How do these multiplication relations of Conditional Probabilities work?

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Could you please show how could I get from the left parts to the right parts for the following two relations? Thank you.

1) P(A|B)*P(B|C) = P(A,B|C)

2) P(A|B)*P(B,C|D) = P(A,B,C|D)

I would like to mention also the source where I found it: https://youtu.be/lzN18y_z6HQ?t=253