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)) & s1 is convergent & lim s1 > 1
  implies s is not summable
proof
  assume that
A1: for n holds s1.n = n-root (abs(s).n) and
A2: s1 is convergent and
A3: lim s1 > 1;
  lim s1 - 1 > 0 by A3,XREAL_1:50;
  then consider m such that
A4: for n st n>=m holds |.s1.n-lim s1.|<lim s1-1 by A2,SEQ_2:def 7;
  now
    let n such that
A5: n>=m+1;
A6: s1.n = n-root (abs(s).n) by A1
      .= n-root |.s.n.| by SEQ_1:12;
    m+1>=m by NAT_1:11;
    then n>=m by A5,XXREAL_0:2;
    then |.n-root |.s.n.| - lim s1.| < lim s1 - 1 by A4,A6;
    then -(lim s1 - 1) < n-root |.s.n.| - lim s1 by SEQ_2:1;
    then 1 - lim s1 + lim s1 < n-root |.s.n.| - lim s1 + lim s1 by XREAL_1:6;
    then
A7: |.s.n.| >= 0 & n-root |.s.n.| |^ n >= 1 by COMPLEX1:46,PREPOWER:11;
    m+1>=1 by NAT_1:11;
    then n>=1 by A5,XXREAL_0:2;
    hence |.s.n.| >= 1 by A7,POWER:4;
  end;
  then s is not convergent or lim s<>0 by Th38;
  hence thesis by Th4;
end;
