
theorem
  for X being set, S being finite Subset of X, n being Element of NAT st
  S is empty or n = 0 holds (S, n)-bag = EmptyBag X
proof
  let X be set, S be finite Subset of X, n be Element of NAT such that
A1: S is empty or n = 0;
  now
    let i be object;
    assume i in X;
    per cases;
    suppose
      i in S;
      hence (S, n)-bag.i = 0 by A1,Th4
        .= (EmptyBag X).i by PBOOLE:5;
    end;
    suppose
      not i in S;
      hence (S, n)-bag.i = 0 by Th3
        .= (EmptyBag X).i by PBOOLE:5;
    end;
  end;
  hence thesis by PBOOLE:3;
end;
