Consider continuous semimartingales $X,Y,Z$, i.e.
$X=X(0) + A + M$,
where $A$ is a process of finite variation and M a local martingale, both starting in $0$
I want to proof: $\langle X + Y , Z \rangle = \langle X,Z\rangle + \langle Y, Z \rangle$
Since $\langle X,B \rangle =0 $ if $B$ is of finite variation, the left hand side reads as:
$$ <M_X+M_Y, M_Z>. $$
$M_K$ denotes the local martingale component in the doob meyer decomposition of each process $K$.
The right hand side gets: $\langle M_X,M_Z \rangle $ + $\langle M_Y,M_Z \rangle $
So I only have to verify this property for the martingale component.
Is that reasonable? How can I go on from here?