reserve r for Real,
  X for set,
  f, g, h for real-valued Function;

theorem Th6:
  for f being Real_Sequence st f is absolutely_summable holds
  |.Sum f.| <= Sum abs f
proof
  let f be Real_Sequence;
  defpred P[Nat] means
   (abs Partial_Sums f).$1 <= (Partial_Sums abs f).$1;
A1: now
    let n be Nat;
    assume P[n];
    then |.(Partial_Sums f).n.| <= (Partial_Sums abs f).n by SEQ_1:12;
    then
    |.(Partial_Sums f).n+f.(n+1).| <= |.(Partial_Sums f).n.|+|.f.(n+1) .|
& |.( Partial_Sums f).n.|+|.f.(n+1).| <= (Partial_Sums abs f).n+|.f.(n+1) .|
by COMPLEX1:56,XREAL_1:6;
    then
    |.(Partial_Sums f).n+f.(n+1).| <= (Partial_Sums abs f).n+|.f.(n+1) .|
    by XXREAL_0:2;
    then |.(Partial_Sums f).(n+1).| <= (Partial_Sums abs f).n+|.f.(n+1).| by
SERIES_1:def 1;
    then (abs Partial_Sums f).(n+1) <= (Partial_Sums abs f).n+|.f.(n+1).| by
SEQ_1:12;
    then
    (abs Partial_Sums f).(n+1) <= (Partial_Sums abs f).n+(abs f).(n+1) by
SEQ_1:12;
    hence P[n+1] by SERIES_1:def 1;
  end;
  (abs Partial_Sums f).0 = |.(Partial_Sums f).0.| by SEQ_1:12
    .= |.f.0.| by SERIES_1:def 1
    .= (abs f).0 by SEQ_1:12;
  then
A2: P[ 0 ] by SERIES_1:def 1;
A3: for n being Nat holds P[n] from NAT_1:sch 2(A2,A1);
  assume
A4: f is absolutely_summable;
  then abs f is summable by SERIES_1:def 4;
  then
A5: Partial_Sums abs f is convergent by SERIES_1:def 2;
  f is summable by A4;
  then
A6: Partial_Sums f is convergent by SERIES_1:def 2;
  then lim abs Partial_Sums f <= lim Partial_Sums abs f by A5,A3,SEQ_2:18;
  then |.lim Partial_Sums f.| <= lim Partial_Sums abs f by A6,SEQ_4:14;
  then |.lim Partial_Sums f.| <= Sum abs f by SERIES_1:def 3;
  hence thesis by SERIES_1:def 3;
end;
