= Solution
Conditional expectation is the <orthogonal projection> from $L^2(\mathcal F)$ onto $L^2(\mathcal G)$. The difference $X-\mathbb E[X\mid\mathcal G]$ is orthogonal to every $\mathcal G$-measurable square-integrable variable, including $\mathbb E[X\mid\mathcal G]-\mathbb E[X\mid\mathcal H]$. The <Pythagorean theorem in an inner-product space> applied to
$$
X-\mathbb E[X\mid\mathcal H]
=(X-\mathbb E[X\mid\mathcal G])
+(\mathbb E[X\mid\mathcal G]-\mathbb E[X\mid\mathcal H])
$$
gives the identity.
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