Conditional expectation with several conditions

Question

Let $ξ_1,ξ_2,...$ be independent identically distributed random variables with finite mathematical expectation, $S_n = ξ_1 + ... + ξ_n, n ∈ \mathbb{N}$. Prove that for all $n \in \mathbb{N}$ the equality $E(ξ_1|S_n,S_{n+1},...) = \frac{S_n}{n}$. I know that $E(\xi_1|S_n) = \frac{S_n}{n}$, but don't quite understand what to do in case, when we have several conditions $S_n, S_{n+1}, ...$. I would be very grateful for the hint.