In Stochastic Equations in Infinite Dimensions (Second Edition) on page 109, the authors state the following:
If $\eta_0,\ldots,\eta_{k-1}$ are random variables with finite second moments and $\mathcal G_0,\ldots,\mathcal G_{k-1}$ an increasing sequence of $\sigma$-fields such that $\eta_i$ are measurable with respect to $\mathcal G_j$, $0\le j\le k-1$, then $$\mathbb E\left(\sum_{j=0}^{k-1}\eta_j-\sum_{j=0}^{k-1}\mathbb E\left(\eta_j\mid\mathcal G_j\right)\right)^2=\sum_{j=0}^{k-1}\left(\mathbb E(\eta_j)^2-\mathbb E(\mathbb E\left(\eta_j\mid\mathcal G_j\right)^2\right)\;.\tag 1$$
I've taken the statement literally; preserving any grammatical, logical and notational error (you can find at least one error of each category).
It's an odd question, but I need to ask it: How can we phrase the statement correctly?
At a first glance, I thought they would mean the following: Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space, $I:=\left\{0,\ldots,k-1\right\}$, $(\mathcal G_i)_{i\in I}$ be a filtration of $\mathcal A$ and $(\eta_i)_{i\in I}\subseteq\mathcal L^2(\operatorname P)$ be $\mathcal F$-adapted. Then it holds $(1)$ (with the parenthesis mismatch being corrected). However, that would be a trivial statement, cause $\operatorname E\left[\eta_i\mid\mathcal G_i\right]$ would be equal to $\eta_i$ and hence both sides of $(1)$ would be $0$.
So, what do they mean?