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For square-integrable random variables, $\mathrm{Cov}(X,Z\mid \mathcal{G})=\mathbb{E}[XZ\mid \mathcal{G}]-\mathbb{E}[X\mid \mathcal{G}]\mathbb{E}[Z\mid \mathcal{G}]$. It is a $\mathcal{G}$-measurable random variable. Replacing $X$ by its conditional expectation given a larger sigma-field leaves this covariance unchanged when $Z$ is measurable with respect to that larger sigma-field. This observation underlies recovery of a martingale-transform integrand by conditional covariance.

Write $\Delta M_n=M_n-M_{n-1}$. Conditional on ${\mathcal{F}}_{n-1}$, the variables $X_{n-1}$, $M_{n-1}$, and $A_n$ are known, while $\mathbb{E}[\Delta M_n\mid {\mathcal{F}}_{n-1}]=0$. HenceThe denominator is positive almost surely, so the recovery of a martingale-transform integrand by conditional covariance gives