Chapter 5.3, Problem 1E

Prove that the multiplication defined 5.24 is a binary operation on $Q.$

$$

Lemma 5.24 Addition and Multiplication in $Q$

The following rules define binary operations on $Q.$

Addition in $Q$ is defined by

$\left[a,b\right]+\left[c,d\right]=\left[ad+bc,bd\right],$

and multiplication in $Q$ is defined by

$\left[a,b\right]·\left[c,d\right]=\left[ac,bd\right].$