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 Th14:
  for V being set, C being finite set for A being Element of
  SubstitutionSet (V, C), a being Element of PFuncs (V, C), B being Element of
SubstitutionSet (V, C) st B = { a } holds A ^ B = {} implies ex b being finite
  set st b in -A & b c= a
proof
  let V, C;
  let A be Element of SubstitutionSet (V, C);
  let a be Element of PFuncs (V, C);
  let B be Element of SubstitutionSet (V, C) such that
A1: B = { a };
  assume
A2: A ^ B = {};
  per cases;
  suppose
A3: A is non empty;
    then reconsider AA = A as finite non empty set;
    defpred P[Element of PFuncs (V, C),set] means $2 in dom $1 /\ dom a & $1.
    $2 <> a.$2;
    defpred P[set] means not contradiction;
A4: ex kk be Function st kk = a & dom kk c= V & rng kk c= C by PARTFUN1:def 3;
A5: now
      let s be Element of PFuncs (V, C) such that
A6:   s in A;
      not s tolerates a
      proof
        assume
A7:     s tolerates a;
        a in B by A1,TARSKI:def 1;
        then s \/ a in { s1 \/ t1 where s1,t1 is Element of PFuncs (V,C) : s1
        in A & t1 in B & s1 tolerates t1 } by A6,A7;
        hence thesis by A2,SUBSTLAT:def 3;
      end;
      then consider x be object such that
A8:   x in dom s /\ dom a and
A9:   s.x <> a.x by PARTFUN1:def 4;
      consider a9 be Function such that
A10:  a9 = a and
A11:  dom a9 c= V and
      rng a9 c= C by PARTFUN1:def 3;
      dom s /\ dom a c= dom a9 by A10,XBOOLE_1:17;
      then dom s /\ dom a c= V by A11;
      then reconsider x as Element of [V] by A8,HEYTING1:def 2;
      take x;
      thus P[s,x] by A8,A9;
    end;
    consider g be Element of Funcs (PFuncs (V, C), [V]) such that
A12: for s be Element of PFuncs (V, C) st s in A holds P[s,g.s] from
    FRAENKEL:sch 27(A5);
    deffunc G(set)=g.$1;
A13: A = [A] by A3,HEYTING1:def 2;
    { G(b) where b is Element of AA : P[b] } is finite from PRE_CIRC:sch
    1;
    then reconsider
    UKA = the set of all  g.b where b is Element of [A]  as
    finite set by A13;
    set f = a|UKA;
A14: dom f c= Involved A
    proof
      let x be object;
      assume x in dom f;
      then x in UKA by RELAT_1:57;
      then consider b be Element of [A] such that
A15:  x = g.b;
      reconsider b as finite Function by A13,Th1;
      reconsider b9 = b as Element of PFuncs (V, C) by A13,SETWISEO:9;
      g.b9 in dom b9 /\ dom a by A13,A12;
      then x in dom b by A15,XBOOLE_0:def 4;
      hence thesis by A13,Def1;
    end;
    rng f c= rng a by RELAT_1:70;
    then rng f c= C by A4;
    then reconsider f9 = f as Element of PFuncs (Involved A, C) by A14,
PARTFUN1:def 3;
    for g be Element of PFuncs (V, C) st g in A holds not f tolerates g
    proof
      let g1 be Element of PFuncs (V, C);
      reconsider g9 = g1 as Function;
      assume
A16:  g1 in A;
      ex z be set st z in dom g1 /\ dom f & g9.z <> f.z
      proof
        set z = g.g1;
        take z;
A17:    z in dom g1 /\ dom a by A12,A16;
        then
A18:    z in dom a by XBOOLE_0:def 4;
        z in the set of all  g.b1 where b1 is Element of [A]  by A13,A16;
        then z in dom a /\ UKA by A18,XBOOLE_0:def 4;
        then
A19:    z in dom f by RELAT_1:61;
        z in dom g1 by A17,XBOOLE_0:def 4;
        hence z in dom g1 /\ dom f by A19,XBOOLE_0:def 4;
        g1.z <> a.z by A12,A16;
        hence thesis by A19,FUNCT_1:47;
      end;
      hence thesis by PARTFUN1:def 4;
    end;
    then f9 in { f1 where f1 is Element of PFuncs (Involved A, C) : for g be
    Element of PFuncs (V, C) st g in A holds not f1 tolerates g };
    hence thesis by RELAT_1:59;
  end;
  suppose
A20: A is empty;
    reconsider K = {} as finite set;
    take K;
    -A = {{}} by A20,Th10;
    hence thesis by TARSKI:def 1;
  end;
end;
