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)	$E \in \mathcal{F}$ is an D1716: Event in $P$

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} $\square$