Here's my proof. I would like to know whether it is right or wrong and if the latter where it needs to be corrected:
(a,b)=[(a,b)][(cf+ed,df)]=[(acf+aed,bdf)].
We want to show its equivalence to:
[(ac,bd)]+[(ae,bf)]=[(acbf+abde,$b^2$df)]
By the equivalence relation [(acf+aed,bdf)]~[(acbf+abde,$b^2$df)] means the following:
(acf+aed)$b^2$df=(bdf)(acbf+abde)=acf$b^2$df+aed$b^2$df=bdfacbf+bdfabde
Since a field is commutative ac$f^2$$b^2$d+ae$d^2b^2$f=ac$f^2b^2$d+ae$d^2b^2$f
Please let me know what can be changed.