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