
theorem IL1:
  for a be light positive Real, n be heavy positive Real holds
  ((1+a) to_power n)*((1-a) to_power n) < (1 + a to_power n)*(1 - a to_power n)
  proof
    let a be light positive Real, n be heavy positive Real;
A1: ((1+a) to_power n) * ((1-a) to_power n) = ((1-a)*(1+a)) to_power n
      by POWER:30
    .= (1 - a*a) to_power n;
A2: (1 + a to_power n)*(1 - a to_power n) = 1 - (a to_power n)*(a to_power n)
    .= 1 - (a*a) to_power n by POWER:30;
    (1 - a*a) to_power n + ((a*a) to_power n) < ((1 - a*a) + a*a) to_power n
      by APB; then
    (1 - a*a) to_power n + ((a*a) to_power n) - ((a*a) to_power n) <
      1 to_power n - (a*a) to_power n by XREAL_1:9;
    hence thesis by A1,A2;
  end;
