
theorem Th3:
  for cseq be Complex_Sequence st (for n be Nat holds
  cseq.n=0c) holds cseq is absolutely_summable & Sum |.cseq.| = 0
proof
  let cseq be Complex_Sequence such that
A1: for n be Nat holds cseq.n=0c;
A2: for n be Nat holds (|.cseq.|).n=0
  proof
    let n be Nat;
    cseq.n=0c by A1;
    hence thesis by COMPLEX1:44,VALUED_1:18;
  end;
A3: for m be Nat holds Partial_Sums (|.cseq.|).m = 0
  proof
    defpred P[Nat] means
    (|.cseq.|).$1 = (Partial_Sums |.cseq.|).$1;
    let m be Nat;
A4: for k be Nat st P[k] holds P[k+1]
    proof
      let k be Nat such that
A5:   (|.cseq.|).k = (Partial_Sums (|.cseq.|)).k;
      thus (|.cseq.|).(k+1) = 0 + (|.cseq.|).(k+1)
        .= (|.cseq.|).k + (|.cseq.|).(k+1) by A2
        .= (Partial_Sums |.cseq.|).(k+1) by A5,SERIES_1:def 1;
    end;
A6: P[0] by SERIES_1:def 1;
    for n be Nat holds P[n] from NAT_1:sch 2(A6,A4);
    hence (Partial_Sums |.cseq.|).m = (|.cseq.|).m .= 0 by A2;
  end;
A7: for p be Real st 0<p ex n be Nat st
   for m be Nat st n<=m holds |.(Partial_Sums |.cseq.|).m-0 .|<p
  proof
    let p be Real such that
A8: 0<p;
    take 0;
    let m be Nat such that
    0<=m;
    |.(Partial_Sums |.cseq.|).m-0 .| = |.0-0 .| by A3
      .= 0 by ABSVALUE:def 1;
    hence thesis by A8;
  end;
  then
A9: Partial_Sums |.cseq.| is convergent by SEQ_2:def 6;
  then
A10: |.cseq.| is summable by SERIES_1:def 2;
  lim Partial_Sums |.cseq.| =0 by A7,A9,SEQ_2:def 7;
  hence thesis by A10,COMSEQ_3:def 9,SERIES_1:def 3;
end;
