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)	$X : \Omega \to \mathbb{R}$ is a D3161: Random real number on $P$
(iii)	\begin{equation} \mathbb{E} |X|^2 < \infty \end{equation}

By definition, $\mathbb{E}(X \mid \mathcal{G})$ is measurable in $\mathcal{G}$. Since $t \mapsto^2$ is a measurable transformation, then $(\mathbb{E}(X \mid \mathcal{G}))^2$ is also measurable in $\mathcal{G}$. Thus, 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} \mathsf{Var}(X \mid \mathcal{G}) & = \mathbb{E}((X - \mathbb{E}(X \mid \mathcal{G}))^2 \mid \mathcal{G}) \\ & = \mathbb{E}(X^2 - 2 X \mathbb{E}(X \mid \mathcal{G}) + (\mathbb{E}(X \mid \mathcal{G}))^2 \mid \mathcal{G}) \\ & = \mathbb{E}(X^2 \mid \mathcal{G}) - 2 (\mathbb{E}(X \mid \mathcal{G})^2 + \mathbb{E}((\mathbb{E}(X \mid \mathcal{G}))^2 \mid \mathcal{G}) \\ & = \mathbb{E}(X^2 \mid \mathcal{G}) - 2 (\mathbb{E}(X \mid \mathcal{G})^2 + (\mathbb{E}(X \mid \mathcal{G}))^2 \\ & = \mathbb{E}(X^2 \mid \mathcal{G}) - (\mathbb{E}(X \mid \mathcal{G}))^2 \\ \end{split} \end{equation} $\square$