
theorem LMC31G:
  for k,m be Nat st 2*k + 1 <= m holds
    2 to_power k <= (2 to_power m) / m
  proof
    let k,m be Nat;
    assume A1: 2*k +1 <= m;
    2*k <= 2*k +1 by NAT_1:11; then
    k+k <= m by A1,XXREAL_0:2; then
    X1: k+k-k <= m -k by XREAL_1:9;
    then m -k in NAT by INT_1:3; then
    reconsider n=m-k as Nat;
    A2: n+k <= 2 to_power n by LMC31E,X1;
    2 to_power (n+k) /2 to_power n
    <= 2 to_power (n+k) / (n+k) by A1,A2,XREAL_1:118; then
    2 to_power (n+k-n) <= 2 to_power (n+k) / (n+k) by POWER:29;
    hence 2 to_power k <= 2 to_power m /m;
  end;
