Proof
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.