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 $x = \dfrac p q \in K$.

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

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

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

So:

... and the job is done.

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