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

theorem
  (for n holds s1.n = n-root(abs(s).n)) & (ex m st for n st m<=n holds
  s1.n >= 1 ) implies s is not summable
proof
  assume
A1: for n holds s1.n = n-root (abs(s).n);
  given m such that
A2: for n st m<=n holds s1.n>=1;
  now
    let n such that
A3: n>=m+1;
    m+1>=1 by NAT_1:11;
    then
A4: n>=1 by A3,XXREAL_0:2;
    m+1>=m by NAT_1:11;
    then
A5: n>=m by A3,XXREAL_0:2;
    s1.n = n-root (abs(s).n) by A1
      .= n-root |.s.n.| by SEQ_1:12;
    then |.s.n.| >= 0 & n-root |.s.n.| |^ n >= 1 by A2,A5,COMPLEX1:46
,PREPOWER:11;
    hence |.s.n.| >= 1 by A4,POWER:4;
  end;
  then s is not convergent or lim s<>0 by Th38;
  hence thesis by Th4;
end;
