reserve V for Universe,
  a,b,x,y,z,x9,y9 for Element of V,
  X for Subclass of V,
  o,p,q,r,s,t,u,a1,a2,a3,A,B,C,D for set,
  K,L,M for Ordinal,
  n for Element of omega,
  fs for finite Subset of omega,
  e,g,h for Function,
  E for non empty set,
  f for Function of VAR,E,
  k,k1 for Element of NAT,
  v1,v2,v3 for Element of VAR,
  H,H9 for ZF-formula;

theorem
  X is closed_wrt_A1-A7 & not n in fs & a in X & a c= X & y in Funcs(fs,
  a) & b c= Funcs(fs \/ {n},a) & b in X implies {x: x in a & {[n,x]} \/ y in b}
  in X
proof
  assume that
A1: X is closed_wrt_A1-A7 and
A2: not n in fs and
A3: a in X and
A4: a c= X and
A5: y in Funcs(fs,a) and
A6: b c= Funcs(fs \/ {n},a) and
A7: b in X;
  y in X by A1,A4,A5,Th14;
  then
A8: {y} in X by A1,Th2;
  set T={{[n,x]} \/ y: x in a};
  set T9= T /\ b;
  T in X by A1,A3,A4,A5,Th15;
  then
A9: T9 in X by A1,A7,Th5;
  then reconsider t9=T9 as Element of V;
  set S={{[n,x]}: {[n,x]} \/ y in b};
  set s={y};
  set R={x\z: x in t9 & z in s};
A10: {[n,x]} \/ y in b implies x in a
  proof
A11: [n,x] in {[n,x]} by TARSKI:def 1;
    assume {[n,x]} \/ y in b;
    then consider g such that
A12: {[n,x]} \/ y = g and
    dom g=fs \/ {n} and
A13: rng g c= a by A6,FUNCT_2:def 2;
    {[n,x]} c= g by A12,XBOOLE_1:7;
    then n in dom g & x=g.n by A11,FUNCT_1:1;
    then x in rng g by FUNCT_1:def 3;
    hence thesis by A13;
  end;
A14: R = S
  proof
    thus R c= S
    proof
      let p be object;
      assume p in R;
      then consider x,z such that
A15:  p=x\z and
A16:  x in t9 and
A17:  z in s;
A18:  x in b by A16,XBOOLE_0:def 4;
      x in T by A16,XBOOLE_0:def 4;
      then consider x9 such that
A19:  x={[n,x9]} \/ y and
      x9 in a;
A20:  {[n,x9]} misses y
      proof
        assume not thesis;
        then consider r being object such that
A21:    r in {[n,x9]} and
A22:    r in y by XBOOLE_0:3;
A23:    ex g st y=g & dom g=fs & rng g c= a by A5,FUNCT_2:def 2;
        r=[n,x9] by A21,TARSKI:def 1;
        hence contradiction by A2,A22,A23,FUNCT_1:1;
      end;
      z=y by A17,TARSKI:def 1;
      then x\z=({[n,x9]}\y) \/ (y\y) by A19,XBOOLE_1:42
        .={[n,x9]} \/ (y\y) by A20,XBOOLE_1:83
        .={[n,x9]} \/ {} by XBOOLE_1:37
        .={[n,x9]};
      hence thesis by A15,A18,A19;
    end;
    let p be object;
    assume p in S;
    then consider x such that
A24: p={[n,x]} and
A25: {[n,x]} \/ y in b;
    reconsider x9={[n,x]} \/ y as Element of V by A7,A25,Th1;
    x in a by A10,A25;
    then x9 in T;
    then
A26: y in s & x9 in t9 by A25,TARSKI:def 1,XBOOLE_0:def 4;
A27: {[n,x]} misses y
    proof
      assume not thesis;
      then consider r being object such that
A28:  r in {[n,x]} and
A29:  r in y by XBOOLE_0:3;
A30:  ex g st y=g & dom g=fs & rng g c= a by A5,FUNCT_2:def 2;
      r=[n,x] by A28,TARSKI:def 1;
      hence contradiction by A2,A29,A30,FUNCT_1:1;
    end;
    x9\y=({[n,x]}\y) \/ (y\y) by XBOOLE_1:42
      .={[n,x]} \/ (y\y) by A27,XBOOLE_1:83
      .={[n,x]} \/ {} by XBOOLE_1:37
      .={[n,x]};
    hence thesis by A24,A26;
  end;
  X is closed_wrt_A6 by A1;
  then R in X by A9,A8;
  then union S in X by A1,A14,Th2;
  then union union S in X by A1,Th2;
  then
A31: union union union S in X by A1,Th2;
  set Z={x: x in a & {[n,x]} \/ y in b};
A32: Z c= union union union S
  proof
    let p be object;
    assume p in Z;
    then consider x such that
A33: p=x and
    x in a and
A34: {[n,x]} \/ y in b;
A35: [n,x] in {[n,x]} by TARSKI:def 1;
A36: {n,x} in {{n,x},{n}} by TARSKI:def 2;
    {[n,x]} in S by A34;
    then {{n,x},{n}} in union S by A35,TARSKI:def 4;
    then
A37: {n,x} in union union S by A36,TARSKI:def 4;
    x in {n,x} by TARSKI:def 2;
    hence thesis by A33,A37,TARSKI:def 4;
  end;
A38: union union union S c= Z \/ {n}
  proof
    let p be object;
    assume p in union union union S;
    then consider C such that
A39: p in C and
A40: C in union union S by TARSKI:def 4;
    consider D such that
A41: C in D and
A42: D in union S by A40,TARSKI:def 4;
    consider E being set such that
A43: D in E and
A44: E in S by A42,TARSKI:def 4;
    consider x such that
A45: E={[n,x]} and
A46: {[n,x]} \/ y in b by A44;
    D=[n,x] by A43,A45,TARSKI:def 1;
    then p in {n,x} or p in {n} by A39,A41,TARSKI:def 2;
    then
A47: p=n or p=x or p in {n} by TARSKI:def 2;
    x in a by A10,A46;
    then p in Z or p in {n} by A46,A47,TARSKI:def 1;
    hence thesis by XBOOLE_0:def 3;
  end;
  per cases;
  suppose
    n in Z;
    then {n} c= Z by ZFMISC_1:31;
    then Z \/ {n} = Z by XBOOLE_1:12;
    hence thesis by A31,A32,A38,XBOOLE_0:def 10;
  end;
  suppose
    not n in Z;
    then
A48: Z misses {n} by ZFMISC_1:50;
    (union union union S) \ {n} c= (Z \/ {n}) \ {n} by A38,XBOOLE_1:33;
    then (union union union S) \ {n} c= (Z\{n}) \/ ({n}\{n}) by XBOOLE_1:42;
    then (union union union S) \ {n} c= Z \/ ({n}\{n}) by A48,XBOOLE_1:83;
    then
A49: (union union union S) \ {n} c= Z \/ {} by XBOOLE_1:37;
    Z \ {n} c= (union union union S) \ {n} by A32,XBOOLE_1:33;
    then Z c= (union union union S) \ {n} by A48,XBOOLE_1:83;
    then
A50: (union union union S) \ {n} = Z by A49;
    n in X by A1,Lm12;
    then {n} in X by A1,Th2;
    hence thesis by A1,A31,A50,Th4;
  end;
end;
