In order to prove that $\langle M,N \rangle$ is the only process which is continuous and has bounded variation such that \begin{align} M_tN_t - \langle M,N \rangle_t \end{align} is a continuous martingale, I try to apply the Doob-Meyer decomposition on the right hand side of the useful equation \begin{align} M_tN_t = \frac{1}{2}\bigg( (M_t+N_t)^2 - M_t^2 - N_t^2 \bigg). \end{align}
So, Doob-Meyer are stating that for $(X_t)$, a cadlag submartingale with $X_0=0$, $\exists!$ predictable process $(A_t)_t$ such that $A_0=0$ and \begin{align} \text{$M_t = X_t - A_t\ \ $ is a uniformly integrable process,} \end{align} i.e. $(X_t - A_t)_t$ is a martingale.
However, for me it is not yet clear how to complete the proof. Can somebody help me?