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 Th25:
  (for n holds seq.n = 0c) implies for m holds (Partial_Sums |.seq .|).m = 0
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) by COMPLEX1:44
      .= |.seq.k.| + |.seq.|.(k+1) by A1
      .= (Partial_Sums |.seq.|).k + |.seq.|.(k+1) by A3,VALUED_1:18
      .= (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);
  hence (Partial_Sums |.seq.|).m = |.seq.|.m .= |.seq.m.| by VALUED_1:18
    .= 0 by A1,COMPLEX1:44;
end;
