
theorem LMC31H:
  for k be Nat st number_e < k holds
    ex N be Nat st for n be Nat st N<=n holds 2 to_power k <= n/log(2,n)
  proof let k be Nat;
    assume
    K1: number_e < k;
    set N = 2 to_power (2*k + 1);
    k*1 <= 2*k by XREAL_1:64; then
    X1: k+ 0 < 2*k + 1 by XREAL_1:8;
    2*k + 1 <= 2 to_power (2*k + 1) by LMC31X; then
    k < 2 to_power (2*k + 1) by X1,XXREAL_0:2; then
    DD: number_e < N by K1,XXREAL_0:2;
    take N;
    let n be Nat;
    assume N<=n; then
    B0: N/log(2,N) <=n/log(2,n) by LMC31H1,DD;
    X0: 0 < 2 to_power (2*k + 1) by POWER:34; then
    X1: 2*k + 1 = log(2,N) by POWER:def 3;
    reconsider m= log(2,N) as Nat by X0,POWER:def 3;
    X3: 2 to_power k <= (2 to_power m) / m by LMC31G,X1;
    (2 to_power m) = N by POWER:def 3,X0;
    hence thesis by B0,XXREAL_0:2,X3;
  end;
