reserve V, C, x, a, b for set;
reserve A, B for Element of SubstitutionSet (V, C);
reserve C for finite set;
reserve A, B for Element of SubstitutionSet (V, C);

theorem
  A = { {} } implies -A = {}
proof
  assume
A1: A = {{}};
  assume -A <> {};
  then consider x1 be object such that
A2: x1 in -A by XBOOLE_0:def 1;
  consider f1 being Element of PFuncs (Involved A, C) such that
  x1 = f1 and
A3: for g be Element of PFuncs (V, C) st g in {{}} holds not f1
  tolerates g by A1,A2;
A4: {} in {{}} by TARSKI:def 1;
  {} is PartFunc of V, C by RELSET_1:12;
  then
A5: {} in PFuncs (V, C) by PARTFUN1:45;
  f1 tolerates {} by PARTFUN1:59;
  hence thesis by A3,A5,A4;
end;
