Ordered Field of Rational Cuts is Isomorphic to Rational Numbers

Theorem
Let $\struct {\RR, +, \times, \le}$ denote the (totally) ordered field of rational cuts.

Let $\struct {\Q, +, \times, \le}$ denote the field of rational numbers.

Then $\struct {\RR, +, \times, \le}$ and $\struct {\Q, +, \times, \le}$ are isomorphic.