
theorem
  for a be light positive Real, n be heavy positive Real holds
  2 to_power n > (1+a) to_power n - (1-a) to_power n > (2*a) to_power n
  proof let a be light positive Real,n be heavy positive Real;
    A1: a < 1 & n > 1 by TA1;
    (1+a) = (1-a) + 2*a; then
    A2: (1+a) to_power n - (1-a) to_power n >
      (1-a) to_power n + (2*a) to_power n - (1-a) to_power n by APB,XREAL_1:9;
    1+1 > 1+a by A1,XREAL_1:6; then
    2 to_power n > (1+a) to_power n by POWER:37; then
    2 to_power n - 0 > (1+a) to_power n - (1-a) to_power n by XREAL_1:14;
    hence thesis by A2;
  end;
