
theorem BPC:
  for a,b be positive Real, n be non positive Real holds
  a to_power n + b to_power n > (a + b) to_power n
  proof
    let a,b be positive Real, n be non positive Real;
    reconsider k = a/b as positive Real;
    k + 1 = a/b + b/b by XCMPLX_1:60 .= (a + b)/b; then
    A1: (k+1)*b = a + b & k*b = a by XCMPLX_1:87;
    A2: ((k+1) to_power n)*(b to_power n) = ((k+1)*b) to_power n &
    (k to_power n)*(b to_power n) = (k*b) to_power n by POWER:30;
    ((k+1) to_power n)*(b to_power n) < ((k to_power n) + 1)*(b to_power n)
      by XREAL_1:68,LMN;
    hence thesis by A1,A2;
  end;
