Proving $E(XE(Y ))=E(X)E(Y )$

Question

Why is $E(XE(Y))=E(X)E(Y)$ where $X$ and $Y$ are random variables

I have not found any rule that can solve this.

Answer 1

$E(Y)$ is a constant with respect to $X$; you're using the linearity of expectation.

Answer 2

$E(X\underbrace{E(Y)}_{\alpha}) = E(\alpha X) = \alpha E(X) = E(X) E(Y) $