If we let $S_n$ denote a simple symmetric random walk, how do I compute $E_0(S_m+S_{n-m}|S_n=y)$ using the fact that $P_0(S_m=x|S_n=y)=P_0(S_{n-m}=y-x|S_n=y)$, where $n>m>0$ and $n$ and $y$ have the same parity?
I know that we can separate $E_0(S_m+S_{n-m}|S_n=y)$ using linearity of expectation, and we get that $E_0(S_m|S_n=y)$ = $mE_0(\xi_1|S_n=y)$, where $\xi_i$ are the increments of the random walk.
However, I'm not sure how to conclude that $E_0(\xi_1|S_n=y)=\frac{y}{n}$, or how the above probabilities come into play.