Conditional bracket isometry for stopped martingale increments ($\mathbb{E}[(M_{\tau }-M_{\sigma }{)}^2\mid {\mathcal{F}}_{\sigma }]=\mathbb{E}([M{]}_{\tau }-[M{]}_{\sigma }\mid {\mathcal{F}}_{\sigma })$)

For an L2-bounded continuous martingale and ordered possibly infinite stopping times $\sigma \le \tau$, interpret the terminal stopped value as $M_{\infty }$. The Doob L2 maximal inequality bounds the expected squared path supremum. Localization of $M^2-[M]$ then gives $\mathbb{E}[M{]}_{\infty }<\infty$, so this square-minus-bracket local martingale has integrable supremum and is uniformly integrable. Apply the optional stopping theorem to it and to $M$. Expanding $(M_{\tau }-M_{\sigma }{)}^2$ makes the cross term vanish conditionally, proving the identity. Bounded ${\mathcal{F}}_{\sigma }$-measurable factors can be multiplied into either side.