theorem Th40:
  for n,m holds ||.Sum(seq,m)-Sum(seq,n).|| <= |. Sum(||.seq.||,
  m)-Sum(||.seq.||,n) .|
proof
  let n,m;
  ||.Sum(seq, m) - Partial_Sums(seq).n.|| <= |.Partial_Sums(||.seq.||).m
  - Partial_Sums(||.seq.||).n.| by Th39;
  then
  ||.Sum(seq, m) - Sum(seq, n).|| <= |.Sum(||.seq.||, m) - Partial_Sums(
  ||.seq.||).n.| by SERIES_1:def 5;
  hence thesis by SERIES_1:def 5;
end;
