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}
$\square$
