Divided by Positive Element of Field of Quotients

Theorem
Let $\left({K, +, \circ}\right)$ be the quotient field of a totally ordered integral domain $\left({D, +, \circ, \le}\right)$.

Then:
 * $\forall z \in K: \exists x, y \in D: z = \dfrac x y, y \in D_+^*$

Proof
By definition:


 * $\forall z \in K: \exists x, y \in D: z = \dfrac x y, y \in D^*$

Suppose $z = x' / y'$ such that $y' \notin D_+^*$.

Then $y' < 0$ as $D$ is totally ordered.

Then:

If $y' < 0$, then $\left({- y'}\right) > 0$ from Properties of Ordered Ring $(4)$.

So all we need to do is set $x = -x', y = -y'$ and the result follows.