Equal Elements of Field of Quotients

Theorem
Let $$\left({D, +, \circ}\right)$$ be an integral domain.

Let $$\left({K, +, \circ}\right)$$ be the quotient field of $$\left({D, +, \circ}\right)$$.

Let $$x = \frac p q \in K$$.

Then:
 * $$\forall k \in \Z^*: x = \frac {p \circ k} {q \circ k}$$

Proof
We have that the quotient field $$\left({K, +, \circ}\right)$$ of an integral domain is its inverse completion.

Thus we have
 * $$\forall x_1, x_2 \in D, y_1, y_2 \in D^*: \frac {x_1} {y_1} = \frac {x_2} {y_2} \iff x_1 \circ y_2 = x_2 \circ y_1$$

So:

$$ $$

... and the job is done.

Note that in order for $$\frac {p \circ k} {q \circ k}$$ to be defined, $$q \circ k \ne 0_D$$, that is, $$k \ne 0_D$$.