langle {x_n} \right \rangle}\right]\!\right] + \left({-\left[\!\left[{\left \langle {x_n} \right \rangle}\right]\!\right]}\right)$ $=$ $\displaystyle \left[\!\left[{\left \langle {x_n - x_n} \right \rangle}\right]\!\right]$ $\displaystyle$ $=$ $\displaystyle \left[\!\left[{\left \langle {0_n} \right \rangle}\right]\!\right]$

Similarly for $\left({-\left[\!\left[{\left \langle {x_n} \right \rangle}\right]\!\right]}\right) + \left[\!\left[{\left \langle {x_n} \right \rangle}\right]\!\right]$.

Thus the inverse of $x \in \left({\R, +}\right)$ is $-x$.

$\blacksquare$