 Localized isometry proof of stochastic-integral quadratic variation

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 $1_F{1}_{(s,t]}H$ 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.