Rule for multiplying rational numbers

The rule for multiplying rational numbers is this:

$\space\frac{a}{b}\cdot \frac{c}{d}=\frac{ac}{bd}$

Can the rule be proven or is it meant to be taken as a given?

Edit: Where $b\neq 0$ and $d\neq 0$.

One knows that $a/b$ may be defined as the number $x$ such that $bx=a$, and that $c/d$ may be defined as the number $y$ such that $dy=c$.

If one wants the multiplication on these objects to be associative and commutative as it is on the integers, one should ask that $ac=(bx)(dy)=(bd)(xy)$ hence that the object $xy$ fits the definition of $(ac)/(bd)$.

$$\rm x\: =\: \dfrac{a}b,\ \ y\: =\: \dfrac{c}d\ \ \Rightarrow\ \ b\ x\: =\: a,\:\ d\ y\: =\: c\ \ \Rightarrow\ \ b\:d\ x\:y\: =\: a\:c\ \ \Rightarrow\ \ x\:y\: =\: \dfrac{a\:c}{b\:d}$$

An approach to make it more clear might be to seperate each rational into a product of its parts ie. $\frac {a}{b}= \frac{a}{1} \cdot \frac{1}{b}$ and $\frac {c}{d}= \frac{c}{1} \cdot \frac{1}{d}$ then use the commutative property to group the "numerator" fractions and the "denominator" fractions seperately: $\frac{a}{b}\cdot \frac{c}{d}=(\frac{a}{1} \cdot \frac{c}{1})\cdot (\frac{1}{b} \cdot \frac{1}{d})=\frac{ac}{1}\cdot\frac{1}{bd}=\frac{ac}{bd}$.