i want to prove this:
$P(A_1 \cap A_2 /B)=P(A_1/B)P(A_2/B) $$\Rightarrow $$P(A_1/A_2 \cap B)=P(A_1/B)$
i did for $A_2 \cap B \neq \emptyset$ .but i am stuck in case $A_2 \cap B = \emptyset$
i want to see why $P(A_1 \cap A_2 /B)=P(A_1/B)P(A_2/B) $means that $A_1$ and $A_2$ are conditionally independent given $B$.
can you help me?