Inverse for Real Multiplication

Theorem
Each element $$x$$ of the set of non-zero real numbers $$\R^*$$ has an inverse element $$\frac 1 x$$ under the operation of real number multiplication:
 * $$\forall x \in \R^*: \exists \frac 1 x \in \R^*: x \times \frac 1 x = 1 = \frac 1 x \times x$$

Proof
We have:

$$ $$

Similarly for $$\frac 1 {\left[\!\left[{\left \langle {x_n} \right \rangle}\right]\!\right]} \times \left[\!\left[{\left \langle {x_n} \right \rangle}\right]\!\right]$$.

So the inverse of $$x \in \left({\R^*, \times}\right)$$ is $$x^{-1} = \frac 1 x$$