We discussed the following version of the Kunita-Watanabe identity in a recent lecture on stochastic analysis, and I don't quite understand how the proof works.
Let $X, Y$ be semimartingales and $H$ an $X$- and $\langle X, Y \rangle$-integrable process. Then one has:
\begin{equation} \langle H \cdot X, Y \rangle = H \cdot \langle X, Y \rangle. \end{equation}
In order to prove this, it suffices to consider local martingales for $X$ and $Y$ (this I understand). We then showed that
$$ ((H \cdot X)_t Y_t - (H \cdot \langle X,Y \rangle )_t)_{t \geq 0} $$
is a martingale. This is supposed to imply the desired result, since the right side of the equation above is a process of locally bounded variation started in 0.
My problem is that I don't understand why this implies the result. Why is this enough?