Theory of Proportions - what is it?

[genericusrnme]

I've recently got back into studying maths in my own time and I've come across this (see attached image)
The middle line.. I put some numbers in there and those equalities do not hold..
I don't see why we can set everything there equal..
Could someone explain this to me or at least point me in the right direction?
I've tried googling theory of proportions but all I've managed to get was chemistry related stuff

I was having a good night of study until that came up as well :(

Thanks in advance

 

[bluskies]

Not sure I understand that step in your picture either, but usually I think the theory of proportions refers to when to ratios are in proportion to each other, you can set them to be equal. Here's something I found with a little searching:

http://answers.yahoo.com/question/index?qid=20111220134210AAxMboC