Currently, I am trying to figure out how to prove two properties of conditional independence.
1) P(A,B|C) = P(A|C)P(B|C)
2) P(A|B,C) = P(A|C)
(P(C) > 0)
As far as I use the first property, I could prove the second as follows:
$$P(A|B,C) = \dfrac{P(A,B,C)}{P(B,C)} = \dfrac{P(A|C)P(B|C)}{P(B|C)} = P(A|C)$$
Then, how could I prove the first one?!
There is nothing to prove. It is not a property, it is a definition. Events $A$ and $B$ are defined as being conditionally independent when given event $C$, if and only if their joint conditional probability when given event $C$ does equal the product of their conditional probabilities given $C$.
$$A\perp B\mid C ~\iff~ \mathsf P(A,B\mid C)=\mathsf P(A\mid C)\;\mathsf P(B\mid C)$$