\begin{align*} P(A) & = P( (A \cap S_1) \cup (A \cup S_2) \cup ... \cup (A \cup S_m) )\\ & = P(A \cap S_1) + P(A \cup S_2) + ... + P(A \cup S_m)\\ & = P(S_1 \cap A) + P(S_2 \cup A) + ... + P(S_m \cup A)\\ & = P(S_1) P(A | S_1) + P(S_2)P(A | S_2) + ... + P(S_m)P(A | S_m)\\ & = \sum_{k=1}^m P(S_k)P(A | S_k) \end{align*}