Hey vmd.
Hint: Use P(A and B) = P(A)P(B) and consider a case where A (in this example) corresponds to two events like P(A) = P(C and D) = P(C)P(D)
Thats the definition of independent: If two events A and B are independent then P(A and B) = P(A)P(B).
You can think of this as P(A|B) = P(A) where B doesn't affect the outcome of A at all (i.e. they really are independent) and the definition of P(A|B) = P(A and B)/P(B) = P(A) which means that P(A and B) = P(A)*P(B). That's the intuitive reason why you have independence in mathematical form.