= Solution
Use two standard results: for every <continuous local martingale> $U$, there is a unique continuous <adapted> increasing <quadratic variation> $[U]$, starting at zero, such that $U^2-U_0^2-[U]$ is a <local martingale>; and a <continuous finite-variation local martingale is constant>.
Define the \b[<quadratic covariation>] by the <polarization identity>
$$
\boxed{[M,N]=\frac14\bigl([M+N]-[M-N]\bigr).}
$$
This is continuous, <adapted>, starts at zero, and is of <finite variation> on compact time intervals. Since $(M+N)^2-(M-N)^2=4MN$, subtraction of the two quadratic-variation identities shows that $MN-M_0N_0-[M,N]$ is a <continuous local martingale>. In the zero-start convention this is exactly $MN-[M,N]\in\mathcal M_{c,\mathrm{loc}}$.
If $A$ is another continuous <adapted> <finite-variation process> with $A_0=0$ and the same <local martingale> property, then $A-[M,N]$ is both a <continuous local martingale> and of <finite variation>. It is constant, and its initial value is zero, so it vanishes. Uniqueness is up to <indistinguishability of stochastic processes>.
\b[The normalization $[M,N]_0=0$ is essential.] If arbitrary nonzero initial constants are allowed in the <martingale> class and this normalization is omitted, adding a constant to the bracket preserves the claimed property. The usual definition of <quadratic covariation> always includes the zero initial value.
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