Let \(A\), \(B\), and \(C\) be events with \(P(C) > 0\). \(A\) and \(B\) are conditionally independent given \(C\) if and only if
\[ P(A \cap B \mid C) = P(A \mid C)P(B \mid C). \]
Show that the above implies \(P(A \mid B\cap C) = P(A \mid C)\).
Let \(A\), \(B\), and \(C\) be events with \(P(C) > 0\). \(A\) and \(B\) are conditionally independent given \(C\) if and only if
\[ P(A \cap B \mid C) = P(A \mid C)P(B \mid C). \]
Show that the above implies \(P(A \mid B\cap C) = P(A \mid C)\).