Multiplication of Positive Cuts preserves Ordering

Theorem
Let $0^*$ denote the rational cut associated with the (rational) number $0$.

Let $\alpha$, $\beta$ and $\gamma$ be cuts such that:


 * $0^* < \alpha < \beta$
 * $0^* < \gamma$

where $<$ denotes the strict ordering on cuts.

Then
 * $\alpha \gamma < \beta \gamma$

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