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 Th35:
  Partial_Sums(||.seq.||) is non-decreasing
proof
  now
    let n be Nat;
    ||.seq.(n+1).|| >= 0 by BHSP_1:28;
    then ||.seq.||.(n+1) >= 0 by BHSP_2:def 3;
    then
    0 + Partial_Sums(||.seq.||).n <= ||.seq.||.(n+1) + Partial_Sums(||.seq
    .||).n by XREAL_1:6;
    hence Partial_Sums(||.seq.||).n <= Partial_Sums(||.seq.||).(n+1) by
SERIES_1:def 1;
  end;
  hence thesis;
end;
