How do I prove this relationship between a positive, bounded random variable and its conditional expectation?

Question

Consider a random variable X such that $0 \le X \le C$, for some positive C and $\mathbb{A} \subseteq \mathbb{B}$ are $\sigma$-algebras. X is defined on $\left( {B,\mathbb{B},{\mathbb{P}}} \right)$. Let $Y = {\rm{E}}\left[ {{\rm{X}}\left| \mathbb{A} \right.} \right]$. I am looking to calculate ${\rm{E}}\left[ {\left( {X - Y} \right)Y} \right]$. Intuition tells me that it should be zero. It is obvious when the expectation is unconditional. Any hints how I should approach this?

Answer 1

$E[(X-Y)Y]=E(E((X-Y)Y|\mathbb A))$ and $E((X-Y)Y|\mathbb A)=Y(E(X|\mathbb A)-Y)=Y(Y-Y)=0$.