reserve X for ComplexUnitarySpace;
reserve g for Point of X;
reserve seq, seq1, seq2 for sequence of X;
reserve Rseq for Real_Sequence;
reserve Cseq,Cseq1,Cseq2 for Complex_Sequence;
reserve z,z1,z2 for Complex;
reserve r for Real;
reserve k,n,m for Nat;

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;
