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
  for n holds ||.Sum(seq, n).|| <= Sum(||.seq.||, n)
proof
  let n;
  ||.Partial_Sums(seq).n.|| <= Partial_Sums(||.seq.||).n by Th37;
  hence thesis by SERIES_1:def 5;
end;
