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 Th39:
  for n, m holds ||.Partial_Sums(seq).m - Partial_Sums(seq).n.||
  <= |.Partial_Sums(||.seq.||).m - Partial_Sums(||.seq.||).n.|
proof
  let n, m;
A1: now
    reconsider d = n, t = m as Integer;
    set PSseq9 = Partial_Sums(||.seq.||);
    set PSseq = Partial_Sums(seq);
    defpred P[Nat] means
||.PSseq.m - PSseq.(m+$1).|| <= |.PSseq9
    .m - PSseq9.(m+$1).|;
A2: PSseq9 is non-decreasing by Th35;
A3: now
      let k;
A4:   ||.seq.(m+k+1).|| >= 0 by BHSP_1:28;
A5:   |.PSseq9.m - PSseq9.(m+(k+1)).| = |.-(PSseq9.(m+(k+1)) - PSseq9. m).|
        .= |.PSseq9.(m+k+1) - PSseq9.m.| by COMPLEX1:52
        .= |.PSseq9.(m+k) + (||.seq.||).(m+k+1) - PSseq9.m.| by SERIES_1:def 1
        .= |.||.seq.(m+k+1).|| + PSseq9.(m+k) - PSseq9.m.| by BHSP_2:def 3
        .= |.(PSseq9.(m+k) - PSseq9.m) + ||.seq.(m+k+1).||.|;
      ||.PSseq.m - PSseq.(m+(k+1)).|| = ||.PSseq.m - (PSseq.(m+k) + seq.(
      m + k +1)).|| by Def1
        .= ||.(PSseq.m - PSseq.(m+k)) - seq.(m+k+1).|| by RLVECT_1:27
        .= ||.(PSseq.m - PSseq.(m+k)) + -seq.(m+k+1).||;
      then ||.PSseq.m - PSseq.(m+(k+1)).|| <= ||.PSseq.m - PSseq.(m+k).|| +
      ||.-seq.(m+k+1).|| by BHSP_1:30;
      then
A6:   ||.PSseq.m - PSseq.(m+(k+1)).|| <= ||.PSseq.m - PSseq.(m+k).|| +
      ||.seq.(m+k+1).|| by BHSP_1:31;
A7:   PSseq9.(m+k) - PSseq9.m >= 0 by A2,SEQM_3:5,XREAL_1:48;
      assume P[k];
      then ||.PSseq.m - PSseq.(m+k).|| + ||.seq.(m+k+1).|| <= |.PSseq9.m -
      PSseq9.(m+k).| + ||.seq.(m+k+1).|| by XREAL_1:6;
      then ||.PSseq.m - PSseq.(m+(k+1)).|| <= |.-(PSseq9.(m+k) - PSseq9.m).|
      + ||.seq.(m+k+1).|| by A6,XXREAL_0:2;
      then ||.PSseq.m - PSseq.(m+(k+1)).|| <= |.PSseq9.(m+k) - PSseq9.m.| +
      ||.seq.(m+k+1).|| by COMPLEX1:52;
      then ||.PSseq.m - PSseq.(m+(k+1)).|| <= (PSseq9.(m+k) - PSseq9.m) + ||.
      seq.(m+k+1).|| by A7,ABSVALUE:def 1;
      hence P[k+1] by A7,A4,A5,ABSVALUE:def 1;
    end;
    assume n >= m;
    then reconsider k = d - t as Element of NAT by INT_1:5;
A8: m + k = n;
    ||.PSseq.m - PSseq.(m+0).|| = ||.09(X).|| by RLVECT_1:15
      .= 0 by BHSP_1:26;
    then
A9: P[0] by COMPLEX1:46;
    for k holds P[k] from NAT_1:sch 2(A9,A3);
    hence thesis by A8;
  end;
  now
    reconsider d = n, t = m as Integer;
    set PSseq9 = Partial_Sums(||.seq.||);
    set PSseq = Partial_Sums(seq);
    defpred P[Nat] means
||.PSseq.(n+$1) - PSseq.n.|| <= |.PSseq9
    .(n+$1) - PSseq9.n.|;
A10: PSseq9 is non-decreasing by Th35;
A11: now
      let k;
A12:  ||.seq.(n+k+1).|| >= 0 by BHSP_1:28;
A13:  |.PSseq9.(n+(k+1)) - PSseq9.n.| = |.PSseq9.(n+k) + (||.seq.||).(
      n+k +1) - PSseq9.n.| by SERIES_1:def 1
        .= |.||.seq.(n+k+1).|| + PSseq9.(n+k) - PSseq9.n.| by BHSP_2:def 3
        .= |.||.seq.(n+k+1).|| + (PSseq9.(n+k) - PSseq9.n).|;
      assume P[k];
      then
A14:  ||.seq.(n+k+1).|| + ||.PSseq.(n+k) - PSseq.n.|| <= ||.seq.(n+k+1)
      .|| + |.PSseq9.(n+k) - PSseq9.n.| by XREAL_1:7;
A15:  PSseq9.(n+k) - PSseq9.n >= 0 by A10,SEQM_3:5,XREAL_1:48;
      ||.PSseq.(n+(k+1)) - PSseq.n.|| = ||.seq.(n+k+1) + PSseq.(n+k) -
      PSseq .n.|| by Def1
        .= ||.seq.(n+k+1) + (PSseq.(n+k) - PSseq.n).|| by RLVECT_1:def 3;
      then
      ||.PSseq.(n+(k+1)) - PSseq.n.|| <= ||.seq.(n+k+1).|| + ||.PSseq.(n+k
      ) - PSseq.n.|| by BHSP_1:30;
      then
      ||.PSseq.(n+(k+1)) - PSseq.n.|| <= ||.seq.(n+k+1).|| + |.PSseq9.(
      n+k)-PSseq9.n.| by A14,XXREAL_0:2;
      then
      ||.PSseq.(n+(k+1)) - PSseq.n.|| <= ||.seq.(n+k+1).|| + (PSseq9.(n+k
      )-PSseq9.n) by A15,ABSVALUE:def 1;
      hence P[k+1] by A15,A12,A13,ABSVALUE:def 1;
    end;
    assume n <= m;
    then reconsider k = t - d as Element of NAT by INT_1:5;
A16: n + k = m;
    ||.PSseq.(n+0) - PSseq.n.|| = ||.09(X).|| by RLVECT_1:15
      .= 0 by BHSP_1:26;
    then
A17: P[0] by COMPLEX1:46;
    for k holds P[k] from NAT_1:sch 2(A17,A11);
    hence thesis by A16;
  end;
  hence thesis by A1;
end;
