Proof of Markov Property I'm trying to understand a simple proof for the markov property which states that:
"$A_1$, $A_3$ are conditionally independent given $A_2$ iff $P(A_3 | A_1 \cap A_2)=P(A_3|A_2)$"
The Proof begins as follows:
"only if"
$$P(A_3 | A_1 \cap A_2) = \dfrac{P(A_1 \cap A_2 \cap A_3)}{P(A_1 \cap A_2)} \\ = \dfrac{P(A_1 \cap A_3 | A_2)P(A_2)}{P(A_1 \cap A_2)} \\ 
= \dfrac{P(A_3|A_2)P(A_1 A_2)P(A_2)}{P(A_1 \cap A_2)}$$
then the proof ends. I'm not really sure why this proves it?
If a question was asked "Prove that for events $A_1$; $A_2$; $A_3$ $\in F$, such that $A_1$ and $A_3$ are conditionally independent given $A_2$ iff the
Markov property holds. Would this suffice as a proof?

To finish off the proof:
$$\dfrac{P(A_3|A_2)P(A_1|A_2)P(A_2)}{P(A_1 \cap A_2)}=\dfrac{P(A_3|A_2)P(A_1 \cap A_2)}{P(A_1 \cap A_2)}=P(A_3|A_2)$$
