Inverse for Real Addition

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:

$$ $$

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 $$-x$$.