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;

theorem Th2:
  (for n holds rseq.n = 0) implies for m holds (Partial_Sums abs( rseq)).m = 0
proof
  defpred P[Nat] means
abs(rseq).$1 = (Partial_Sums abs(rseq)).$1;
  assume
A1: for n holds rseq.n = 0;
A2: for k st P[k] holds P[k+1]
  proof
    let k be Nat such that
A3: P[k];
    thus abs(rseq).(k+1) = 0 + abs(rseq).(k+1)
      .= |.0.| + abs(rseq).(k+1) by ABSVALUE:def 1
      .= |.rseq.k.| + abs(rseq).(k+1) by A1
      .= (Partial_Sums abs(rseq)).k + abs(rseq).(k+1) by A3,SEQ_1:12
      .= (Partial_Sums abs(rseq)).(k+1) by SERIES_1:def 1;
  end;
  let m;
A4: P[0] by SERIES_1:def 1;
  for n holds P[n] from NAT_1:sch 2(A4,A2);
  hence (Partial_Sums abs(rseq)).m = abs(rseq).m .= |.rseq.m.| by SEQ_1:12
    .= |.0.| by A1
    .= 0 by ABSVALUE:def 1;
end;
