Rational Multiplication is Closed/Proof 2

Theorem
The operation of multiplication on the set of rational numbers $\Q$ is well-defined and closed:
 * $\forall x, y \in \Q: x \times y \in \Q$

Proof
From the definition of rational numbers, there exists four integers $p$, $q$, $r$, $s$, where:


 * $\displaystyle q \ne 0$
 * $\displaystyle s \ne 0$
 * $\displaystyle \frac p q = x$
 * $\displaystyle \frac r s = y$

We have that:


 * $\displaystyle p \times r \in \Z$
 * $\displaystyle q \times s \in \Z$

Since $q \ne 0$ and $s \ne 0$, we have that $q \times s \ne 0$.

Therefore, by the definition of rational numbers:


 * $\displaystyle x \times y = \frac {p \times r} {q \times s} \in \Q$.

Hence the result.