
theorem
  for x be Nat st 1 < x holds
  ex N,c be Nat st
  for n be Nat st N <= n holds n to_power x <= c * (x to_power n)
  proof
    let x be Nat;
    assume AS1:1 < x;
    consider N0 be Element of NAT
    such that
    P1: for n be Nat
    st N0 <=n holds x/log(2,x) < n /log(2,n) by LMC31;
    set N=N0+2;
    reconsider N as Nat;
    reconsider c = 1 as Element of NAT;
    take N,c;
    let n be Nat;
    assume AS2: N<= n;
    N0 <= N by NAT_1:12;
    then N0 <= n by XXREAL_0:2,AS2; then
    E1: x/log(2,x) < n /log(2,n) by P1;
    1+1 <= x by AS1,NAT_1:13;
    then log(2,2) <= log(2,x) by PRE_FF:10; then
    P2: 0 < log(2,x) by POWER:52;
    2 <= N by NAT_1:11; then
    2 <= n by XXREAL_0:2,AS2; then
    log(2,2) <= log(2,n) by PRE_FF:10; then
    P3: 0 < log(2,n) by POWER:52; then
    x/log(2,x) * log(2,n) < n /log(2,n) * log(2,n) by XREAL_1:68,E1;
    then
    x/log(2,x) * log(2,n) < n by P3,XCMPLX_1:87; then
    log(2,n)*(x/log(2,x))*log(2,x) < n *log(2,x) by XREAL_1:68,P2;
    then log(2,n)*((x/log(2,x))*log(2,x)) < n *log(2,x); then
    PP4:log(2,n)*x < n *log(2,x) by P2,XCMPLX_1:87;
    P5: 2 to_power (log(2,n)*x)
    = 2 to_power (log(2,n)) to_power x by POWER:33
    .= n to_power x by POWER:def 3,AS2;
    2 to_power (n *log(2,x)) = 2 to_power (log(2,x)) to_power n by POWER:33
    .= x to_power n by AS1,POWER:def 3;
    hence thesis by PP4,P5,POWER:39;
  end;
