Let A be any event and note that

\begin{equation*} A \cap A^c = \emptyset. \end{equation*}

But \(A \cup A^c = S\text{.}\) So, by subadditivity 4.3.2

\begin{equation*} 1 = P(S) = P(A \cup A^c) = P(A) + P(A^c) \end{equation*}

as desired.