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}

From result R3562: Real conditional variance partition into conditional moments, we have \begin{equation} \text{Var}(X \mid \mathcal{G}) = \mathbb{E}(X^2 \mid \mathcal{G}) - (\mathbb{E}(X \mid \mathcal{G}))^2 \end{equation} Taking expectations of each side and applying result R4782: Expectation of conditional expectation for a random real number, we have \begin{equation} \begin{split} \mathbb{E}(\text{Var}(X \mid \mathcal{G})) & = \mathbb{E}(\mathbb{E}(X^2 \mid \mathcal{G})) - \mathbb{E}((\mathbb{E}(X \mid \mathcal{G}))^2) \\ & = \mathbb{E} X^2 - \mathbb{E}((\mathbb{E}(X \mid \mathcal{G}))^2) \\ \end{split} \end{equation} Next, applying result R2262: Real variance partition into first and second moments to the random variable $\mathbb{E}(X \mid \mathcal{G})$, we have \begin{equation} \begin{split} \text{Var}(\mathbb{E}(X \mid \mathcal{G})) & = \mathbb{E}((\mathbb{E}(X \mid \mathcal{G}))^2) - (\mathbb{E} (\mathbb{E}(X \mid \mathcal{G})))^2 \\ & = \mathbb{E}((\mathbb{E}(X \mid \mathcal{G}))^2) - (\mathbb{E} X)^2 \end{split} \end{equation} Finally, again applying result R2262: Real variance partition into first and second moments, we conclude \begin{equation} \begin{split} \mathbb{E}(\text{Var}(X \mid \mathcal{G})) + \mathsf{Var}(\mathbb{E}(X \mid \mathcal{G})) & = \mathbb{E} X^2 - \mathbb{E}((\mathbb{E}(X \mid \mathcal{G}))^2) + \mathbb{E}((\mathbb{E}(X \mid \mathcal{G}))^2) - (\mathbb{E} X)^2 \\ & = \mathbb{E} X^2 - (\mathbb{E} X)^2 \\ & = \text{Var} X \end{split} \end{equation} $\square$