## probability – Assume that \$ A \$ and \$ B \$ are independent events. For a \$ C \$ event such as \$ P (C)> \$ 0, prove that \$ A \$ event donated \$ C \$

assume $$A$$ and $$B$$ are independent events. For an event $$C$$ such as $$P (C)> 0$$ , prove that the event of $$A$$ given $$C$$ is independent of the event of $$B$$ given $$C$$

We have A and B are independent so $$P (AB) = P (A) cdot P (B)$$

We must show that $$P ((A mid C) cap (B mid C)) = P (A mid C) cdot P (B mid C)$$

My procedure was like that
$$P ((A mid C) cap (B mid C)) = P ((A mid C) mid (B mid C)) cdot P (B mid C)$$
$$= frac {P (AB mid C)} {P (B mid C)}$$

I played until I got this
$$frac {P (AC)} {P (C)} cdot frac {P (B mid AC)} {P (B mid C)}$$
Now the first part gives us $$P (A mid C)$$ . I could not get from the second part the missing part that is $$P (B mid C)$$.

Is my procedure correct? If so, how can I find the second part?