Inverse for Real Multiplication

Theorem
Each element $x$ of the set of non-zero real numbers $\R_{\ne 0}$ has an inverse element $\dfrac 1 x$ under the operation of real number multiplication:
 * $\forall x \in \R_{\ne 0}: \exists \dfrac 1 x \in \R_{\ne 0}: x \times \dfrac 1 x = 1 = \dfrac 1 x \times x$

Proof
We have:

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

So the inverse of $x \in \left({\R_{\ne 0}, \times}\right)$ is $x^{-1} = \dfrac 1 x$.