
theorem NE1:
  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 = 1
  proof
    let a,b be positive Real, n be Real;
    (a+b) to_power n = (a to_power n) + (b to_power n) implies n = 1
    proof
      assume
      A1: (a+b) to_power n = (a to_power n) + (b to_power n); then
      A2: not n is heavy positive by LME;
      reconsider n as positive Real by A1,BPC;
      n is light positive or n = 1 by A2,XXREAL_0:1;
      hence thesis by A1,BPA;
    end;
    hence thesis;
  end;
