reserve X for non empty CUNITSTR;
reserve a, b for Complex;
reserve x, y for Point of X;
reserve X for ComplexUnitarySpace;
reserve x, y, z, u, v for Point of X;
reserve seq, seq1, seq2, seq3 for sequence of X;
reserve  n for Nat;

theorem
  for seq be Complex_Sequence st (for n be Nat holds seq.n=0c
  ) holds seq is summable & Sum seq = 0c
proof
  let seq be Complex_Sequence such that
A1: for n be Nat holds seq.n=0c;
A2: for m be Nat holds Partial_Sums (seq).m = 0c
  proof
    defpred P[Nat] means seq.$1 = (Partial_Sums seq).$1;
    let m be Nat;
A3: for k be Nat st P[k] holds P[k+1]
    proof
      let k be Nat such that
A4:   seq.k = (Partial_Sums (seq)).k;
      thus seq.(k+1) = 0c + (seq).(k+1) .= seq.k + seq.(k+1) by A1
        .= (Partial_Sums seq).(k+1) by A4,SERIES_1:def 1;
    end;
A5: P[0] by SERIES_1:def 1;
    for n be Nat holds P[n] from NAT_1:sch 2(A5,A3);
    hence (Partial_Sums (seq)).m = seq.m .= 0c by A1;
  end;
A6: for p be Real
    st 0<p ex n be Nat st for m be Nat
  st n<=m holds |.((Partial_Sums seq).m-0c).|<p
  proof
    let p be Real such that
A7: 0<p;
    take 0;
    let m be Nat such that
    0<=m;
    thus thesis by A2,A7,COMPLEX1:44;
  end;
  then
A8: Partial_Sums (seq) qua Complex_Sequence is convergent by COMSEQ_2:def 5;
  then lim (Partial_Sums (seq)) = 0c by A6,COMSEQ_2:def 6;
  hence thesis by A8,COMSEQ_3:def 7,def 8;
end;
