Addition of Cuts is Commutative

Theorem
Let $\alpha$ and $\beta$ be cuts.

Let the operation of $\alpha + \beta$ be the sum of $\alpha$ and $\beta$.

Then:
 * $\alpha + \beta = \beta + \alpha$

Proof
$\alpha + \beta$ is the set of all rational numbers of the form $p + q$ such that $p \in \alpha$ and $q \in \beta$.

Similarly, $\beta + \alpha$ is the set of all rational numbers of the form $q + p$ such that $p \in \alpha$ and $q \in \beta$.

From Rational Addition is Commutative we have that:
 * $p + q = q + p$

The result follows.