I am studying the construction of the field of fractions from an integral domain. The multiplication operation on this field work as follows

$$(a,b)(c,d)=(ac,bd)$$

Also, $$(a,b)=(a_1,b_1)Leftrightarrow ab_1=ba_1$$

To show that the multiplication operation is well-defined in my book appears the following:

Suppose that $(a_1,b_1)=(a,b)$ and $(c_1,d_1)=(c,d)$. We have $(a_1,b_1)(c_1,d_1)=(a_1c_1,b_1d_1)$ and begin{eqnarray*} acb_1 d_1 -a_1c_1bd&=&(acb_1d_1-a_1cbd_1)+(a_1cbd_1-a_1c_1bd) \

&=& cd_1(ab_1-a_1b)+a_1b(cd_1-c_1d)\

&=& cd_1(0+a_1b(0)\

&=& 0end{eqnarray*}

My problem is that I do not know exactly where the term “$a_1cbd_1$” comes from. I have not been able to understand the proof at all for this detail, could you please help me?