reserve n,m,k for Nat;
reserve a,p,r for Real;
reserve s,s1,s2,s3 for Real_Sequence;

theorem Th34:
  for n,m st n<=m holds |.Partial_Sums(s).m - Partial_Sums(s).n.|
  <= |.Partial_Sums(|.s.|).m - Partial_Sums(|.s.|).n.|
proof
  let n,m such that
A1: n<=m;
  reconsider u=n, v=m as Integer;
  set s2=Partial_Sums(abs(s));
  set s1=Partial_Sums(s);
  defpred X[Nat] means
   |.s1.(n+$1) - s1.n.| <= |.s2.(n+$1) - s2.n.|;
  now
    let k;
    |.s.k.|>=0 by COMPLEX1:46;
    hence abs(s).k >= 0 by SEQ_1:12;
  end;
  then
A2: s2 is non-decreasing by Th16;
A3: for k st X[k] holds X[k+1]
  proof
    let k;
A4: |.s.(n+k+1).| >= 0 by COMPLEX1:46;
    assume |.s1.(n+k) - s1.n.| <= |.s2.(n+k) - s2.n.|;
    then
A5: |.s.(n+k+1).| + |.s1.(n+k) - s1.n.| <= |.s.(n+k+1).| + |.s2.(n+k
    ) - s2.n.| by XREAL_1:7;
A6: |.s2.(n+(k+1)) - s2.n.| = |.s2.(n+k) + (abs(s)).(n+k+1) - s2.n.| by Def1
      .= |.|.s.(n+k+1).| + s2.(n+k) - s2.n.| by SEQ_1:12
      .= |.|.s.(n+k+1).| + (s2.(n+k) - s2.n).|;
    s2.(n+k)>=s2.n by A2,SEQM_3:5;
    then
A7: s2.(n+k) - s2.n >= 0 by XREAL_1:48;
    |.s1.(n+(k+1)) - s1.n.| = |.s.(n+k+1) + s1.(n+k) - s1.n.| by Def1
      .= |.s.(n+k+1) + (s1.(n+k) - s1.n).|;
    then |.s1.(n+(k+1))-s1.n.| <= |.s.(n+k+1).|+|.s1.(n+k) - s1.n.| by
COMPLEX1:56;
    then |.s1.(n+(k+1))-s1.n.| <= |.s.(n+k+1).|+|.s2.(n+k)-s2.n.| by A5,
XXREAL_0:2;
    then |.s1.(n+(k+1))-s1.n.| <= |.s.(n+k+1).|+(s2.(n+k)-s2.n) by A7,
ABSVALUE:def 1;
    hence thesis by A7,A4,A6,ABSVALUE:def 1;
  end;
  reconsider k = v-u as Element of NAT by A1,INT_1:5;
A8: n+k = m;
A9: X[0];
  for k holds X[k] from NAT_1:sch 2(A9,A3);
  hence thesis by A8;
end;
