 reserve n,i,k,m for Nat;
 reserve p for Prime;

theorem
  for x being Real holds
    ex y being non zero Nat st x < ln.(ln.(y + 1))
  proof
    let x be Real;
    set N = [/ exp_R.(exp_R.x) \];
A1: exp_R.(exp_R.x) > 0 by SIN_COS:54; then
    N > 0 by INT_1:def 7; then
    N in NAT by INT_1:3; then
    reconsider N as non zero Nat by SIN_COS:54,INT_1:def 7;
    take N;
A3: exp_R.x > 0 by SIN_COS:54;
    ln.(exp_R.(exp_R.x)) < ln.(N + 1) by A1,LogMono,INT_1:32; then
    exp_R.x < ln.(N + 1) by LogExp; then
    ln.(exp_R.x) < ln.(ln.(N + 1)) by A3,LogMono;
    hence thesis by LogExp;
  end;
