Real Numbers form Ordered Field

Theorem
The set of real numbers $\R$ forms a totally ordered field under addition and multiplication: $\struct {\R, +, \times, \le}$.

Proof
From Real Numbers form Field, we have that $\struct {\R, +, \times}$ forms a field.

Then we have that Ordering on Real Numbers is Total Ordering.

Hence $\struct {\R, +, \times, \le}$ is a totally ordered field.