What Simon said appears to support your view, but I wonder if he oversimplified the situation for the sake of brevity.
Because if you look at Betfair's corporate website:
http://corporate.betfair.com/corporate/company-history.html
you see that:
So unless CS/LCG have actually taken over Tradefair, they are separate companies.
Just that, as a "white label partner", CS/LCG runs Betfair's SB operation (known as Tradefair').
Betfair's corporate address is not the same as LCG/CS. I can only think that the reason that CS/LCG's address is shown on Tradefair's website is that CS/LCG run the SB operation for TF; i.e. it's an operational address, rather than a corporate address for Tradefair.
(Also the registered address of LCG).
From the Tradefair website:
Well tricky corporate stuff that I don't pretend to fully understand, but perhaps LCG own the trading name/brand Tradefair, but they are only doing it on behalf of Betfair, which is a separate company, under this partnership agreement.
It's all a bit hazy really, but what I think supports me in my view is that LCG's website talks about CS and then them aquiring Prospreads (Furesbetting). It does not mention Tradefair by name, just refers to having opened 2 SB platforms. One of these could be Tradefair, but it would be as a "white label partner" and not as a wholly owned subsidiary, like CS or LCG. That's my theory, anyway.