= Localized isometry proof of stochastic-integral quadratic variation
{title2=$[H\mathbin\cdot M]=H^2\mathbin\cdot[M]$}
For a <continuous local martingale> and a locally bounded <predictable process>, stop at a localizing time, an integrand bound, a time bound, and level bounds for the centered martingale and its bracket. The stopped martingale is bounded and its bracket and integrand are deterministically bounded. The <Itô isometry>, applied to $\mathbf1_F\mathbf1_{(s,t]}H$ for $F\in\mathcal F_s$, gives the conditional second-moment identity. Expanding the square proves that $(H\cdot M)^2-H^2\cdot[M]$ is a martingale after stopping. <Uniqueness of an increasing square compensator> identifies the bracket, and increasing stopping times patch the identities into the displayed local result.
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