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
  |.seq .| is non-increasing & (for n holds rseq1.n = 2 to_power n * |.
seq.|.(2 to_power n)) implies (seq is absolutely_summable iff rseq1 is summable
  )
proof
  assume |.seq.| is non-increasing & for n holds rseq1.n = 2 to_power n * |.
  seq.|.(2 to_power n);
  then for n holds |.seq.| is non-increasing & |.seq.|.n >= 0 & rseq1.n = 2
  to_power n * |.seq.|.(2 to_power n) by Lm3;
  then |.seq.| is summable iff rseq1 is summable by SERIES_1:31;
  hence thesis;
end;
