Conditional expectation proof question

[oyth94]

Here is a proof question: For two random variables X and Y, we can define E(X|Y) to be the function of Y that satisfies E(Xg(X)) = E(E(X|Y)g(Y)) for any function g. Using this definition show that E(X₁ + X₂|Y) = E(X₁|Y) + E(X₂|Y)

So what I did was I plugged into X = X₁ + X₂
E(E(X₁ + X₂)|Y)g(Y))
= E(X₁g(Y)) + E(X₂g(Y))
= E(E(X₁|y)g(Y) + E(X₂|Y)g(Y))
= E(g(Y) [E(X₁|Y) + E(X₂|Y)]

am I on the right track? what do I do after that?

 

[tarantella]

Are the $y$ supposed to be $Y$?