Existence of Unique Inverse Element for Multiplication of Cuts

Theorem
Let $0^*$ be the rational cut associated with the (rational) number $0$:
 * $0^* = \set {r \in \Q: r < 0}$

Let $\alpha$ be a cut such that $\alpha \ne 0^*$.

Then for every cut $\beta$, there exists a unique cut $\gamma$ such that:
 * $\alpha \gamma = \beta$

where $\alpha \gamma$ denotes the operation of product of $\alpha$ and $\gamma$.

In this context, $\gamma$ can be expressed as $\beta / \alpha$.