reserve W for Universe,
  H for ZF-formula,
  x,y,z,X for set,
  k for Variable,
  f for Function of VAR,W,
  u,v for Element of W;
reserve F for Function,
  A,B,C for Ordinal,
  a,b,b1,b2,c for Ordinal of W,
  fi for Ordinal-Sequence,
  phi for Ordinal-Sequence of W,
  H for ZF-formula;
reserve psi for Ordinal-Sequence;
reserve L for DOMAIN-Sequence of W,
  n for Element of NAT,
  f for Function of VAR,L.a;
reserve x1 for Variable;

theorem
  omega in W & (for a,b st a in b holds L.a c= L.b) & (for a st a <> {}
  & a is limit_ordinal holds L.a = Union (L|a)) implies for H ex phi st phi is
increasing & phi is continuous & for a st phi.a = a & {} <> a for f holds Union
  L,(Union L)!f |= H iff L.a,f |= H
proof
  assume that
A1: omega in W and
A2: for a,b st a in b holds L.a c= L.b and
A3: for a st a <> {} & a is limit_ordinal holds L.a = Union (L|a);
  set M = Union L;
A4: union rng L = M by CARD_3:def 4;
  defpred RT[ZF-formula] means ex phi st phi is increasing & phi is continuous
  & for a st phi.a = a & {} <> a for f holds M,M!f |= $1 iff L.a,f |= $1;
A5: dom L = On W by Def2;
A6: for H st H is universal & RT[the_scope_of H] holds RT[H]
  proof
    deffunc D(Ordinal of W, Ordinal-Sequence) = union($2,$1);
    let H;
    set x0 = bound_in H;
    set H9 = the_scope_of H;
    defpred P[set,set] means ex f being Function of VAR,M st $1 = f & ((ex m
being Element of M st m in L.$2 & M,f/(x0,m) |= 'not' H9) or $2 = 0-element_of
    W & not ex m being Element of M st M,f/(x0,m) |= 'not' H9);
    assume H is universal;
    then
A7: H = All(bound_in H,the_scope_of H) by ZF_LANG:44;
A8: for h being Element of Funcs(VAR,M) qua non empty set ex a st P[h,a]
    proof
      let h be Element of Funcs(VAR,M) qua non empty set;
      reconsider f = h as Element of Funcs(VAR,M);
      reconsider f as Function of VAR,M;
      now
        per cases;
        suppose
          for m being Element of M holds not M,f/(x0,m) |= 'not' H9;
          hence thesis;
        end;
        suppose
A9:       not for m being Element of M holds not M,f/(x0,m) |= 'not' H9;
          thus thesis
          proof
            consider m being Element of M such that
A10:        M,f/(x0,m) |= 'not' H9 by A9;
            consider X be set such that
A11:        m in X and
A12:        X in rng L by A4,TARSKI:def 4;
            consider x being object such that
A13:        x in dom L and
A14:        X = L.x by A12,FUNCT_1:def 3;
            reconsider x as Ordinal by A13;
            reconsider b = x as Ordinal of W by A5,A13,ORDINAL1:def 9;
            take b, f;
            thus thesis by A10,A11,A14;
          end;
        end;
      end;
      hence thesis;
    end;
    consider rho being Function such that
A15: dom rho = Funcs(VAR,M) qua non empty set and
A16: for h being Element of Funcs(VAR,M) qua non empty set ex a st a
    = rho.h & P[h,a] & for b st P[h,b] holds a c= b from ALFA9Universe(A8);
    defpred SI[Ordinal of W,Ordinal of W] means $2 = sup (rho.:Funcs(VAR,L.$1)
    );
A17: for a ex b st SI[a, b]
    proof
      let a;
      set X = rho.:Funcs(VAR,L.a);
A18:  X c= W
      proof
        let x be object;
        assume x in X;
        then consider h being object such that
        h in dom rho and
A19:    h in Funcs(VAR,L.a) and
A20:    x = rho.h by FUNCT_1:def 6;
        Funcs(VAR,L.a) c= Funcs(VAR,M) by Th16,FUNCT_5:56;
        then reconsider h as Element of Funcs(VAR,M) qua non empty set by A19;
        ex a st a = rho.h & (ex f being Function of VAR,M st h = f & ((
        ex m being Element of M st m in L.a & M,f/(x0,m) |= 'not' H9) or a =
0-element_of W & not ex m being Element of M st M,f/(x0,m) |= 'not' H9)) & for
b st ex f being Function of VAR,M st h = f & ((ex m being Element of M st m in
L.b & M,f/(x0,m) |= 'not' H9) or b = 0-element_of W & not ex m being Element of
        M st M,f/(x0,m) |= 'not' H9) holds a c= b by A16;
        hence thesis by A20;
      end;
      Funcs(omega,L.a) in W by A1,CLASSES2:61;
      then
A21:  card Funcs(omega,L.a) in card W by CLASSES2:1;
      card VAR = card omega & card(L.a) = card(L.a) by Th17,CARD_1:5;
      then card Funcs(VAR,L.a) = card Funcs(omega,L.a) by FUNCT_5:61;
      then card X in card W by A21,CARD_1:67,ORDINAL1:12;
      then X in W by A18,CLASSES1:1;
      then reconsider b = sup X as Ordinal of W by Th19;
      take b;
      thus thesis;
    end;
    consider si being Ordinal-Sequence of W such that
A22: for a holds SI[a, si.a] from OrdSeqOfUnivEx(A17);
    deffunc C(Ordinal of W, Ordinal of W) = succ((si.succ $1) \/ $2);
    consider ksi being Ordinal-Sequence of W such that
A23: ksi.0-element_of W = si.0-element_of W and
A24: for a holds ksi.(succ a) = C(a,ksi.a) and
A25: for a st a <> 0-element_of W & a is limit_ordinal holds ksi.a =
    D(a,ksi|a) from UOSExist;
    defpred P[Ordinal] means si.$1 c= ksi.$1;
    given phi such that
A26: phi is increasing and
A27: phi is continuous and
A28: for a st phi.a = a & {} <> a for f holds M,M!f |= the_scope_of H
    iff L.a,f |= the_scope_of H;
A29: ksi is increasing
    proof
      let A,B;
      assume that
A30:  A in B and
A31:  B in dom ksi;
      A in dom ksi by A30,A31,ORDINAL1:10;
      then reconsider a = A, b = B as Ordinal of W by A31,ORDINAL1:def 9;
      defpred Theta[Ordinal of W] means a in $1 implies ksi.a in ksi.$1;
A32:  Theta[c] implies Theta[succ c]
      proof
        assume that
A33:    a in c implies ksi.a in ksi.c and
A34:    a in succ c;
A35:    a c= c by A34,ORDINAL1:22;
A36:    ksi.a in ksi.c or ksi.a = ksi.c
        proof
          per cases;
          suppose
            a <> c;
            then a c< c by A35;
            hence thesis by A33,ORDINAL1:11;
          end;
          suppose
            a = c;
            hence thesis;
          end;
        end;
A37:    ksi.c c= (si.succ c) \/ ksi.c by XBOOLE_1:7;
        ksi.succ c = succ((si.succ c) \/ ksi.c) & (si.succ c) \/ ksi.c
        in succ((si. succ c) \/ ksi.c) by A24,ORDINAL1:22;
        hence thesis by A37,A36,ORDINAL1:10,12;
      end;
A38:  for b st b <> 0-element_of W & b is limit_ordinal & for c st c in
      b holds Theta[c] holds Theta[b]
      proof
        ksi.succ a = succ((si.succ a) \/ ksi.a) by A24;
        then (si.succ a) \/ ksi.a in ksi.succ a by ORDINAL1:6;
        then
A39:    ksi.a in ksi.succ a by ORDINAL1:12,XBOOLE_1:7;
        let b such that
A40:    b <> 0-element_of W and
A41:    b is limit_ordinal and
        for c st c in b holds Theta[c] and
A42:    a in b;
        succ a in dom ksi & succ a in b by A41,A42,ORDINAL1:28,ORDINAL4:34;
        then
A43:    ksi.succ a in rng(ksi|b) by FUNCT_1:50;
        ksi.b = union(ksi|b,b) by A25,A40,A41
          .= Union (ksi|b) by Th14
          .= union rng (ksi|b) by CARD_3:def 4;
        hence thesis by A43,A39,TARSKI:def 4;
      end;
A44:  Theta[0-element_of W] by ORDINAL4:33;
      Theta[c] from UniverseInd(A44,A32,A38);
      then ksi.a in ksi.b by A30;
      hence thesis;
    end;
A45: 0-element_of W = {} by ORDINAL4:33;
A46: ksi is continuous
    proof
      let A,B;
      assume that
A47:  A in dom ksi and
A48:  A <> 0 and
A49:  A is limit_ordinal and
A50:  B = ksi.A;
      A c= dom ksi by A47,ORDINAL1:def 2;
      then
A51:  dom (ksi|A) = A by RELAT_1:62;
      reconsider a = A as Ordinal of W by A47,ORDINAL1:def 9;
A52:  B = union(ksi|a,a) by A25,A45,A48,A49,A50
        .= Union (ksi|a) by Th14
        .= union rng (ksi|a) by CARD_3:def 4;
A53:  B c= sup (ksi|A)
      proof
        let C;
        assume C in B;
        then consider X such that
A54:    C in X and
A55:    X in rng (ksi|A) by A52,TARSKI:def 4;
        rng(ksi|A) c= rng ksi by RELAT_1:70;
        then X in rng ksi by A55;
        then reconsider X as Ordinal;
        X in sup (ksi|A) by A55,ORDINAL2:19;
        hence thesis by A54,ORDINAL1:10;
      end;
A56:  ksi|A is increasing by A29,ORDINAL4:15;
A57:  sup (ksi|A) c= B
      proof
        let C;
        assume C in sup (ksi|A);
        then consider D being Ordinal such that
A58:    D in rng (ksi|A) and
A59:    C c= D by ORDINAL2:21;
        consider x being object such that
A60:    x in dom (ksi|A) and
A61:    D = (ksi|A).x by A58,FUNCT_1:def 3;
        reconsider x as Ordinal by A60;
A62:    succ x in A by A49,A60,ORDINAL1:28;
        then
A63:    (ksi|A).succ x in rng (ksi|A) by A51,FUNCT_1:def 3;
        x in succ x by ORDINAL1:6;
        then D in (ksi|A).succ x by A51,A56,A61,A62;
        then D in B by A52,A63,TARSKI:def 4;
        hence thesis by A59,ORDINAL1:12;
      end;
      sup (ksi|A) is_limes_of ksi|A by A48,A49,A51,A56,ORDINAL4:8;
      hence thesis by A53,A57,XBOOLE_0:def 10;
    end;
A64: a <> 0-element_of W & a is limit_ordinal implies si.a c= sup (si|a)
    proof
      defpred C[object] means $1 in Free 'not' H9;
      assume that
A65:  a <> 0-element_of W and
A66:  a is limit_ordinal;
A67:  si.a = sup (rho.:Funcs(VAR,L.a)) by A22;
      let A;
      assume A in si.a;
      then consider B such that
A68:  B in rho.:Funcs(VAR,L.a) and
A69:  A c= B by A67,ORDINAL2:21;
      consider x being object such that
A70:  x in dom rho and
A71:  x in Funcs(VAR,L.a) and
A72:  B = rho.x by A68,FUNCT_1:def 6;
      reconsider h = x as Element of Funcs(VAR,M) qua non empty set by A15,A70;
      consider a1 being Ordinal of W such that
A73:  a1 = rho.h and
A74:  ex f being Function of VAR,M st h = f & ((ex m being Element
of M st m in L.a1 & M,f/(x0,m) |= 'not' H9) or a1 = 0-element_of W & not ex m
      being Element of M st M,f/(x0,m) |= 'not' H9) and
A75:  for b st ex f being Function of VAR,M st h = f & ((ex m being
Element of M st m in L.b & M,f/(x0,m) |= 'not' H9) or b = 0-element_of W & not
      ex m being Element of M st M,f/(x0,m) |= 'not' H9) holds a1 c= b by A16;
      consider f being Function of VAR,M such that
A76:  h = f and
A77:  (ex m being Element of M st m in L.a1 & M,f/(x0,m) |= 'not' H9
) or a1 = 0-element_of W & not ex m being Element of M st M,f/(x0,m) |= 'not'
      H9 by A74;
      defpred P[object,object] means
      for a st f.$1 in L.a holds f.$2 in L.a;
A78:  now
A79:    for x,y being object holds P[x,y] or P[y,x]
        proof
          let x,y be object;
          given a such that
A80:      f.x in L.a & not f.y in L.a;
          let b such that
A81:      f.y in L.b;
          a in b or a = b or b in a by ORDINAL1:14;
          then L.a c= L.b or L.b c= L.a by A2;
          hence thesis by A80,A81;
        end;
        assume Free 'not' H9 <> {};
        then
A82:    Free 'not' H9 <> {};
A83:    L.a = Union (L|a) & Union (L|a) = union rng (L|a ) by A3,A45,A65,A66,
CARD_3:def 4;
A84:    for x,y,z being object st P[x,y] & P[y,z] holds P[x,z];
        consider z being object such that
A85:    z in Free 'not' H9 &
for y being object st y in Free 'not' H9 holds P[z,y
        ] from CARD_2:sch 2(A82,A79,A84);
        reconsider z as Variable by A85;
A86:    dom (L|a) c= a by RELAT_1:58;
A87:    ex s being Function st f = s & dom s = VAR & rng s c= L.a by A71,A76,
FUNCT_2:def 2;
        then f.z in rng f by FUNCT_1:def 3;
        then consider X such that
A88:    f.z in X and
A89:    X in rng (L|a) by A87,A83,TARSKI:def 4;
        consider x being object such that
A90:    x in dom (L|a) and
A91:    X = (L|a).x by A89,FUNCT_1:def 3;
A92:    (L|a).x = L.x by A90,FUNCT_1:47;
        reconsider x as Ordinal by A90;
        a in On W by ORDINAL1:def 9;
        then x in On W by A90,A86,ORDINAL1:10;
        then reconsider x as Ordinal of W by ORDINAL1:def 9;
        take x;
        thus x in a by A90,A86;
        let y be Variable;
        assume y in Free 'not' H9;
        hence f.y in L.x by A85,A88,A91,A92;
      end;
      now
        assume
A93:    Free 'not' H9 = {};
        take b = 0-element_of W;
        thus b in a by A45,A65,ORDINAL3:8;
        thus for x1 st x1 in Free 'not' H9 holds f.x1 in L.b by A93;
      end;
      then consider c such that
A94:  c in a and
A95:  for x1 st x1 in Free 'not' H9 holds f.x1 in L.c by A78;
A96:  si.c = sup (rho.:Funcs(VAR,L.c)) by A22;
      c in dom si & dom (si|a) = dom si /\ a by ORDINAL4:34,RELAT_1:61;
      then
A97:  c in dom (si|a) by A94,XBOOLE_0:def 4;
      si.c = (si|a).c by A94,FUNCT_1:49;
      then si.c in rng (si|a) by A97,FUNCT_1:def 3;
      then
A98:  si.c in sup (si|a) by ORDINAL2:19;
      deffunc F(object) = f.$1;
      set e = the Element of L.c;
      deffunc G(object) = e;
      consider v being Function such that
A99: dom v = VAR & for x being object st x in VAR
holds (C[x] implies v.x = F(x)
      ) & (not C[x] implies v.x = G(x)) from PARTFUN1:sch 1;
A100: rng v c= L.c
      proof
        let x be object;
        assume x in rng v;
        then consider y being object such that
A101:   y in dom v and
A102:   x = v.y by FUNCT_1:def 3;
        reconsider y as Variable by A99,A101;
        y in Free 'not' H9 or not y in Free 'not' H9;
        then x = f.y & f.y in L.c or x = e by A95,A99,A102;
        hence thesis;
      end;
      then reconsider v as Function of VAR,L.c by A99,FUNCT_2:def 1,RELSET_1:4;
A103: v in Funcs(VAR,L.c) by A99,A100,FUNCT_2:def 2;
      Funcs(VAR,L.c) c= Funcs(VAR,M) by Th16,FUNCT_5:56;
      then reconsider
      v9 = v as Element of Funcs(VAR,M) qua non empty set by A103;
      consider a2 being Ordinal of W such that
A104: a2 = rho.v9 and
A105: ex f being Function of VAR,M st v9 = f & ((ex m being Element
of M st m in L.a2 & M,f/(x0,m) |= 'not' H9) or a2 = 0-element_of W & not ex m
      being Element of M st M,f/(x0,m) |= 'not' H9) and
A106: for b st ex f being Function of VAR,M st v9 = f & ((ex m being
Element of M st m in L.b & M,f/(x0,m) |= 'not' H9) or b = 0-element_of W & not
      ex m being Element of M st M,f/(x0,m) |= 'not' H9) holds a2 c= b by A16;
      consider f9 being Function of VAR,M such that
A107: v9 = f9 and
A108: (ex m being Element of M st m in L.a2 & M,f9/(x0,m) |= 'not'
H9) or a2 = 0-element_of W & not ex m being Element of M st M,f9/(x0,m) |=
      'not' H9 by A105;
A109: v = M!v by Th16,ZF_LANG1:def 1;
A110: now
        given m being Element of M such that
A111:   m in L.a1 and
A112:   M,f/(x0,m) |= 'not' H9;
A113:   now
          let x1;
A114:     f/(x0,m).x0 = m by FUNCT_7:128;
A115:     x1 <> x0 implies f/(x0,m).x1 = f.x1 & (M!v)/(x0,m).x1 = (M!v).
          x1 by FUNCT_7:32;
          assume x1 in Free 'not' H9;
          hence f/(x0,m).x1 = (M!v)/(x0,m).x1 by A99,A109,A114,A115,FUNCT_7:128
;
        end;
        then M,(M!v)/(x0,m) |= 'not' H9 by A112,ZF_LANG1:75;
        then consider m9 being Element of M such that
A116:   m9 in L.a2 & M,f9/(x0,m9) |= 'not' H9 by A109,A107,A108;
        now
          let x1;
A117:     f/(x0,m9).x0 = m9 by FUNCT_7:128;
A118:     x1 <> x0 implies f/(x0,m9).x1 = f.x1 & (M!v)/(x0,m9).x1 = (M!v
          ).x1 by FUNCT_7:32;
          assume x1 in Free 'not' H9;
          hence f/(x0,m9).x1 = f9/(x0,m9).x1 by A99,A109,A107,A117,A118,
FUNCT_7:128;
        end;
        then
A119:   a1 c= a2 by A75,A76,A116,ZF_LANG1:75;
        a2 c= a1 by A109,A106,A111,A112,A113,ZF_LANG1:75;
        hence a1 = a2 by A119;
      end;
      now
        assume
A120:   not ex m being Element of M st M,f/(x0,m) |= 'not' H9;
        now
          let m be Element of M;
          now
            let x1;
A121:       f/(x0,m).x0 = m by FUNCT_7:128;
A122:       x1 <> x0 implies f/(x0,m).x1 = f.x1 & (M!v)/(x0,m).x1 = (M!v
            ).x1 by FUNCT_7:32;
            assume x1 in Free 'not' H9;
            hence f/(x0,m).x1 = (M!v)/(x0,m).x1 by A99,A109,A121,A122,
FUNCT_7:128;
          end;
          hence not M,(M!v)/(x0,m) |= 'not' H9 by A120,ZF_LANG1:75;
        end;
        hence a1 = a2 by A77,A109,A107,A108,A120;
      end;
      then B in rho.:Funcs(VAR,L.c) by A15,A72,A73,A74,A76,A103,A104,A110,
FUNCT_1:def 6;
      then B in si.c by A96,ORDINAL2:19;
      then B in sup (si|a) by A98,ORDINAL1:10;
      hence thesis by A69,ORDINAL1:12;
    end;
A123: a <> 0-element_of W & a is limit_ordinal & (for b st b in a holds P[
    b]) implies P[a]
    proof
      assume that
A124: a <> 0-element_of W & a is limit_ordinal and
A125: for b st b in a holds si.b c= ksi.b;
      thus si.a c= ksi.a
      proof
A126:   si.a c= sup (si|a) by A64,A124;
        let A;
        assume A in si.a;
        then consider B such that
A127:   B in rng (si|a) and
A128:   A c= B by A126,ORDINAL2:21;
        consider x being object such that
A129:   x in dom (si|a) and
A130:   B = (si|a).x by A127,FUNCT_1:def 3;
        reconsider x as Ordinal by A129;
A131:   a in On W by ORDINAL1:def 9;
        x in dom si by A129,RELAT_1:57;
        then x in On W;
        then reconsider x as Ordinal of W by ORDINAL1:def 9;
A132:   si.x = B by A129,A130,FUNCT_1:47;
A133:   si.x c= ksi.x by A125,A129;
        dom ksi = On W by FUNCT_2:def 1;
        then ksi.x in ksi.a by A29,A129,A131;
        hence thesis by A128,A132,A133,ORDINAL1:12,XBOOLE_1:1;
      end;
    end;
A134: P[a] implies P[succ a]
    proof
      assume si.a c= ksi.a;
      ksi.succ a = succ((si.succ a) \/ (ksi.a)) & (si.succ a) \/ (ksi.a)
      in succ(( si.succ a) \/ (ksi.a)) by A24,ORDINAL1:6;
      then si.succ a in ksi.succ a by ORDINAL1:12,XBOOLE_1:7;
      hence si.succ a c= ksi.succ a by ORDINAL1:def 2;
    end;
A135: P[0-element_of W] by A23;
A136: P[a] from UniverseInd(A135,A134,A123);
A137: now
      assume x0 in Free H9;
      thus thesis
      proof
        take gamma = phi*ksi;
        ex f being Ordinal-Sequence st f = phi*ksi & f is increasing by A26,A29
,ORDINAL4:13;
        hence gamma is increasing & gamma is continuous by A27,A29,A46,
ORDINAL4:17;
        let a such that
A138:   gamma.a = a and
A139:   {} <> a;
        let f;
        a in dom gamma by ORDINAL4:34;
        then
A140:   phi.(ksi.a) = gamma.a by FUNCT_1:12;
        a in dom ksi by ORDINAL4:34;
        then
A141:   a c= ksi.a by A29,ORDINAL4:10;
        ksi.a in dom phi by ORDINAL4:34;
        then
A142:   ksi.a c= phi.(ksi.a) by A26,ORDINAL4:10;
        then
A143:   ksi.a = a by A138,A141,A140;
A144:   phi.a = a by A138,A142,A141,A140,XBOOLE_0:def 10;
        thus M,M!f |= H implies L.a,f |= H
        proof
          assume
A145:     M,M!f |= H;
          now
            let g be Function of VAR,L.a such that
A146:       for k st g.k <> f.k holds x0 = k;
            now
              let k;
              M!f = f & M!g = g by Th16,ZF_LANG1:def 1;
              hence (M!g).k <> (M!f).k implies x0 = k by A146;
            end;
            then M,(M!g) |= H9 by A7,A145,ZF_MODEL:16;
            hence L.a,g |= H9 by A28,A139,A144;
          end;
          hence thesis by A7,ZF_MODEL:16;
        end;
        assume that
A147:   L.a,f |= H and
A148:   not M,M!f |= H;
        consider m being Element of M such that
A149:   not M,(M!f)/(x0,m) |= H9 by A7,A148,ZF_LANG1:71;
A150:   si.a c= ksi.a by A136;
A151:   si.a = sup (rho.:Funcs(VAR,L.a)) by A22;
        reconsider h = M!f as Element of Funcs(VAR,M) qua non empty set by
FUNCT_2:8;
        consider a1 being Ordinal of W such that
A152:   a1 = rho.h and
A153:   ex f being Function of VAR,M st h = f & ((ex m being Element
of M st m in L.a1 & M,f/(x0,m) |= 'not' H9) or a1 = 0-element_of W & not ex m
        being Element of M st M,f/(x0,m) |= 'not' H9) and
        for b st ex f being Function of VAR,M st h = f & ((ex m being
Element of M st m in L.b & M,f/(x0,m) |= 'not' H9) or b = 0-element_of W & not
ex m being Element of M st M,f/(x0,m) |= 'not' H9) holds a1 c= b by A16;
A154:   M!f = f by Th16,ZF_LANG1:def 1;
        M,(M!f)/(x0,m) |= 'not' H9 by A149,ZF_MODEL:14;
        then consider m being Element of M such that
A155:   m in L.a1 and
A156:   M,(M!f)/(x0,m) |= 'not' H9 by A153;
        f in Funcs(VAR,L.a) by FUNCT_2:8;
        then a1 in rho.:Funcs(VAR,L.a) by A15,A152,A154,FUNCT_1:def 6;
        then a1 in si.a by A151,ORDINAL2:19;
        then L.a1 c= L.a by A2,A143,A150;
        then reconsider m9 = m as Element of L.a by A155;
        (M!f)/(x0,m) = M!(f/(x0,m9)) by A154,Th16,ZF_LANG1:def 1;
        then not M,M!(f/(x0,m9)) |= H9 by A156,ZF_MODEL:14;
        then not L.a,f/(x0,m9) |= H9 by A28,A139,A144;
        hence contradiction by A7,A147,ZF_LANG1:71;
      end;
    end;
    now
      assume
A157: not x0 in Free H9;
      thus thesis
      proof
        take phi;
        thus phi is increasing & phi is continuous by A26,A27;
        let a such that
A158:   phi.a = a & {} <> a;
        let f;
        thus M,M!f |= H implies L.a,f |= H
        proof
A159:     for k st (M!f).k <> (M!f).k holds x0 = k;
          assume M,M!f |= H;
          then M,M!f |= H9 by A7,A159,ZF_MODEL:16;
          then L.a,f |= H9 by A28,A158;
          hence thesis by A7,A157,ZFMODEL1:10;
        end;
A160:   for k st f.k <> f.k holds x0 = k;
        assume L.a,f |= H;
        then L.a,f |= H9 by A7,A160,ZF_MODEL:16;
        then M,M!f |= H9 by A28,A158;
        hence thesis by A7,A157,ZFMODEL1:10;
      end;
    end;
    hence thesis by A137;
  end;
A161: for H st H is conjunctive & RT[the_left_argument_of H] & RT[
  the_right_argument_of H] holds RT[H]
  proof
    let H;
    assume H is conjunctive;
    then
A162: H = (the_left_argument_of H) '&' (the_right_argument_of H) by ZF_LANG:40;
    given fi1 being Ordinal-Sequence of W such that
A163: fi1 is increasing and
A164: fi1 is continuous and
A165: for a st fi1.a = a & {} <> a for f holds M,M!f |=
    the_left_argument_of H iff L.a,f |= the_left_argument_of H;
    given fi2 being Ordinal-Sequence of W such that
A166: fi2 is increasing and
A167: fi2 is continuous and
A168: for a st fi2.a = a & {} <> a for f holds M,M!f |=
    the_right_argument_of H iff L.a,f |= the_right_argument_of H;
    take phi = fi2*fi1;
    ex fi st fi = fi2*fi1 & fi is increasing by A163,A166,ORDINAL4:13;
    hence phi is increasing & phi is continuous by A163,A164,A167,ORDINAL4:17;
    let a such that
A169: phi.a = a and
A170: {} <> a;
    a in dom fi1 by ORDINAL4:34;
    then
A171: a c= fi1.a by A163,ORDINAL4:10;
    let f;
A172: a in dom phi by ORDINAL4:34;
A173: a in dom fi2 by ORDINAL4:34;
A174: now
      assume fi1.a <> a;
      then a c< fi1.a by A171;
      then
A175: a in fi1.a by ORDINAL1:11;
A176: phi.a = fi2.(fi1.a) by A172,FUNCT_1:12;
      fi1.a in dom fi2 by ORDINAL4:34;
      then fi2.a in fi2.(fi1.a) by A166,A175;
      hence contradiction by A166,A169,A173,A176,ORDINAL1:5,ORDINAL4:10;
    end;
    then
A177: fi2.a = a by A169,A172,FUNCT_1:12;
    thus M,M!f |= H implies L.a,f |= H
    proof
      assume
A178: M,M!f |= H;
      then M,M!f |= the_right_argument_of H by A162,ZF_MODEL:15;
      then
A179: L.a,f |= the_right_argument_of H by A168,A170,A177;
      M,M!f |= the_left_argument_of H by A162,A178,ZF_MODEL:15;
      then L.a,f |= the_left_argument_of H by A165,A170,A174;
      hence thesis by A162,A179,ZF_MODEL:15;
    end;
    assume
A180: L.a,f |= H;
    then L.a,f |= the_right_argument_of H by A162,ZF_MODEL:15;
    then
A181: M,M!f |= the_right_argument_of H by A168,A170,A177;
    L.a,f |= the_left_argument_of H by A162,A180,ZF_MODEL:15;
    then M,M!f |= the_left_argument_of H by A165,A170,A174;
    hence thesis by A162,A181,ZF_MODEL:15;
  end;
A182: for H st H is atomic holds RT[H]
  proof
A183: dom id On W = On W;
    then reconsider phi = id On W as Sequence by ORDINAL1:def 7;
A184: rng id On W = On W;
    reconsider phi as Ordinal-Sequence;
    reconsider phi as Ordinal-Sequence of W by A183,A184,FUNCT_2:def 1
,RELSET_1:4;
    let H such that
A185: H is being_equality or H is being_membership;
    take phi;
    thus
A186: phi is increasing
    proof
      let A,B;
      assume
A187: A in B & B in dom phi;
      then phi.A = A by FUNCT_1:18,ORDINAL1:10;
      hence thesis by A187,FUNCT_1:18;
    end;
    thus phi is continuous
    proof
      let A,B;
      assume that
A188: A in dom phi and
A189: A <> 0 & A is limit_ordinal and
A190: B = phi.A;
A191: A c= dom phi by A188,ORDINAL1:def 2;
      phi|A = phi*(id A) by RELAT_1:65
        .= id((dom phi) /\ A) by FUNCT_1:22
        .= id A by A191,XBOOLE_1:28;
      then
A192: rng(phi|A) = A;
      phi.A = A by A188,FUNCT_1:18;
      then
A193: sup(phi|A) = B by A190,A192,ORDINAL2:18;
A194: phi|A is increasing by A186,ORDINAL4:15;
      dom (phi|A) = A by A191,RELAT_1:62;
      hence thesis by A189,A193,A194,ORDINAL4:8;
    end;
    let a such that
    phi.a = a and
    {} <> a;
    let f;
    thus M,M!f |= H implies L.a,f |= H
    proof
      assume
A195: M,M!f |= H;
A196: M!f = f by Th16,ZF_LANG1:def 1;
A197: now
        assume H is being_membership;
        then
A198:   H = (Var1 H) 'in' (Var2 H) by ZF_LANG:37;
        then (M!f).Var1 H in (M!f).Var2 H by A195,ZF_MODEL:13;
        hence thesis by A196,A198,ZF_MODEL:13;
      end;
      now
        assume H is being_equality;
        then
A199:   H = (Var1 H) '=' (Var2 H) by ZF_LANG:36;
        then (M!f).Var1 H = (M!f).Var2 H by A195,ZF_MODEL:12;
        hence thesis by A196,A199,ZF_MODEL:12;
      end;
      hence thesis by A185,A197;
    end;
    assume
A200: L.a,f |= H;
A201: M!f = f by Th16,ZF_LANG1:def 1;
A202: now
      assume H is being_membership;
      then
A203: H = (Var1 H) 'in' (Var2 H) by ZF_LANG:37;
      then f.Var1 H in f.Var2 H by A200,ZF_MODEL:13;
      hence thesis by A201,A203,ZF_MODEL:13;
    end;
    now
      assume H is being_equality;
      then
A204: H = (Var1 H) '=' (Var2 H) by ZF_LANG:36;
      then f.Var1 H = f.Var2 H by A200,ZF_MODEL:12;
      hence thesis by A201,A204,ZF_MODEL:12;
    end;
    hence thesis by A185,A202;
  end;
A205: for H st H is negative & RT[the_argument_of H] holds RT[H]
  proof
    let H;
    assume H is negative;
    then
A206: H = 'not' the_argument_of H by ZF_LANG:def 30;
    given phi such that
A207: phi is increasing & phi is continuous and
A208: for a st phi.a = a & {} <> a for f holds M,M!f |= the_argument_of
    H iff L.a,f |= the_argument_of H;
    take phi;
    thus phi is increasing & phi is continuous by A207;
    let a such that
A209: phi.a = a & {} <> a;
    let f;
    thus M,M!f |= H implies L.a,f |= H
    proof
      assume M,M!f |= H;
      then not M,M!f |= the_argument_of H by A206,ZF_MODEL:14;
      then not L.a,f |= the_argument_of H by A208,A209;
      hence thesis by A206,ZF_MODEL:14;
    end;
    assume L.a,f |= H;
    then not L.a,f |= the_argument_of H by A206,ZF_MODEL:14;
    then not M,M!f |= the_argument_of H by A208,A209;
    hence thesis by A206,ZF_MODEL:14;
  end;
  thus RT[H] from ZF_LANG:sch 1(A182,A205,A161,A6);
end;
