reserve n,m,k for Nat;
reserve a,p,r for Real;
reserve s,s1,s2,s3 for Real_Sequence;

theorem
  (for n holds s.n<>0 & s1.n=abs(s).(n+1)/abs(s).n) & s1 is convergent &
  lim s1 < 1 implies s is absolutely_summable
proof
  assume that
A1: for n holds s.n<>0 & s1.n=abs(s).(n+1)/abs(s).n and
A2: s1 is convergent & lim s1 < 1;
  now
    let n;
    abs(s).n = |.s.n.| & s.n <> 0 by A1,SEQ_1:12;
    hence abs(s).n > 0 by COMPLEX1:47;
    thus s1.n=abs(s).(n+1)/abs(s).n by A1;
  end;
  then abs(s) is summable by A2,Th26;
  hence thesis;
end;
