= Solution
As with <quadratic variation>, the increasing process must be normalized by $A_0=0$. Suppose $A,A'$ have this initial value and both $M^2-A$ and $M^2-A'$ are <continuous local martingales>. Since $M^2$ and these differences are continuous, both $A$ and $A'$ are continuous. Their difference is a <finite-variation process>, and
$$
A-A'=(M^2-A')-(M^2-A)
$$
is a <continuous local martingale>. The result proved in part (a) makes this difference constant. Its initial value is zero, so \b[$A$ and $A'$ are indistinguishable]. This is the <uniqueness of an increasing square compensator>; it uses no existence argument.
If no initial-value normalization is imposed, the literal uniqueness assertion is false: whenever $A$ works, $A+c$ works for every deterministic constant $c$, since subtracting a constant preserves the <local martingale> property. The exact unnormalized conclusion is $A_t-A'_t=A_0-A'_0$ for all $t$, up to <indistinguishability of stochastic processes>.
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