This is probably a standard question, apologies for that in advance.
Let $(\Omega, {\cal F}, {\cal F}_t,\mathbb{P})$ be a filtered probability space on which a Wiener process $\{W(\omega,t)\}_{\omega \in \Omega, t \geq 0}$ can be modelled. The latter process is adapted to the filtration ${\cal F}_t$. Let $\tau:\Omega \to \mathbb{R}^+$ be a stopping time. How does one precisely prove that the process $\{W_\tau(\omega,t)\}_{\omega \in \Omega, t \geq 0}$ defined by $$W_\tau(\omega,t)=W(\omega,\,t+\tau(\omega))-W(\omega,\,\tau(\omega))$$ is again a Wiener process?