Real Numbers form Ordered Field

Theorem
The set of real numbers $\R$ forms an 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.

From Ordering Properties of Real Numbers we have that $\struct {\R, +, \times, \le}$ is a ordered field.