Using results

(i)	R5101: Disjoint additivity of probability measure for binary union
(ii)	R2060: Probability of set difference
(iii)	R3719: Probability of complement event

we have \begin{equation} \mathbb{P}(E \cup E^{\complement}) = \mathbb{P}(E) + \mathbb{P}(E^{\complement}) = \mathbb{P}(E) + (1 - \mathbb{P}(E)) = 1 \end{equation} $\square$