
theorem Th12:
  for n being Nat st n > 0 holds
  [\ log (2, 2*n) /] +1 >= [\ log (2, 2*n + 1) /]
proof
  let n be Nat;
  set l22n = log (2, 2*n);
  set l22np1 = log (2, 2*n + 1);
  assume
A1: n > 0;
  then 0+1 <= n by NAT_1:13;
  then 1 < 1 * n + n by XREAL_1:8;
  then 2 * n + 1 < 2 * n + 2 * n by XREAL_1:8;
  then
A2: log (2, 2*n+1) <= log (2, 2*(2*n)) by Th10;
  log (2,2*(2*n)) = log (2,2)+l22n by A1,POWER:53
    .= l22n + 1 by POWER:52;
  then [\l22np1/] <= [\l22n+1/] by A2,Th9;
  hence thesis by INT_1:28;
end;
