I have the following problem:
Let $B_t$ a standard Brownian motion and define $$T_a:=\inf\{t>0:B_t=a\}$$ Define $$\overline{B}_t=B_{1+t}-B_1$$ I need to show that $$\overline{T}_{-B_1}=B^2_1\overline{T}_1$$
I am able to prove that $\overline{B}_t$ is a Brownian motion independent of $B_1$ and that $T_a=a^2T_1$ in law (just using scaling property). I don't know how to prove that the same result is true for $a=-B_1$ that is random. Can someone help me? I need a rigorous proof.