Hence, with A = {\(a_1, a_2, ..., a_{|A|}\)}, breaking up the disjoint probabilities as above gives
\begin{align*}
P(A) & = P( \{ a_1 \} \cup \{ a_2 \} \cup ... \cup \{ a_{|A|} \} )\\
& = P(\{ a_1 \}) + P(\{ a_2 \} ) + ... + P(\{ a_{|A|} \} )\\
& = \frac{1}{{|S|}} + \frac{1}{{|S|}} + ... + \frac{1}{{|S|}}\\
& = \frac{|A|}{{|S|}}
\end{align*}
as desired.