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 & rseq1.n=|.seq.|.(n+1)/|.seq.|.n) & rseq1 is
  convergent & lim rseq1 < 1 implies seq is absolutely_summable
proof
  assume that
A1: for n holds seq.n<>0c & rseq1.n=|.seq.|.(n+1)/|.seq.|.n and
A2: rseq1 is convergent & lim rseq1 < 1;
  now
    let n;
A3: seq.n<>0c & |.seq.|.n =|.seq.n.| by A1,VALUED_1:18;
    hence |.seq.|.n <>0 & rseq1.n=|.seq.|.(n+1)/|.seq.|.n by A1,COMPLEX1:47;
    thus (|.seq.|).n <>0 & rseq1.n=|.(|.seq.|).|.(n+1)/|.seq.|.n by A1,A3,
COMPLEX1:47;
    thus (|.seq.|).n <>0 & rseq1.n=abs((|.seq.|)).(n+1)/abs((|.seq.|)).n by A1
,A3,COMPLEX1:47;
  end;
  then |.seq.| is absolutely_summable by A2,SERIES_1:37;
  hence thesis;
end;
