Remark

For a noninductive argument for Proposition 4, note first that if a point of the sample space is not a menber of any of the sets Ei then its probability does not contribute anything to either side of the equality. On the other hand, suppose that a point is in exactly m of the events Ei, where m>0. Then since it is in its probability is counted once in ; also as this point is contained in subsets of the type , its probability is counted

times on the right of the equality sign in Proposition 4. Thus, for m>0, we must show that
However, since , the preceding is equivalent to
and the latter equation follows from the binomial theorem since