theorem Th14:
  for K,L st K = {} & L = {} holds K =>> L = {[{},{}]}
proof
  let K,L;
  assume that
A1: K = {} and
 L = {};
A2: {} = {}.A;
A3: K = {}.DISJOINT_PAIRS A by A1;
A4: now
    let f;
    thus FinPairUnion(K,pair_diff A.:(f,incl DISJOINT_PAIRS A)) =
    the_unity_wrt FinPairUnion A by A3,NORMFORM:18,SETWISEO:31
      .= [{},{}] by A2,NORMFORM:19;
  end;
A5: { FinPairUnion(K,pair_diff A.:(f,incl DISJOINT_PAIRS A)) : f.:K c= L } =
  {[{},{}]}
  proof
    thus { FinPairUnion(K,pair_diff A.:(f,incl DISJOINT_PAIRS A)) : f.:K c= L
    } c= {[{},{}]}
    proof
      let x be object;
      assume x in { FinPairUnion(K,pair_diff A.:(f,incl DISJOINT_PAIRS A)) :
      f.:K c= L };
      then
      ex f st x = FinPairUnion(K,pair_diff A.:(f,incl DISJOINT_PAIRS A)) &
      f.:K c= L;
      then x = [{},{}] by A4;
      hence thesis by TARSKI:def 1;
    end;
    thus {[{},{}]} c= { FinPairUnion(K,pair_diff A.:(f,incl DISJOINT_PAIRS A))
    : f.: K c= L }
    proof
      set f9 = the (Element of Funcs(DISJOINT_PAIRS A, [:Fin A,Fin A:]));
      let x be object;
      assume x in {[{},{}]};
      then x = [{},{}] by TARSKI:def 1;
      then
A6:   x = FinPairUnion(K,pair_diff A.:(f9,incl DISJOINT_PAIRS A)) by A4;
      f9.:K c= L by A1;
      hence thesis by A6;
    end;
  end;
  [{},{}] is Element of DISJOINT_PAIRS A by Th12;
  hence thesis by A5,ZFMISC_1:46;
end;
