reserve i, k, m, n for Nat,
  r, s for Real,
  rn for Real,
  x, y , z, X for set,
  T, T1, T2 for non empty TopSpace,
  p, q for Point of T,
  A, B, C for Subset of T,
  A9 for non empty Subset of T,
  pq for Element of [:the carrier of T,the carrier of T:],
  pq9 for Point of [:T,T:],
  pmet,pmet1 for Function of [:the carrier of T,the carrier of T:],REAL,
  pmet9,pmet19 for RealMap of [:T,T:] ,
  f,f1 for RealMap of T,
  FS2 for Functional_Sequence of [:the carrier of T,the carrier of T:],REAL,
  seq for Real_Sequence;

theorem Th13:
  for k holds |.Partial_Sums(seq).k.|<=Partial_Sums(abs(seq)).k
proof
  set PS=Partial_Sums(seq),absPS=Partial_Sums(abs(seq));
  defpred P[Nat] means |.PS.$1.|<=absPS.$1;
A1: for k st P[k] holds P[k+1]
  proof
    let k;
    assume P[k];
    then
A2: |.PS.k.|+|.seq.(k+1).| <=absPS.k+|.seq.(k+1).| by XREAL_1:7;
    PS.(k+1)=PS.k+seq.(k+1) by SERIES_1:def 1;
    then
A3: |.PS.(k+1).|<=|.PS.k.|+|.seq.(k+1).| by COMPLEX1:56;
    (abs seq).(k+1)= |.seq.(k+1).| by SEQ_1:12;
    then |.PS.(k+1).|<=absPS.k+(abs seq).(k+1) by A3,A2,XXREAL_0:2;
    hence thesis by SERIES_1:def 1;
  end;
  absPS.0=(abs seq).0 & (abs seq).0=|.seq.0 .| by SEQ_1:12,SERIES_1:def 1;
  then
A4: P[ 0 ] by SERIES_1:def 1;
  for k holds P[k] from NAT_1:sch 2(A4,A1);
  hence thesis;
end;
