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) & (ex m st for n st n>=m holds abs(s).(n+1)/abs(s
  ).n >= 1) implies s is not summable
proof
  assume
A1: for n holds s.n<>0;
  given m such that
A2: for n st n>=m holds abs(s).(n+1)/abs(s).n >= 1;
A3: now
    defpred X[Nat] means |.s.(m+$1).|>=|.s.m.|;
A4: for k st X[k] holds X[k+1]
    proof
      let k such that
A5:   |.s.(m+k).|>=|.s.m.|;
      abs(s).(m+k+1)/abs(s).(m+k) >= 1 by A2,NAT_1:11;
      then |.s.(m+k+1).|/abs(s).(m+k) >= 1 by SEQ_1:12;
      then
A6:   |.s.(m+k+1).|/|.s.(m+k).| >= 1 by SEQ_1:12;
      s.(m+k) <> 0 by A1;
      then |.s.(m+k).| > 0 by COMPLEX1:47;
      then |.s.(m+k+1).|>=|.s.(m+k).| by A6,XREAL_1:191;
      hence thesis by A5,XXREAL_0:2;
    end;
    let n;
    assume n>=m;
    then consider k be Nat such that
A7: n=m+k by NAT_1:10;
    reconsider k as Element of NAT by ORDINAL1:def 12;
A8: n=m+k by A7;
A9: X[0];
    for k holds X[k] from NAT_1:sch 2(A9,A4);
    hence |.s.n.|>=|.s.m.| by A8;
  end;
  s.m <> 0 by A1;
  then |.s.m.| > 0 by COMPLEX1:47;
  then s is not convergent or lim s <> 0 by A3,Th38;
  hence thesis by Th4;
end;
