reserve a, b, r, M2 for Real;
reserve Rseq,Rseq1,Rseq2 for Real_Sequence;
reserve k, n, m, m1, m2 for Nat;
reserve X for RealUnitarySpace;
reserve g for Point of X;
reserve seq, seq1, seq2 for sequence of X;

theorem
  for n holds Partial_Sums(||.seq.||).n >= 0
proof
  let n;
  ||.(seq.0).|| >= 0 by BHSP_1:28;
  then
A1: ||.seq.||.0 >= 0 by BHSP_2:def 3;
  Partial_Sums(||.seq.||).0 <= Partial_Sums(||.seq.||).n by Th35,SEQM_3:11;
  hence thesis by A1,SERIES_1:def 1;
end;
