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

theorem Th29:
  (for n holds s.n>=0 & s1.n = n-root (s.n)) & (ex m st for n st m
  <=n holds s1.n>=1) implies s is not summable
proof
  assume that
A1: for n holds s.n>=0 & s1.n = n-root (s.n) and
A2: ex m st for n st m<=n holds s1.n>=1;
  consider m such that
A3: for n st m<=n holds s1.n>=1 by A2;
A4: for n st m+1<=n holds s.n>=1
  proof
    let n such that
A5: m+1<=n;
    1<=1+m by NAT_1:11;
    then
A6: 1<=n by A5,XXREAL_0:2;
A7: s.n >= 0 by A1;
    m<=m+1 by NAT_1:11;
    then m<=n by A5,XXREAL_0:2;
    then s1.n >= 1 by A3;
    then
A8: n-root (s.n) >= 1 by A1;
    now
      per cases by A8,XXREAL_0:1;
      suppose
        n-root (s.n) = 1;
        then s.n = 1 |^ n by A6,A7,POWER:4;
        hence thesis;
      end;
      suppose
        n-root (s.n) > 1;
        then (n-root (s.n)) to_power n > 1 by A5,POWER:35;
        hence thesis by A6,A7,POWER:4;
      end;
    end;
    hence thesis;
  end;
  for k ex n st k<=n & not |.s.n - 0.| < 1
  proof
    let k;
    take n = m + 1 + k;
    not s.n < 1 by A4,NAT_1:11;
    hence thesis by NAT_1:11,SEQ_2:1;
  end;
  then s is not convergent or not lim s = 0 by SEQ_2:def 7;
  hence thesis by Th4;
end;
