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
  for V being set, C being finite set, A being Element of Fin PFuncs (V,
  C ) st A = {} holds A =>> A = {{}}
proof
  deffunc G(set)={};
  let V be set, C be finite set, A be Element of Fin PFuncs (V, C);
  defpred P[Function] means dom $1 = A;
  deffunc F(Element of PFuncs (A, A))= union {$1.i \ i where i is Element of
  PFuncs (V, C) : i in A};
A1: ex a being Element of PFuncs (A, A) st P[a]
  proof
    reconsider e = id A as Element of PFuncs (A, A) by PARTFUN1:45;
    take e;
    thus thesis;
  end;
A2: { {} where f is Element of PFuncs (A, A) : P[f] } = {{}} from LATTICE3:
  sch 1 (A1);
  assume
A3: A = {};
  now
    let f be Element of PFuncs (A, A);
    not ex x be object st x in {f.i \ i where i is Element of PFuncs (V, C) :
    i in A}
    proof
      given x be object such that
A4:   x in {f.i \ i where i is Element of PFuncs (V, C) : i in A};
      ex x1 being Element of PFuncs (V, C) st x = f.x1 \ x1 & x1 in A by A4;
      hence contradiction by A3;
    end;
    hence {f.i \ i where i is Element of PFuncs (V, C) : i in A} = {} by
XBOOLE_0:def 1;
  end;
  then
A5: for v being Element of PFuncs (A, A) st P[v] holds F(v) = G(v) by
ZFMISC_1:2;
A6: { F(f) where f is Element of PFuncs (A, A) : P[f]} = { G(f) where f is
  Element of PFuncs (A, A) : P[f] } from FRAENKEL:sch 6 (A5);
  {} is PartFunc of V,C by RELSET_1:12;
  then {} in PFuncs (V,C) by PARTFUN1:45;
  then {{}} c= PFuncs (V,C) by ZFMISC_1:31;
  hence thesis by A6,A2,XBOOLE_1:28;
end;
