
theorem
  for x be Nat st 1 < x holds
    ex N be Nat st for n be Nat st N<=n holds 4 < n/log(x,n)
  proof
    let x be Nat;
    assume AS: 1 < x;
    log(2,x) >= log(2,2) by PRE_FF:10,AS,NAT_1:23; then
    AS2: 1 <= log(2,x) by POWER:52;
    consider N be Nat such that LL1: for n be Nat st N<=n
    holds 4 < n/log(2,n) by LMC31HC;
    take N;
    let n be Nat;
    assume N <= n;then
    CL1: 4 < n/log(2,n) by LL1;
    then 0 <> n;then
    log(2,n) = log(2,x)*log(x,n) by POWER:56,AS;then
    4*log(2,x) < n/(log(x,n)*log(2,x)) *log(2,x) by AS2,XREAL_1:68,CL1;
    then
    4*log(2,x) < ( n/log(x,n)) *(1/log(2,x)) *log(2,x) by XCMPLX_1:103;
    then
    4*log(2,x) < ( n/log(x,n)) * (log(2,x)* (1/log(2,x)));then
    TT11:4*log(2,x) < ( n/log(x,n))*1 by XCMPLX_1:106,AS2;
    4*1 <= 4*log(2,x) by XREAL_1:64,AS2;
    hence thesis by TT11,XXREAL_0:2;
  end;
