Let $P = (\Omega, \mathcal{F}, \mathbb{P})$ be a D1159: Probability space such that

(i)	$\mathcal{G} \subseteq \mathcal{F}$ is a D470: Subsigma-algebra of $\mathcal{F}$ on $\Omega$
(ii)	$X, Y : \Omega \to \mathbb{R}$ are each a D3161: Random real number on $P$
(iii)	\begin{equation} \mathbb{E} |X|^2, \mathbb{E} |Y|^2 < \infty \end{equation}

By definition, $\mathbb{E}(X \mid \mathcal{G})$ and $\mathbb{E}(Y \mid \mathcal{G})$ are measurable in $\mathcal{G}$. Since we have \begin{equation} \begin{split} (X - \mathbb{E}(X \mid \mathcal{G})) (Y - \mathbb{E}(Y \mid \mathcal{G})) = X Y - X \mathbb{E}(Y \mid \mathcal{G}) - Y \mathbb{E}(X \mid \mathcal{G}) + \mathbb{E}(X \mid \mathcal{G}) \mathbb{E}(Y \mid \mathcal{G}) \end{split} \end{equation} then, applying results

(i)	R4784: Basic real linearity of basic real conditional expectation
(ii)	R4781: Conditional expectation of known random real number

we have \begin{equation} \begin{split} \text{Cov}(X, Y \mid \mathcal{G}) & = \mathbb{E}( (X - \mathbb{E}(X \mid \mathcal{G})) (Y - \mathbb{E}(Y \mid \mathcal{G})) \mid \mathcal{G}) \\ & = \mathbb{E}(X Y \mid \mathcal{G}) - \mathbb{E}(X \mid \mathcal{G}) \mathbb{E}(Y \mid \mathcal{G}) - \mathbb{E}(Y \mid \mathcal{G}) \mathbb{E}(X \mid \mathcal{G}) + \mathbb{E}(\mathbb{E}(X \mid \mathcal{G}) \mathbb{E}(Y \mid \mathcal{G}) \mid \mathcal{G}) \\ & = \mathbb{E}(X Y \mid \mathcal{G}) - 2 \mathbb{E}(X \mid \mathcal{G}) \mathbb{E}(Y \mid \mathcal{G}) + \mathbb{E}(\mathbb{E}(X \mid \mathcal{G}) \mathbb{E}(Y \mid \mathcal{G}) \mid \mathcal{G}) \\ & = \mathbb{E}(X Y \mid \mathcal{G}) - 2 \mathbb{E}(X \mid \mathcal{G}) \mathbb{E}(Y \mid \mathcal{G}) + \mathbb{E}(X \mid \mathcal{G}) \mathbb{E}(Y \mid \mathcal{G}) \\ & = \mathbb{E}(X Y \mid \mathcal{G}) - \mathbb{E}(X \mid \mathcal{G}) \mathbb{E}(Y \mid \mathcal{G}) \\ \end{split} \end{equation} This is what was required to be shown. $\square$