Rational Multiplication is Closed/Proof 2

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


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

We have that:


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

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

Therefore, by the definition of rational numbers:


 * $x \times y = \dfrac {p \times r} {q \times s} \in \Q$

Hence the result.