theorem Th45:
  ASeq is disjoint_valued implies Partial_Sums(P * ASeq) is
  convergent & lim Partial_Sums(P * ASeq) = upper_bound Partial_Sums(P * ASeq)
   & lim
  Partial_Sums(P * ASeq) = P.Union ASeq
proof
  assume ASeq is disjoint_valued;
  then (P * Partial_Union ASeq) = Partial_Sums(P * ASeq) by Th44;
  hence thesis by Th41;
end;
