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;
