\begin{definition}[Definition:Multiplication of Positive Cuts]
Let $0^*$ denote the rational cut associated with the (rational) number $0$.
Let $\alpha$ and $\beta$ be positive cuts, that is, cuts such that $\alpha \ge 0^*$ and $\beta \ge 0^*$, where $\ge$ denotes the ordering on cuts.
Let the operation of '''multiplication''' be defined on $\alpha$ and $\beta$ as:
:$\gamma := \alpha \beta$
where $\gamma$ is the set of all rational numbers $r$ such that either:
:$r < 0$
or
:$\exists p \in \alpha, q \in \beta: r = p q$
where $p \ge 0$ and $q \ge 0$.
In this context, $\gamma$ is known as the '''product of $\alpha$ and $\beta$'''.
\end{definition}