So, as far as I understand you have that if $0\leq t_0 you know that $W_{t_k}-W_{t_{k-1}},k=\overline{1,n}$ are independent random variables (this I will denote by A). And now you are to prove that if $W_t-W_s$ is independent from $\mathcal F_s$(this I will denote by B). First, I will prove that B follows from A. Let $0. We know that $W_t-W_s$ is independent from $W_{t_k}-W_{t_{k-1}},k=\overline{1,n}$ (in the next I will denote this fact as $W_t-W_s\bigsqcup W_{t_k}-W_{t_{k-1}}$). By definition it means that $W_t-W_s\bigsqcup \sigma \{W_{t_k}-W_{t_{k-1}},k=\overline{1,n}\}=\sigma\{W_{t_k},k=\overline{1,n}\}$. Hence $W_t-W_s \bigsqcup\, \bigcup\limits_{0\leq q_1<\ldots. The set on the left is $\pi$-class (stable under finite intersection), hence (it is well known theorem about independence of sigma-algebras generated by independent $\pi$-classes are themselves independent) $W_t-W_s\bigsqcup \sigma \{ \bigcup\limits_{0\leq q_1<\ldots. The last fact, you can prove yourself. If you want to prove that B entails A, try to use characteristic functions. And the criteria of independence which uses c.f.. Note, that the fact you want to prove is true for much bigger class of stochastic processes. The reference here could be any normal lectures on Levy processes, for instance, ``Lectures on Levy processes'' by R.Schilling. Unfortunately, I could not find it by googling. If you want, I can send you the .pdf.