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?

Need an algorithm to determine if the expressions A> B> C> D are legal

I need to set the priority of the parameters of the function, such as the function Func (A, B, C, D, E)
Others can write rules to set this priority. I need an algorithm to check if this rule is legal or not. I do not know which keywords should I search on Google?
A> B> C> D> E is legal.
A> B> C> A it's not legal.
A> (B && C && D)> (B && C)> E is legal.
A> (B && C)> (B && C && D)> E it's not legal.
Please enlighten me and refer to links or documents.