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

Proof
By definition:


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

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 an Ordered Ring no. 4.

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