
theorem
  for x be Nat st 1 < x holds
  not ex N,c be Nat st
  for n be Nat st N <= n holds x to_power n <= c * (n to_power x)
  proof
    let x be Nat;
    assume AS: 1 < x;
    assume CNT: ex N,c be Nat st
    for n be Nat st N <= n holds
    x to_power n <= c * (n to_power x);
    ex N,c be Nat st for n be Nat st N <= n holds
    2 to_power n <= c * (n to_power x)
    proof
      consider N,c be Nat such that CNT2:
      for n be Nat st N <= n holds
      x to_power n <= c * (n to_power x) by CNT;
      take N,c;
      let n be Nat;
      assume N <= n;then
      LCX1: x to_power n <= c * ( n to_power x) by CNT2;
      1+1 <= x by AS,INT_1:7;then
      2 to_power n <= x to_power n by LEMC01;
      hence thesis by LCX1,XXREAL_0:2;
    end;
    hence contradiction by AS,N2POWINPOLY;
  end;
