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
  (for n holds seq.n<>0c) & (ex m st for n st n>=m holds |.seq.|.(n+1)/
  |.seq.|.n >= 1) implies seq is not absolutely_summable
proof
  assume that
A1: for n holds seq.n<>0c and
A2: ex m st for n st n>=m holds |.seq.|.(n+1)/|.seq.|.n >= 1;
  consider m such that
A3: for n st n>=m holds |.seq.|.(n+1)/|.seq.|.n >= 1 by A2;
A4: now
    let n;
    seq.n<>0c by A1;
    then |.seq.n.|<>0 by COMPLEX1:47;
    hence |.seq.|.n <>0 by VALUED_1:18;
  end;
  for n st n>=m holds abs((|.seq.|)).(n+1)/abs((|.seq.|)).n >= 1 by A3;
  hence |.seq.| is not summable by A4,SERIES_1:39;
end;
