Strict Positivity Property induces Total Ordering

Theorem
Let $\left({D, +, \times}\right)$ be an ordered integral domain where $P$ is the positivity property.

Then there exists an ordering $\le$ on $\left({D, +, \times}\right)$ which is compatible with the ring structure of $\left({D, +, \times}\right)$.

It follows that $\left({D, +, \times, \le}\right)$ is a totally ordered ring.