Let $E[z | v]$ denote the conditional expectation of the random variable $z$ conditional on the random variable $v$.
Assume that $s = x + \epsilon$, where $\epsilon$ is a random normal variable independent from $x$, and $E[\epsilon | s] = 0$. Thus $E[x | s] = s$. Also, assume $y = a_0 + a_1 x + a_2 e$, where $e$ is an independent (from all other random variables) random normal variable as well. Now I'd like to show that $E[x | s, y] = b_0 + b_1 s + b_2 y$ for some constants $b_0$, $b_1$, and $b_2$. It seems like this linearity should be obvious, but I'm not sure how to prove it. Do I need additional assumptions? How do I prove this?