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 Th37:
  for n holds ||.Partial_Sums(seq).n.|| <= Partial_Sums(||.seq.||) .n
proof
  defpred P[Nat] means
||.Partial_Sums(seq).$1.|| <= Partial_Sums(
  ||.seq.||).$1;
A1: now
    let n;
    assume P[n];
    then
A2: ||.Partial_Sums(seq).n.|| + ||.seq.(n+1).|| <= Partial_Sums(||.seq.||)
    .n + ||.seq.(n+1).|| by XREAL_1:6;
    Partial_Sums(seq).(n+1) = Partial_Sums(seq).n + seq.(n+1) by Def1;
    then
    ||.Partial_Sums(seq).(n+1).|| <= ||.Partial_Sums(seq).n.|| + ||.seq.(n
    +1).|| by BHSP_1:30;
    then
    ||.Partial_Sums(seq).(n+1).|| <= Partial_Sums(||.seq.||).n + ||.seq.(n
    +1).|| by A2,XXREAL_0:2;
    then
    ||.Partial_Sums(seq).(n+1).|| <= Partial_Sums(||.seq.||).n + ||.seq.||
    .(n+1) by BHSP_2:def 3;
    hence P[n+1] by SERIES_1:def 1;
  end;
  Partial_Sums(||.seq.||).0 = ||.seq.||.0 by SERIES_1:def 1
    .= ||.(seq.0).|| by BHSP_2:def 3;
  then
A3: P[0] by Def1;
  thus for n holds P[n] from NAT_1:sch 2(A3,A1);
end;
