I know that for a continuous local martinagle $M$ we have $\langle M\rangle^\tau = \langle M^\tau\rangle$ for any stopping times. Now if $M,N$ are two local martingale I know that there exists again a process $\langle M,N\rangle$ (continuous, adapted and of bounded variation) such that $MN-\langle M,N\rangle$ is a continuous local martingale. Furthermore I know $\langle M,N\rangle = \frac{1}{4}(\langle M+N\rangle -\langle M-N\rangle)$. Now I want to prove:
$\langle M,N\rangle^\tau=\langle M^\tau,N^\tau\rangle = \langle M^\tau,N\rangle = \langle M,N^\tau\rangle$
The first equation is clear, since: $\langle M,N\rangle^\tau=\frac{1}{4}(\langle M+N\rangle -\langle M-N\rangle)^\tau=\frac{1}{4}(\langle M+N\rangle^\tau -\langle M-N\rangle^\tau)=\frac{1}{4}(\langle M^\tau+N^\tau\rangle -\langle M^\tau-N^\tau\rangle)=\langle M^\tau,N^\tau\rangle $ Where I used the definition of $\langle M,N\rangle$ and the property $\langle M^\tau\rangle = \langle M\rangle^\tau$. Unfortunately I am not able to prove the second equality. The third will be then the same, I guess. Thank you for your help
hulik