reserve Al for QC-alphabet,
     PHI for Consistent Subset of CQC-WFF(Al),
     PSI for Subset of CQC-WFF(Al),
     p,q,r,s for Element of CQC-WFF(Al),
     A for non empty set,
     J for interpretation of Al,A,
     v for Element of Valuations_in(Al,A),
     m,n,i,j,k for Element of NAT,
     l for CQC-variable_list of k,Al,
     P for QC-pred_symbol of k,Al,
     x,y for bound_QC-variable of Al,
     z for QC-symbol of Al,
     Al2 for Al-expanding QC-alphabet;
reserve J2 for interpretation of Al2,A,
        Jp for interpretation of Al,A,
        v2 for Element of Valuations_in(Al2,A),
        vp for Element of Valuations_in(Al,A);

theorem Th10:
  ex Al2 being Al-expanding QC-alphabet,
     PSI being Consistent Subset of CQC-WFF(Al2)
  st PHI c= PSI & PSI is with_examples
proof
  deffunc S(Nat) = $1-th_FCEx(Al);
  deffunc PSI(Nat) = $1-th_EF(Al,PHI);
  set Al2 = union the set of all S(n) ;
  set PSI = union the set of all PSI(n) ;
A1: PHI c= PSI
  proof
    PHI = PSI(0) by Def9;
    then PHI in the set of all PSI(n) ;
    hence PHI c= PSI by ZFMISC_1:74;
  end;
A2: Al c= Al2 & for n holds S(n) c= Al2
  proof
    Al = S(0) by Def7;
    then Al in the set of all S(n) ;
    hence Al c= Al2 by ZFMISC_1:74;
    let n;
    S(n) in the set of all S(k) ;
    hence S(n) c= Al2 by ZFMISC_1:74;
  end;
  reconsider Al2 as non empty set by A2;
  set Al2sym = union the set of all QC-symbols(S(n)) ;
    NAT c= Al2sym & Al2 = [:NAT,Al2sym:]
  proof
    for s being object st s in Al2 holds s in [:NAT,Al2sym:]
    proof
      let s be object such that
A3:   s in Al2;
      consider P being set such that
A4:   s in P & P in the set of all S(n)  by A3,TARSKI:def 4;
      consider n being Element of NAT such that
A5:   P = S(n) by A4;
A6:   for y being set st y in QC-symbols(S(n)) holds y in Al2sym
      proof
        let y be set such that
A7:      y in QC-symbols(S(n));
         QC-symbols(S(n)) in the set of all QC-symbols(S(k)) ;
         hence y in Al2sym by A7,TARSKI:def 4;
      end;
      s in [:NAT,QC-symbols(S(n)):] by A5,A4,QC_LANG1:5;
      then ex k , y being object st k in NAT & y in QC-symbols(S(n)) &
        s = [k,y] by ZFMISC_1:def 2;
      then ex k being set,y being set st k in NAT &y in Al2sym & s=[k,y] by A6;
      hence thesis by ZFMISC_1:87;
    end;
    then
A8: Al2 c= [:NAT,Al2sym:];
    QC-symbols(Al) = QC-symbols(S(0)) by Def7;
    then QC-symbols(Al) in the set of all QC-symbols(S(n)) ;
    then NAT c= QC-symbols(Al) & QC-symbols(Al) c= Al2sym
     by QC_LANG1:3,ZFMISC_1:74;
    hence NAT c= Al2sym;
    for x being object st x in [:NAT,Al2sym:] holds x in Al2
    proof
      let x be object such that
A9:   x in [:NAT,Al2sym:];
      consider m,y being object such that
A10:   m in NAT & y in Al2sym & x = [m,y] by A9,ZFMISC_1:def 2;
      consider P being set such that
A11:  y in P & P in the set of all QC-symbols(S(n))  by A10,TARSKI:def 4;
      consider n being Element of NAT such that
A12:  P = QC-symbols(S(n)) by A11;
      [m,y] in [:NAT,QC-symbols(S(n)):] by A10,A11,A12,ZFMISC_1:87;
      then
A13:  [m,y] in S(n) by QC_LANG1:5;
      S(n) c= Al2 by A2;
      hence thesis by A10,A13;
    end;
    then [:NAT, Al2sym:] c= Al2;
    hence Al2 = [:NAT, Al2sym:] by A8,XBOOLE_0:def 10;
  end;
  then reconsider Al2 as QC-alphabet by QC_LANG1:def 1;
  reconsider Al2 as Al-expanding QC-alphabet by A2,QC_TRANS:def 1;
  for p being object st p in PSI holds p in CQC-WFF(Al2)
  proof
    let p be object such that
A14: p in PSI;
    consider P being set such that
A15: p in P & P in the set of all PSI(n)  by A14,TARSKI:def 4;
    consider n being Element of NAT such that
A16: P = PSI(n) by A15;
    Al2 is S(n)-expanding QC-alphabet by A2,QC_TRANS:def 1;
    then p is Element of CQC-WFF(Al2) by QC_TRANS:7,A15,A16;
    hence thesis;
  end;
  then reconsider PSI as Subset of CQC-WFF(Al2) by TARSKI:def 3;
  PSI is Consistent
  proof
    defpred C[Nat] means PSI($1) is Consistent & PSI($1) is Al2-Consistent;
A17: C[0]
    proof
A18:  PSI(0) = PHI by Def9;
      PHI is Al2-Consistent by QC_TRANS:23;
      then S(0) = Al & for S being Subset of CQC-WFF(Al2) st PSI(0) = S holds
      S is Consistent by A18,Def7,QC_TRANS:def 2;
      hence thesis by A18,QC_TRANS:def 2;
    end;
A19: for n being Nat holds C[n] implies C[n+1]
    proof
      let n be Nat;
A20:  FCEx(S(n)) = S(n+1) by Th5;
      reconsider Al2 as S(n+1)-expanding QC-alphabet by A2,QC_TRANS:def 1;
      assume C[n];
      then reconsider PSIn = PSI(n) as Consistent Subset of CQC-WFF(S(n));
      PSI(n+1) = PSIn \/ Example_Formulae_of(S(n)) by Def9;
      then reconsider PSIn1 = PSI(n+1) as Consistent Subset of CQC-WFF(S(n+1))
       by A20,Th9;
      PSIn1 is Al2-Consistent by QC_TRANS:23;
      hence thesis;
    end;
A21: for n being Nat holds C[n] from NAT_1:sch 2(A17,A19);
A22: for n holds PSI(n) c= PSI
    proof
      let n;
      let p be object such that
A23:  p in PSI(n);
      PSI(n) in the set of all PSI(k);
      hence p in PSI by A23,TARSKI:def 4;
    end;
A24: for n holds PSI(n) in bool CQC-WFF(Al2)
    proof
      let n;
      PSI(n) c= PSI & PSI c= CQC-WFF(Al2) by A22;
      then PSI(n) c= CQC-WFF(Al2);
      hence thesis;
    end;
    consider f being Function such that
A25: dom f = NAT & for n holds f.n = PSI(n) from FUNCT_1:sch 4;
    for y being object st y in rng f holds y in bool CQC-WFF(Al2)
    proof
      let y be object such that
A26:  y in rng f;
      consider x being object such that
A27:   x in dom f & y = f.x by A26,FUNCT_1:def 3;
      reconsider x as Element of NAT by A25,A27;
      f.x = PSI(x) by A25;
      hence thesis by A24,A27;
    end;
    then reconsider f as sequence of bool CQC-WFF(Al2)
     by A25,FUNCT_2:2,TARSKI:def 3;
    set PSIp = union rng f;
    f in Funcs(NAT,bool CQC-WFF(Al2)) by FUNCT_2:8;
    then union rng f c= union (bool CQC-WFF(Al2)) by ZFMISC_1:77,FUNCT_2:92;
    then reconsider PSIp as Subset of CQC-WFF(Al2) by ZFMISC_1:81;
    for n,m being Nat
     st m in dom f & n in dom f & n < m holds f.n is Consistent &
      f.n c= f.m
    proof
      let nn,mm be Nat such that
A28:   mm in dom f & nn in dom f & nn < mm;
      reconsider n = nn, m= mm as Element of NAT by ORDINAL1:def 12;
      f.n is Subset of CQC-WFF(Al2) & f.n = PSI(n) & PSI(n) is Al2-Consistent
       by A21,A25;
      hence f.nn is Consistent by QC_TRANS:def 2;
      defpred S[Nat] means
       $1 <= m implies ex k st k=m-$1 & PSI(k) c= PSI(m);
A29:  S[0];
A30:  for k being Nat holds S[k] implies S[k+1]
      proof
        let k be Nat;
        assume
A31:     S[k];
        set j1 = m-k;
        set j2 = m-(k+1);
        per cases;
        suppose
A32:      k+1 <= m;
          then k <= m by NAT_1:13;
          then reconsider j1,j2 as Element of NAT by A32,NAT_1:21;
          PSI(j2+1) = (the EF-Sequence of Al,PHI).(j2)
                    \/ Example_Formulae_of(j2-th_FCEx(Al)) by Def9;
          then PSI(j2) c= PSI(j1) & PSI(j1) c= PSI(m)
           by A31,A32,NAT_1:13,XBOOLE_1:7;
          hence thesis by XBOOLE_1:1;
        end;
        suppose not k+1<=m;
          hence thesis;
        end;
      end;
A33:  for k being Nat holds S[k] from NAT_1:sch 2(A29,A30);
      set k = m-n;
      reconsider k as Element of NAT by A28,NAT_1:21;
      S[k] & k <= k+n by A33,NAT_1:11;
      then PSI(n) c= PSI(m) & f.n = PSI(n) & f.m = PSI(m) by A25;
      hence f.nn c= f.mm;
    end;
    then reconsider PSIp as Consistent Subset of CQC-WFF(Al2) by HENMODEL:11;
    for y being object st y in the set of all PSI(n)
      ex x being object st x in dom f & y = f.x
    proof
      let P be object such that
A34:   P in the set of all PSI(n) ;
      consider n such that
A35:   P = PSI(n) by A34;
      n in dom f & f.n = P by A25,A35;
      hence thesis;
    end;
    then
A36: the set of all PSI(n)  c= rng f by FUNCT_1:9;
    for y being object st y in rng f holds y in the set of all PSI(n)
    proof
      let y be object such that
A37:  y in rng f;
      consider x being object such that
A38:   x in dom f & y = f.x by A37,FUNCT_1:def 3;
      reconsider x as Element of NAT by A25,A38;
      f.x = PSI(x) by A25;
      hence thesis by A38;
    end;
    then rng f c= the set of all PSI(n) ;
    then PSIp = PSI by A36,XBOOLE_0:def 10;
    hence thesis;
  end;
  then reconsider PSI as Consistent Subset of CQC-WFF(Al2);
  set S = the set of all S(n) ;
  S(0) in S;
  then reconsider S as non empty set;
A39: for a,b being set st a in S & b in S holds ex c being set st
   a \/ b c= c & c in S
  proof
    let a,b be set such that
A40: a in S & b in S;
    consider i such that
A41: a = S(i) by A40;
    consider j such that
A42: b = S(j) by A40;
    per cases;
    suppose j <= i;
      then S(j) c= S(i) by Th6;
      hence thesis by A40,A41,A42,XBOOLE_1:8;
    end;
    suppose not j <= i;
      then S(i) c= S(j) by Th6;
      hence thesis by A40,A41,A42,XBOOLE_1:8;
    end;
  end;
A43: for p being Element of CQC-WFF(Al2) holds ex n st
   p is Element of CQC-WFF(S(n))
  proof
    defpred P[Element of CQC-WFF(Al2)] means ex n st $1 is Element of
     CQC-WFF(S(n));
A44: P[VERUM(Al2)]
    proof
      reconsider Al2 as (S(0))-expanding QC-alphabet by A2,QC_TRANS:def 1;
      VERUM(S(0)) in CQC-WFF(S(0));
      then Al2-Cast(VERUM(S(0))) in CQC-WFF(S(0)) by QC_TRANS:def 3;
      then VERUM(Al2) in CQC-WFF(S(0)) by QC_TRANS:8;
      hence thesis;
    end;
A45: for k being Nat
    for P being QC-pred_symbol of k,Al2, l being CQC-variable_list
    of k,Al2 holds P[P!l]
    proof
      let k be Nat;
      let P be QC-pred_symbol of k,Al2, l be CQC-variable_list of k,Al2;
      ex n st rng l c= bound_QC-variables(S(n)) & P is QC-pred_symbol of k,S(n)
      proof
A46:    rng l c= bound_QC-variables(Al2) & {P} c= QC-pred_symbols(Al2)
         by ZFMISC_1:31;
        bound_QC-variables(Al2) c= QC-variables(Al2)
         & QC-variables(Al2) c= [:NAT,QC-symbols(Al2):] by QC_LANG1:4;
        then
A47:     bound_QC-variables(Al2) c= [:NAT,QC-symbols(Al2):] &
         QC-pred_symbols(Al2) c= [:NAT,QC-symbols(Al2):]
         by QC_LANG1:6;
        then rng l c= [:NAT,QC-symbols(Al2):] & {P} c= [:NAT,QC-symbols(Al2):]
         by A46;
        then rng l c= Al2 & {P} c= Al2 by QC_LANG1:5;
        then rng l \/ {P} c= union S & rng l \/ {P} is finite by XBOOLE_1:8;
        then consider a being set such that
A48:     a in S & rng l \/ {P} c= a by A39,COHSP_1:6,13;
        consider n such that
A49:     a = S(n) by A48;
        take n;
A50:    rng l c= rng l \/ {P} & {P} c= rng l \/ {P} by XBOOLE_1:7;
        for s being object st s in rng l holds s in bound_QC-variables(S(n))
        proof
          let s be object such that
A51:       s in rng l;
          s in bound_QC-variables(Al2) by A51;
          then s in [:{4}, QC-symbols(Al2):] by QC_LANG1:def 4;
          then consider s1,s2 being object such that
A52:       s1 in {4} & s2 in QC-symbols(Al2) & s = [s1,s2] by ZFMISC_1:def 2;
          rng l c= S(n) & {P} c= S(n) by A48,A49,A50;
          then s in S(n) by A51;
          then s in [:NAT,QC-symbols(S(n)):] by QC_LANG1:5;
          then consider s3,s4 being object such that
A53:        s3 in NAT & s4 in QC-symbols(S(n)) & s = [s3,s4] by ZFMISC_1:def 2;
          s = [s1,s4] by A52,A53,XTUPLE_0:1;
          then s in [:{4},QC-symbols(S(n)):] by A52,A53,ZFMISC_1:def 2;
          hence thesis by QC_LANG1:def 4;
        end;
        hence rng l c= bound_QC-variables(S(n));
        thus P is QC-pred_symbol of k,S(n)
        proof
          P in [:NAT,QC-symbols(Al2):] by A47;
          then consider p1,p2 being object such that
A54:       p1 in NAT & p2 in QC-symbols(Al2) & P = [p1,p2] by ZFMISC_1:def 2;
          rng l c= S(n) & P in S(n) by A48,A49,A50,ZFMISC_1:31;
          then P in [:NAT,QC-symbols(S(n)):] by QC_LANG1:5;
          then reconsider p2 as QC-symbol of S(n) by A54,ZFMISC_1:87;
A55:      P`1 =(the_arity_of P)+7 by QC_LANG1:def 8 .= k + 7 by QC_LANG1:11;
          reconsider p1 as Element of NAT by A54;
          P`1=7 + the_arity_of P & P`1=p1 by A54,QC_LANG1:def 8;
          then 7 <= p1 by NAT_1:11;
          then [p1,p2] in
            {[m,x] where m is Nat, x is QC-symbol of S(n):
               7 <= m};
          then reconsider P as QC-pred_symbol of S(n) by A54,QC_LANG1:def 7;
          the_arity_of P = k by A55,QC_LANG1:def 8;
          then P in {Q where Q is QC-pred_symbol of S(n): the_arity_of Q=k};
          hence thesis by QC_LANG1:def 9;
        end;
      end;
      then consider n such that
A56:   rng l c= bound_QC-variables(S(n)) & P is QC-pred_symbol of k,S(n);
      l is CQC-variable_list of k,S(n) by A56,FINSEQ_1:def 4,XBOOLE_1:1;
      then consider l2 being CQC-variable_list of k,S(n), P2 being
       QC-pred_symbol of k,S(n) such that
A57:   l2 = l & P = P2 by A56;
      reconsider Al2 as (S(n))-expanding QC-alphabet by A2,QC_TRANS:def 1;
A58:  Al2-Cast(P2) = P & Al2-Cast(l2) = l by A57,QC_TRANS:def 5,def 6;
      P2!l2 = Al2-Cast(P2!l2) by QC_TRANS:def 3 .= P!l by A58,QC_TRANS:8;
      hence thesis;
    end;
A59: for r being Element of CQC-WFF(Al2) st P[r] holds P['not' r]
    proof
      let r be Element of CQC-WFF(Al2);
      assume P[r];
      then consider n such that
A60:   r is Element of CQC-WFF(S(n));
      consider r2 being Element of CQC-WFF(S(n)) such that
A61:   r = r2 by A60;
      reconsider Al2 as (S(n))-expanding QC-alphabet by A2,QC_TRANS:def 1;
      'not' r2 = Al2-Cast('not' r2) by QC_TRANS:def 3
       .= 'not' Al2-Cast(r2) by QC_TRANS:8 .= 'not' r by A61,QC_TRANS:def 3;
      hence thesis;
    end;
A62: for r,s being Element of CQC-WFF(Al2) st P[r] & P[s] holds P[r '&' s]
    proof
      let r,s be Element of CQC-WFF(Al2);
      assume P[r] & P[s];
      then consider n,m such that
A63:   r is Element of CQC-WFF(S(n)) & s is Element of CQC-WFF(S(m));
      per cases;
      suppose n <= m;
        then reconsider Sm=S(m) as S(n)-expanding QC-alphabet
         by Th6,QC_TRANS:def 1;
        r is Element of CQC-WFF(Sm) by A63,QC_TRANS:7;
        then consider r2,s2 being Element of CQC-WFF(Sm) such that
A64:     r2 = r & s2 = s by A63;
        reconsider Al2 as Sm-expanding QC-alphabet by A2,QC_TRANS:def 1;
A65:    r = Al2-Cast(r2) & s = Al2-Cast(s2) by A64,QC_TRANS:def 3;
        r2 '&' s2 = Al2-Cast(r2 '&' s2) by QC_TRANS:def 3
         .= r '&' s by A65,QC_TRANS:8;
        hence thesis;
      end;
      suppose not n <= m;
        then reconsider Sn=S(n) as S(m)-expanding QC-alphabet
         by Th6,QC_TRANS:def 1;
        s is Element of CQC-WFF(Sn) by A63,QC_TRANS:7;
        then consider r2,s2 being Element of CQC-WFF(Sn) such that
A66:     r2 = r & s2 = s by A63;
        reconsider Al2 as Sn-expanding QC-alphabet by A2,QC_TRANS:def 1;
A67:    r = Al2-Cast(r2) & s = Al2-Cast(s2) by A66,QC_TRANS:def 3;
        r2 '&' s2 = Al2-Cast(r2 '&' s2) by QC_TRANS:def 3
         .= r '&' s by A67,QC_TRANS:8;
        hence thesis;
      end;
    end;
    for x being bound_QC-variable of Al2, r being Element of CQC-WFF(Al2) st
     P[r] holds P[All(x,r)]
    proof
      let x be bound_QC-variable of Al2, r be Element of CQC-WFF(Al2);
      x in QC-variables(Al2) & QC-variables(Al2) c= [:NAT,QC-symbols(Al2):]
       by QC_LANG1:4;
      then x in [:NAT,QC-symbols(Al2):] & Al2 = [:NAT,QC-symbols(Al2):]
       by QC_LANG1:5;
      then {x} c= union S & {x} is finite by ZFMISC_1:31;
      then consider a being set such that
A68:   a in S & {x} c= a by A39,COHSP_1:6,13;
      consider n such that
A69:   a = S(n) by A68;
      assume P[r];
      then consider m such that
A70:   r is Element of CQC-WFF(S(m));
      x in bound_QC-variables(Al2);
      then x in [:{4},QC-symbols(Al2):] by QC_LANG1:def 4;
      then consider x1,x2 being object such that
A71:   x1 in {4} & x2 in QC-symbols(Al2) & x = [x1,x2] by ZFMISC_1:def 2;
A72:  x in S(n) by A68,A69,ZFMISC_1:31;
      per cases;
      suppose
A73:    n <= m;
        then reconsider Sm=S(m) as S(n)-expanding QC-alphabet
         by Th6,QC_TRANS:def 1;
        S(n) c= S(m) by A73,Th6;
        then x in S(m) by A72;
        then x in [:NAT, QC-symbols(S(m)):] by QC_LANG1:5;
        then consider x3,x4 being object such that
A74:     x3 in NAT & x4 in QC-symbols(S(m)) & x = [x3,x4] by ZFMISC_1:def 2;
        x = [x1,x4] by A71,A74,XTUPLE_0:1;
        then x in [:{4},QC-symbols(Sm):] by A71,A74,ZFMISC_1:def 2;
        then x is bound_QC-variable of Sm by QC_LANG1:def 4;
        then consider x2 being bound_QC-variable of Sm, r2 being Element of
         CQC-WFF(Sm) such that
A75:     x2 = x & r2 = r by A70;
        reconsider Al2 as Sm-expanding QC-alphabet by A2,QC_TRANS:def 1;
A76:    r = Al2-Cast(r2) & x = Al2-Cast(x2) by A75,QC_TRANS:def 3,def 4;
        All(x2,r2) = Al2-Cast(All(x2,r2)) by QC_TRANS:def 3
         .= All(x,r) by A76,QC_TRANS:8;
        hence thesis;
      end;
      suppose not n <= m;
        then reconsider Sn=S(n) as S(m)-expanding QC-alphabet
         by Th6,QC_TRANS:def 1;
        x in [:NAT,QC-symbols(Sn):] by A72,QC_LANG1:5;
        then consider x3,x4 being object such that
A77:     x3 in NAT & x4 in QC-symbols(Sn) & x = [x3,x4] by ZFMISC_1:def 2;
        x = [x1,x4] by A71,A77,XTUPLE_0:1;
        then x in [:{4},QC-symbols(Sn):] by A71,A77,ZFMISC_1:def 2;
        then x is bound_QC-variable of Sn & r is Element of CQC-WFF(Sn)
         by A70,QC_TRANS:7,QC_LANG1:def 4;
        then consider x2 being bound_QC-variable of Sn, r2 being Element of
         CQC-WFF(Sn) such that
A78:     x2 = x & r2 = r;
        reconsider Al2 as Sn-expanding QC-alphabet by A2,QC_TRANS:def 1;
A79:    r = Al2-Cast(r2) & x = Al2-Cast(x2) by A78,QC_TRANS:def 3,def 4;
        All(x2,r2) = Al2-Cast(All(x2,r2)) by QC_TRANS:def 3
         .= All(x,r) by A79,QC_TRANS:8;
        hence thesis;
      end;
    end;
    then
A80: for r,s being Element of CQC-WFF(Al2), x being bound_QC-variable of
     Al2, k being Nat, l being CQC-variable_list of k,Al2,
     P being QC-pred_symbol of k,Al2 holds P[VERUM(Al2)] & P[P!l] & (P[r]
     implies P['not' r]) & (P[r] & P[s] implies P[r '&' s]) &
     (P[r] implies P[All(x,r)]) by A44,A45,A59,A62;
    for p being Element of CQC-WFF(Al2) holds P[p] from CQC_LANG:sch 1(A80);
    hence thesis;
  end;
  PSI is with_examples
  proof
    for x being bound_QC-variable of Al2, p being Element of CQC-WFF(Al2) holds
     ex y be bound_QC-variable of Al2 st PSI |- ('not' Ex(x,p) 'or' (p.(x,y)))
    proof
      let x be bound_QC-variable of Al2, p be Element of CQC-WFF(Al2);
      ex n st (x is bound_QC-variable of S(n) & p is Element of CQC-WFF(S(n)))
      proof
        consider m such that
A81:     p is Element of CQC-WFF(S(m)) by A43;
        x in QC-variables(Al2) & QC-variables(Al2) c= [:NAT,QC-symbols(Al2):]
         by QC_LANG1:4;
        then x in [:NAT,QC-symbols(Al2):] & Al2 = [:NAT,QC-symbols(Al2):]
         by QC_LANG1:5;
        then {x} c= union S & {x} is finite by ZFMISC_1:31;
        then consider a being set such that
A82:     a in S & {x} c= a by A39,COHSP_1:6,13;
        consider n such that
A83:     a = S(n) by A82;
        x in bound_QC-variables(Al2);
        then x in [:{4},QC-symbols(Al2):] by QC_LANG1:def 4;
        then consider x1,x2 being object such that
A84:     x1 in {4} & x2 in QC-symbols(Al2) & x = [x1,x2] by ZFMISC_1:def 2;
A85:    x in S(n) by A82,A83,ZFMISC_1:31;
        per cases;
        suppose
A86:      n <= m;
          then reconsider Sm = S(m) as S(n)-expanding QC-alphabet by
           Th6,QC_TRANS:def 1;
          S(n) c= S(m) by A86,Th6;
          then x in S(m) by A85;
          then x in [:NAT, QC-symbols(S(m)):] by QC_LANG1:5;
          then consider x3,x4 being object such that
A87:       x3 in NAT & x4 in QC-symbols(S(m)) & x = [x3,x4] by ZFMISC_1:def 2;
          x = [x1,x4] by A84,A87,XTUPLE_0:1;
          then x in [:{4},QC-symbols(Sm):] by A84,A87,ZFMISC_1:def 2;
          then x is bound_QC-variable of Sm by QC_LANG1:def 4;
          hence thesis by A81;
        end;
        suppose not n <= m;
          then reconsider Sn = S(n) as S(m)-expanding QC-alphabet by
           Th6,QC_TRANS:def 1;
          x in [:NAT, QC-symbols(S(n)):] by A85,QC_LANG1:5;
          then consider x3,x4 being object such that
A88:       x3 in NAT & x4 in QC-symbols(S(n)) & x = [x3,x4] by ZFMISC_1:def 2;
          x = [x1,x4] by A84,A88,XTUPLE_0:1;
          then x in [:{4},QC-symbols(Sn):] by A84,A88,ZFMISC_1:def 2;
          then x is bound_QC-variable of Sn & p is Element of CQC-WFF(Sn)
           by A81,QC_TRANS:7,QC_LANG1:def 4;
          hence thesis;
        end;
      end;
      then consider n such that
A89:   x is bound_QC-variable of S(n) & p is Element of CQC-WFF(S(n));
A90:  FCEx(S(n)) = S(n+1) by Th5;
A91:  PSI(n+1) = PSI(n) \/ Example_Formulae_of(S(n)) by Def9;
      consider x2 being bound_QC-variable of S(n), p2 being Element of
       CQC-WFF(S(n)) such that
A92:   x2 = x & p2 = p by A89;
      Example_Formula_of(p2,x2) in Example_Formulae_of(S(n));
      then
A93:  Example_Formula_of(p2,x2) in PSI(n+1) by A91,XBOOLE_0:def 3;
      S(n) c= S(n+1) by Th6,NAT_1:11;
      then reconsider Sn1 = S(n+1) as S(n)-expanding QC-alphabet
       by QC_TRANS:def 1;
      set y2 = Example_of(p2,x2);
      reconsider y2 as bound_QC-variable of Sn1 by Th5;
      reconsider Al2 as Sn1-expanding QC-alphabet by A2,QC_TRANS:def 1;
      y2 is bound_QC-variable of Al2 by TARSKI:def 3, QC_TRANS:4;
      then consider y being bound_QC-variable of Al2 such that
A94:   y = y2;
A95:  Sn1-Cast(p2) = p & Sn1-Cast(x2) = x by A92,QC_TRANS:def 3,def 4;
      then
A96:  Al2-Cast(Sn1-Cast(p2)) = p & Al2-Cast(Sn1-Cast(x2)) = x
       by QC_TRANS:def 3,def 4;
A97: Al2-Cast(Ex(Sn1-Cast(x2),Sn1-Cast(p2))) = Ex(x,p) by A96,Th7;
      reconsider p as Element of CQC-WFF(Al2);
      reconsider x as bound_QC-variable of Al2;
A98: (Sn1-Cast(p2)).(Sn1-Cast(x2),y2) = p.(x,y) by A94,A95,QC_TRANS:17;
A99: Example_Formula_of(p2,x2)
       = Al2-Cast(('not' Ex(Sn1-Cast(x2), Sn1-Cast(p2))) 'or'
        ((Sn1-Cast(p2)).(Sn1-Cast(x2),y2))) by A90,QC_TRANS:def 3
       .= Al2-Cast('not' Ex(Sn1-Cast(x2), Sn1-Cast(p2))) 'or'
        Al2-Cast((Sn1-Cast(p2)).(Sn1-Cast(x2),y2)) by Th7
       .= 'not' Al2-Cast(Ex(Sn1-Cast(x2), Sn1-Cast(p2))) 'or'
        Al2-Cast((Sn1-Cast(p2)).(Sn1-Cast(x2),y2)) by QC_TRANS:8
       .= 'not' Ex(x,p) 'or' p.(x,y) by A97,A98,QC_TRANS:def 3;
      set example = 'not' Ex(x,p) 'or' p.(x,y);
      reconsider example as Element of CQC-WFF(Al2);
      reconsider PSI as Consistent Subset of CQC-WFF(Al2);
      PSI(n+1) in the set of all PSI(m) ;
      then PSI(n+1) c= PSI by ZFMISC_1:74;
      hence thesis by A93,A99,GOEDELCP:21;
    end;
    hence thesis by GOEDELCP:def 2;
  end;
  hence thesis by A1;
end;
