Result
R1194: Indicator function with respect to set complement shows that $I_{E^{\complement}} = 1 - I_E$. Thus, applying
R1816: Complex-linearity of complex conditional expectation, we have
\begin{equation}
\mathbb{P}(E^{\complement} \mid \mathcal{G})
= \mathbb{E}(I_{E^{\complement}} \mid \mathcal{G})
= \mathbb{E}(1 - I_E \mid \mathcal{G})
= 1 - \mathbb{E}(I_E \mid \mathcal{G})
= 1 - \mathbb{P}(E \mid \mathcal{G})
\end{equation}
$\square$
