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 Partial_Sums(||.seq.||).n >= 0
proof
  let n;
  ||.(seq.0).|| >= 0 by CSSPACE:44;
  then ||.seq.||.0 >= 0 by CLVECT_2:def 3;
  then Partial_Sums(||.seq.||).0 >= 0 by SERIES_1:def 1;
  hence thesis by Th35,SEQM_3:11;
end;
