I think I got the definition of the conditional expectation now, but I'm still having some problems with actual calculations...

Let $(X,Y,Z)$ be a real gaussian vector. X and Y centered and independent. I need to show that

$\mathbb{E}(Z|X,Y)=\mathbb{E}(Z|X)+\mathbb{E}(Z|Y)-\mathbb{E}(Z)$

what sounds completely intuitive for me, and it works for all the special cases where Z is either independend from X and/or Y. But I don't know how to proof that. At the proof using the def I'm stuck at how the $\mathbb{1}_{A}$ for $A\in\sigma(X,Y)$ looks like.... If you have any good tips on how to calculate/proof such equations, pls tell me :)

EDIT: Ok, I guess if i can assume that $A=\{X\in B,Y\in C\}$ for $B,C$ Borel sets on $\mathbb{R}$ I can proof it. But is it sufficient to show def of the conditional exp. only for a generator of the sigma algebra?

Thanks in advise