
theorem L1:
  for k be Nat
  ex N be Nat st
  for x be Nat st N <= x holds x to_power k < 2 to_power x
  proof
    let k be Nat;
    consider N0 be Nat such that
    P1: for x be Nat st N0 <= x holds k < x /log(2,x) by LMC31;
    set N=N0+2;
    take N;
    now let x be Nat;
      assume AS2: N<= x;
      N0 <= N by NAT_1:12; then
      N0 <= x by XXREAL_0:2,AS2; then
      E1:k < x /log(2,x) by P1;
      2 <= N by NAT_1:11; then
      2 <= x by XXREAL_0:2,AS2; then
      log(2,2) <= log(2,x) by PRE_FF:10; then
      P3: 0 < log(2,x) by POWER:52; then
      k*log(2,x) < x /log(2,x) * log(2,x) by XREAL_1:68,E1; then
      P4: k*log(2,x) < x by P3,XCMPLX_1:87;
      2 to_power (log(2,x)*k) = 2 to_power (log(2,x)) to_power k by POWER:33
      .= x to_power k by AS2,POWER:def 3;
      hence x to_power k < 2 to_power x by P4,POWER:39;
    end;
    hence thesis;
  end;
