Let $(B_t)$ be a standard Brownian motion and $\{ \mathcal{F}_t \}$ be the filtration generated by the Brownian motion.
For a stopping time $\tau$, we know that $\{B_{\tau + t} - B_{\tau}\}_{t \geq 0}$ is a Brownian motion independent of $\{ \mathcal{F}^{+}_{\tau} \}$.
For a fixed $a>0$, let $\tau'$ be defined by $$ \tau' := \inf \{ t \geq 0 : B_{\tau + t} - B_{\tau} = a \}.$$
I don't understand why $\tau'$ is also independent of $\{ \mathcal{F}^{+}_{\tau} \}$. Any ideas?