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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.
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