Let $B_t$ denote a standard Brownian Motion, let $T$ denote the first passage time to the level $z>0$ and define $$X_t:= B_t 1_{t<T}+(2B_T-B_t)1_{t \geq T}$$ In the book "Stochastic Integration and Differential Equations" the following claim is made about $X_t$: "Since B and -B have the same distribution, it follows from Theorem 32 that X is also a standard Brownian Motion"
Now Theorem 32 states that for a Levy process ($U_t$) and a a.s. finite stopping time $T$ if we define $Y_t:= U_{T+t}-U_{T}$, then $Y_t$ is again a Levy process adapted to $\mathcal{H}_t:=\mathcal{F}_{T+t}$. Further $Y$ is independent of $\mathcal{F}_T$ and has the same distribution as $U$.
My question: I do not really see how to apply this Theorem to arrive at the above conclusion regarding the Brownian motion. For instance, I think I need a claim about $B_t-B_T$ on $\{T\geq t\}$, however the theorem would only provide a statement regarding $B_{s+T}-B_T$. Note that the latter describes a constant time difference, i.e. $s+T-T=s$ is not random, whereas I seem to need something applying to the distribution referring to a random time difference, i.e. $t-T$ is random.
I would be glad if someone could explain in detail the required steps to arrive at the conclusion via an application of the stated Theorem (I do not need an alternative derivation).
A rigorous proof is given using the fact needed by OP - specialized to Brownian motion in $\mathbb{R}^d$ - in Schilling, Partzsch and Boettcher's Brownian motion Th. 6.12. p. 72.
I noticed the edition I referenced had significant typos. This third edition (Th.6.13. p. 73) seems to have them corrected.