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 Th12:
  for n st X is closed_wrt_A1-A7 & a in X & b in X holds {{[n,x]}
  \/ y: x in a & y in b} in X
proof
  let n;
  assume that
A1: X is closed_wrt_A1-A7 and
A2: a in X and
A3: b in X;
A4: Funcs({n},a) in X by A1,A2,Th9;
  then reconsider F=Funcs({n},a) as Element of V;
  set Z={{[n,x]} \/ y: x in a & y in b};
  set Y={x \/ y: x in F & y in b};
A5: Y=Z
  proof
    thus Y c= Z
    proof
      let p be object;
      assume p in Y;
      then consider x,y such that
A6:   p=x \/ y and
A7:   x in F and
A8:   y in b;
      consider g such that
A9:   x=g and
A10:  dom g={n} and
A11:  rng g c= a by A7,FUNCT_2:def 2;
      n in dom g by A10,TARSKI:def 1;
      then
A12:  g.n in rng g by FUNCT_1:def 3;
      then reconsider z=g.n as Element of V by A2,A11,Th1;
      p={[n,z]} \/ y by A6,A9,A10,GRFUNC_1:7;
      hence thesis by A8,A11,A12;
    end;
    let p be object;
    assume p in Z;
    then consider x,y such that
A13: p={[n,x]} \/ y and
A14: x in a and
A15: y in b;
    reconsider g={[n,x]} as Function;
    rng g={x} by RELAT_1:9;
    then dom g={n} & rng g c= a by A14,RELAT_1:9,ZFMISC_1:31;
    then
A16: {[n,x]} in F by FUNCT_2:def 2;
    then reconsider z={[n,x]} as Element of V by A4,Th1;
    p=z \/ y by A13;
    hence thesis by A15,A16;
  end;
  X is closed_wrt_A5 by A1;
  hence thesis by A3,A4,A5;
end;
