reserve rseq, rseq1, rseq2 for Real_Sequence;
reserve seq, seq1, seq2 for Complex_Sequence;
reserve k, n, n1, n2, m for Nat;
reserve p, r for Real;
reserve z for Complex;
reserve Nseq,Nseq1 for increasing sequence of NAT;

theorem Th24:
  (for n holds seq.n = 0c) implies for m holds (Partial_Sums seq). m = 0c
proof
  defpred P[Nat] means seq.$1 = (Partial_Sums seq).$1;
  assume
A1: for n holds seq.n = 0c;
A2: for k being Nat st P[k] holds P[k+1]
  proof
    let k be Nat such that
A3: P[k];
    thus seq.(k+1) = 0c + seq.(k+1)
      .= (Partial_Sums seq).k + seq.(k+1) by A1,A3
      .= (Partial_Sums seq).(k+1) by SERIES_1:def 1;
  end;
  let m;
A4: P[0] by SERIES_1:def 1;
  for n being Nat holds P[n] from NAT_1:sch 2(A4,A2);
  then seq = Partial_Sums seq;
  hence thesis by A1;
end;
