
theorem
  for seq be Complex_Sequence st seq is absolutely_summable & Sum |.seq
  .|=0 holds for n be Nat holds seq.n =0c
proof
  let seq be Complex_Sequence such that
A1: seq is absolutely_summable and
A2: Sum |.seq.|=0;
A3: for n be Nat holds (Partial_Sums |.seq.|).n <= Sum |.seq.|
  proof
    let n be Nat;
    (Partial_Sums(|.seq.|)).n <= lim Partial_Sums |.seq.| by A1,SEQ_4:37;
    hence thesis by SERIES_1:def 3;
  end;
A4: now
    assume ex n be Nat st (|.seq.|).n <> 0;
    then consider n1 be Nat such that
A5: (|.seq.|).n1 <> 0;
A6: for n be Nat holds 0 <= Partial_Sums(|.seq.|).n
    proof
      let n be Nat;
      (|.seq.|).0 = |.(seq.0).| by VALUED_1:18;
      then
A7:   0 <=(|.seq.|).0 by COMPLEX1:46;
      n=n+0 & Partial_Sums(|.seq.|).0 = (|.seq.|).0 by SERIES_1:def 1;
      hence thesis by A7,SEQM_3:5;
    end;
    Partial_Sums(|.seq.|).n1 >0
    proof
      now
        per cases;
        case
          n1=0;
          then Partial_Sums(|.seq.|).(n1) <> 0 by A5,SERIES_1:def 1;
          hence thesis by A6;
        end;
        case
A8:       n1<>0;
          set nn=n1-1;
          (|.seq.|).n1 = |.(seq.n1).| by VALUED_1:18;
          then
A9:       nn+1 =n1 & 0 <= (|.seq.|).n1 by COMPLEX1:46;
          0 + 1 <= n1 by A8,INT_1:7;
          then
A10:      nn in NAT by INT_1:5;
          then 0 <= Partial_Sums(|.seq.|).nn by A6;
          then (|.seq.|).(n1)+0 <= (|.seq.|).(n1)+Partial_Sums(|.seq.|).nn by
XREAL_1:7;
          hence thesis by A5,A10,A9,SERIES_1:def 1;
        end;
      end;
      hence thesis;
    end;
    hence contradiction by A2,A3;
  end;
  for n be Nat holds seq.n =0c
  proof
    let n be Nat;
    0 = (|.seq.|).n by A4
      .= |.(seq.n).| by VALUED_1:18;
    hence thesis by COMPLEX1:45;
  end;
  hence thesis;
end;
