
theorem L2:
  for k be Nat
  ex N be Nat st
  for x be Nat st N <= x holds 1/ (2 to_power x) < 1/ (x to_power k)
  proof
    let k be Nat;
    consider N0 be Nat such that
    P1: for x be Nat st N0 <= x
    holds x to_power k < 2 to_power x by L1;
    set N=N0+2;
    take N;
    now let x be Nat;
      assume AS: N <=x;
      N0 <= N by NAT_1:12; then
      N0 <= x by XXREAL_0:2,AS; then
      E1:x to_power k < 2 to_power x by P1;
      0 < x to_power k by POWER:34,AS;
      hence 1/ (2 to_power x) < 1/ (x to_power k)
      by XREAL_1:76,E1;
    end;
    hence thesis;
  end;
