reserve rseq, rseq1, rseq2 for Real_Sequence;
reserve seq, seq1, seq2 for Complex_Sequence;
reserve k, n, n1, n2, m for Nat;
reserve p, r for Real;
reserve z for Complex;
reserve Nseq,Nseq1 for increasing sequence of NAT;

theorem Th31:
  |.Partial_Sums(seq).m- Partial_Sums(seq).n.| <= |.
  Partial_Sums(|.seq.|).m- Partial_Sums(|.seq.|).n.|
proof
A1: for n,k be Nat holds 0 <= Partial_Sums(|.seq.|).(n+k)-
  Partial_Sums(|.seq.|).n
  proof
    let n;
    defpred P[Nat] means 0 <= Partial_Sums(|.seq.|).(n+$1)-
    Partial_Sums(|.seq.|).n;
A2: now
      let k;
A3:   Partial_Sums(|.seq.|).(n+(k+1))- Partial_Sums(|.seq.|).n =
      Partial_Sums(|.seq.|).(n+k)+|.seq.|.(n+k+1) - Partial_Sums(|.seq.|).n by
SERIES_1:def 1
        .=Partial_Sums(|.seq.|).(n+k)+|.seq.(n+k+1).| - Partial_Sums(|.seq.|
      ).n by VALUED_1:18
        .=Partial_Sums(|.seq.|).(n+k)- Partial_Sums(|.seq.|).n +|.seq.(n+k+1
      ).|;
A4:   0 <= |.seq.(n+k+1).| by COMPLEX1:46;
      assume P[k];
      hence P[k+1] by A3,A4;
    end;
A5: P[0];
    thus for k be Nat holds P[k] from NAT_1:sch 2(A5,A2);
  end;
A6: for n,k be Nat holds |. Partial_Sums(|.seq.|).(n+k)-
  Partial_Sums(|.seq.|).n.|
  = Partial_Sums(|.seq.|).(n+k)- Partial_Sums(|.seq.|).n by A1,ABSVALUE:def 1;
A7:
    for n,m st n <= m holds |.Partial_Sums(seq).m- Partial_Sums(seq).n
  .| <= |. Partial_Sums(|.seq.|).m- Partial_Sums(|.seq.|).n.|
  proof
    let n,m be Nat;
    assume n <= m;
    then consider k be Nat such that
A8: m=n+k by NAT_1:10;
A9: for k be Nat holds |.Partial_Sums(seq).(n+k)- Partial_Sums
    (seq).n.| <= |. Partial_Sums(|.seq.|).(n+k)- Partial_Sums(|.seq.|).n.|
    proof
      defpred P[Nat] means |.Partial_Sums(seq).(n+$1)- Partial_Sums
(seq).n.| <= |. Partial_Sums(|.seq.|).(n+$1)- Partial_Sums(|.seq.|).n.|;
A10:  now
        let k be Nat;
        assume P[k];
        then
A11:    |.Partial_Sums(seq).(n+k)- Partial_Sums(seq).n+seq.(n+k+1).| <=
        |. Partial_Sums(seq).(n+k)- Partial_Sums(seq).n.| +|.seq.(n+k+1).| & |.
        Partial_Sums(seq).(n+k)- Partial_Sums(seq).n.| +|.seq.(n+k+1).| <= |.
Partial_Sums(|.seq.|).(n+k)- Partial_Sums(|.seq.|).n .| +|.seq.(n+k+1).| by
COMPLEX1:56,XREAL_1:6;
A12:    |.Partial_Sums(seq).(n+(k+1))- Partial_Sums(seq).n .| =|.(
        Partial_Sums(seq).(n+k)+seq.(n+k+1))- Partial_Sums(seq).n.|
        by SERIES_1:def 1
          .=|.Partial_Sums(seq).(n+k)- Partial_Sums(seq).n+seq.(n+k+1).|;
        |. Partial_Sums(|.seq.|).(n+k)- Partial_Sums(|.seq.|).n .| +|.
seq.(n+k+1).| =Partial_Sums(|.seq.|).(n+k)- Partial_Sums(|.seq.|).n +|.seq.(n+k
        +1).| by A6
          .=Partial_Sums(|.seq.|).(n+k)+|.seq.(n+k+1).| - Partial_Sums(|.seq
        .|).n
          .=Partial_Sums(|.seq.|).(n+k)+|.seq.|.(n+k+1) - Partial_Sums(|.seq
        .|).n by VALUED_1:18
          .=Partial_Sums(|.seq.|).(n+(k+1))- Partial_Sums(|.seq.|).n by
SERIES_1:def 1
          .=|. Partial_Sums(|.seq.|).(n+(k+1))- Partial_Sums(|.seq.|).n .|
        by A6;
        hence P[k+1] by A12,A11,XXREAL_0:2;
      end;
A13:  P[0];
      thus for k be Nat holds P[k] from NAT_1:sch 2(A13,A10 );
    end;
    thus thesis by A9,A8;
  end;
  for n, m holds |.Partial_Sums(seq).m- Partial_Sums(seq).n.| <= |.
  Partial_Sums(|.seq.|).m- Partial_Sums(|.seq.|).n .|
  proof
    let n,m;
    m <= n implies |.Partial_Sums(seq).m- Partial_Sums(seq).n.| <= |.
    Partial_Sums(|.seq.|).m- Partial_Sums(|.seq.|).n .|
    proof
      assume m <= n;
      then
A14:  |.Partial_Sums(seq).n- Partial_Sums(seq).m.| <= |. Partial_Sums(
      |.seq.|).n- Partial_Sums(|.seq.|).m .| by A7;
      |. Partial_Sums(|.seq.|).n- Partial_Sums(|.seq.|).m .| =|.-(
      Partial_Sums(|.seq.|).n- Partial_Sums(|.seq.|).m) .| by COMPLEX1:52
        .=|. Partial_Sums(|.seq.|).m- Partial_Sums(|.seq.|).n .|;
      hence thesis by A14,COMPLEX1:60;
    end;
    hence thesis by A7;
  end;
  hence thesis;
end;
