Absolute Value of Cut is Greater Than or Equal To Zero Cut

Definition
Let $\alpha$ be a cut.

Let $\size \alpha$ denote the absolute value of $\alpha$.

Then:
 * $\size \alpha \ge 0^*$

where:
 * $0^*$ denotes the rational cut associated with the (rational) number $0$
 * $\ge$ denotes the ordering on cuts.

Proof
Let $\alpha \ge 0^*$.

Then by definition $\size \alpha = \alpha \ge 0^*$.

Let $\alpha < 0^*$.

Then:
 * $\exists \beta: \beta + \alpha = 0^*$

Thus:
 * $\alpha = -\beta$

and it follows that $\beta > 0^*$.

The result follows.