
theorem LME:
  for a,b be positive Real, n be Real holds
  (a + b) to_power n > a to_power n + b to_power n iff n is heavy positive
  proof
    let a,b be positive Real, n be Real;
    not n is heavy positive implies (a + b) to_power n <= a to_power n + b
      to_power n
    proof
      assume
      not n is heavy positive; then
      n <= 1 by TA1; then
      per cases by XXREAL_0:1;
      suppose n = 1;
        hence thesis;
      end;
      suppose
        B1: n < 1;
        per cases;
        suppose n is positive; then
          n is light positive by B1,TA1;
          hence thesis by BPA;
        end;
        suppose n is non positive;
          hence thesis by BPC;
        end;
      end;
    end;
    hence thesis by APB;
  end;
