Any tips on how to prove the following for any Borel functions $f$, $A\in \mathcal F_{t_1}$ and Brownian motion $B$, $t_1<t_2<T$?
$$E((B(t_2)-B(t_1))f(B(T))1_A)=E(\int_{t_1}^{t_2}\frac{B(T)-B(u)}{T-u}duf(B(T))1_A)$$
The conditional probability distribution of $B(T)$ at time $t$ is of course normal with expectation $B(t)$ and variance $T-t$. I also have computed the quadratic variation of $E(f(B(T))|B(t))$ if it may help in the task.