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 Th9:
  PHI \/ Example_Formulae_of(Al) is Consistent Subset of CQC-WFF(FCEx(Al))
proof
  reconsider Al2 = FCEx(Al) as Al-expanding QC-alphabet;
  for s being object st s in CQC-WFF(Al) holds s in CQC-WFF(Al2)
  proof
    let s be object;
    assume s in CQC-WFF(Al);
    then reconsider s as Element of CQC-WFF(Al);
    s is Element of CQC-WFF(Al2) by QC_TRANS:7;
    hence thesis;
  end;
  then
A1: CQC-WFF(Al) c= CQC-WFF(Al2);
  then PHI c= CQC-WFF(Al2) & Example_Formulae_of(Al) c= CQC-WFF(Al2);
  then reconsider B = PHI \/ Example_Formulae_of(Al) as Subset of CQC-WFF(Al2)
   by XBOOLE_1:8;
  B is Consistent
  proof
    assume B is Inconsistent;
    then consider CHI2 being Subset of CQC-WFF(Al2) such that
A2:  CHI2 c= B & CHI2 is finite & CHI2 is Inconsistent by HENMODEL:7;
    reconsider CHI2 as finite Subset of CQC-WFF(Al2) by A2;
    consider Al1 being countable QC-alphabet such that
A3:  CHI2 is finite Subset of CQC-WFF(Al1) & Al2 is Al1-expanding
     by QC_TRANS:20;
    reconsider Al2 as Al1-expanding QC-alphabet by A3;
    consider CHI1 being Subset of CQC-WFF(Al1) such that
A4:  CHI1 = CHI2 by A3;
    reconsider CHI1 as finite Subset of CQC-WFF(Al1) by A4;
    set PHI1 = CHI1 /\ PHI;
    PHI1 c= CHI1 & CHI1 c= CQC-WFF(Al1) by XBOOLE_1:18;
    then reconsider PHI1 as Subset of CQC-WFF(Al1) by XBOOLE_1:1;
    reconsider Al2 as Al-expanding QC-alphabet;
    PHI is Subset of CQC-WFF(Al2) by A1,XBOOLE_1:1;
    then consider PHIp being Subset of CQC-WFF(Al2) such that
A5:  PHIp = PHI;
    set PHI2 = CHI2 /\ PHIp;
    reconsider PHI2 as Subset of CQC-WFF(Al2);
    PHI1 is Consistent
    proof
      reconsider Al2 as Al1-expanding QC-alphabet;
      PHI is Al2-Consistent by QC_TRANS:23;
      then PHIp is Consistent & PHI2 c= PHIp by A5,QC_TRANS:def 2,XBOOLE_1:18;
      then PHI2 is Consistent & PHI2 = PHI1 by A4,A5,QC_TRANS:22;
      then PHI2 is Al1-Consistent by QC_TRANS:18;
      hence PHI1 is Consistent by A4,A5,QC_TRANS:def 2;
    end;
    then reconsider PHI1 as Consistent Subset of CQC-WFF(Al1);
    still_not-bound_in PHI1 is finite;
    then consider CZ being Consistent Subset of CQC-WFF(Al1), JH being
     Henkin_interpretation of CZ such that
A6:  JH,valH(Al1) |= PHI1 by GOEDELCP:34;
    consider A2 being non empty set, J2 being interpretation of Al2,A2,
     v2 being Element of Valuations_in (Al2,A2) such that
A7: J2,v2 |= PHI2 by A4,A5,A6,QC_TRANS:21;
    set Ex2 = CHI2 /\ Example_Formulae_of(Al);
    reconsider Ex2 as Subset of CQC-WFF(Al2);
A8: CHI2 = CHI2 /\ (PHIp \/ Example_Formulae_of(Al)) by A2,A5,XBOOLE_1:28
     .= PHI2 \/ Ex2 by XBOOLE_1:23;
    ex A being non empty set, J being interpretation of Al2,A, v being Element
     of Valuations_in(Al2,A) st J,v |= PHI2 \/ Ex2
    proof
      defpred P[set] means ex A being non empty set, J being interpretation of
       Al2,A, v being Element of Valuations_in(Al2,A), Ex3 being Subset of
       CQC-WFF(Al2) st Ex3 = $1 & J,v |= PHI2 \/ Ex3;
A9:  Ex2 is finite;
      J2,v2 |= PHI2 \/ {}CQC-WFF(Al2) by A7; then
A10:  P[{}];
A11:  for b,B being set st b in Ex2 & B c= Ex2 & P[B] holds P[B \/ {b}]
      proof
        let b,B be set such that
A12:    b in Ex2 & B c= Ex2 & P[B];
        reconsider B as Subset of CQC-WFF(Al2) by A12;
        consider A being non empty set, J being interpretation of Al2,A, v
         being Element of Valuations_in(Al2,A) such that
A13:     J,v |= PHI2 \/ B by A12;
        Ex2 c= Example_Formulae_of(Al) by XBOOLE_1:18;
        then b in Example_Formulae_of(Al) by A12;
        then consider p,x such that
A14:     b = Example_Formula_of(p,x);
        set fc = Example_of(p,x);
        set x2 = Al2-Cast(x);
        set p2 = Al2-Cast(p);
        reconsider fc,x2 as bound_QC-variable of Al2;
        reconsider p2 as Element of CQC-WFF(Al2);
        reconsider b as Element of CQC-WFF(Al2) by A14;
A15:    J,v |= b implies thesis
        proof
          assume
A16:       J,v |= b;
          for q2 being Element of CQC-WFF(Al2) st q2 in PHI2 \/ (B \/ {b})
           holds J,v |= q2
          proof
            let q2 be Element of CQC-WFF(Al2);
            assume q2 in PHI2 \/ (B \/ {b});
            then q2 in (PHI2 \/ B) \/ {b} by XBOOLE_1:4;
            then
A17:        q2 in (PHI2 \/ B) or q2 in {b} by XBOOLE_0:def 3;
            per cases;
            suppose q2 in (PHI2 \/ B);
              hence thesis by A13,CALCUL_1:def 11;
            end;
            suppose not q2 in (PHI2 \/ B);
              hence thesis by A16,A17,TARSKI:def 1;
            end;
          end;
          hence thesis by CALCUL_1:def 11;
        end;
        per cases;
        suppose not J,v |= Ex(x2,p2);
          then J,v |= 'not' Ex(x2,p2) by VALUAT_1:17;
          hence thesis by A14,A15,Th8;
        end;
        suppose J,v |= Ex(x2,p2);
          then consider a being Element of A such that
A18:       J,v.(x2|a) |= p2 by GOEDELCP:9;
          reconsider vp = v.(fc|a) as Element of Valuations_in(Al2,A);
A19:      for p2 being Element of CQC-WFF(Al2) st p2 is Element of CQC-WFF(Al)
          holds v|(still_not-bound_in p2) = vp|(still_not-bound_in p2)
          proof
            let p2 be Element of CQC-WFF(Al2);
            assume p2 is Element of CQC-WFF(Al);
            then consider pp being Element of CQC-WFF(Al) such that
A20:         pp = p2;
            not [the free_symbol of Al,[x,p]] in QC-symbols(Al) by Def2;
            then not [4,[the free_symbol of Al,[x,p]]] in
             [:{4},QC-symbols(Al):] by ZFMISC_1:87;
            then not fc in bound_QC-variables(Al) by QC_LANG1:def 4;
            then not fc in still_not-bound_in pp;
            then not fc in still_not-bound_in Al2-Cast(pp) by QC_TRANS:12;
            then not fc in still_not-bound_in p2 by A20,QC_TRANS:def 3;
            hence v|(still_not-bound_in p2) = vp|(still_not-bound_in p2)
             by CALCUL_1:26;
          end;
          p2 = p by QC_TRANS:def 3;
          then v|(still_not-bound_in p2) = vp|(still_not-bound_in p2) by A19;
          then v.(x2|a)|(still_not-bound_in p2) =
           vp.(x2|a)|(still_not-bound_in p2) by SUBLEMMA:64;
          then J,vp.(x2|a) |= p2 & vp.fc = a by A18,SUBLEMMA:49,68;
          then
A21:      J,vp |= p2.(x2,fc) by CALCUL_1:24;
          for q2 being Element of CQC-WFF(Al2) st q2 in PHI2 \/ (B \/ {b})
           holds J,vp |= q2
          proof
            let q2 be Element of CQC-WFF(Al2);
            assume q2 in PHI2 \/ (B \/ {b});
            then q2 in (PHI2 \/ B) \/ {b} by XBOOLE_1:4;
            then
A22:        q2 in (PHI2 \/ B) or q2 in {b} by XBOOLE_0:def 3;
            per cases;
            suppose
A23:          q2 in PHI2;
              then
A24:           q2 in PHI2 \/ B by XBOOLE_0:def 3;
              PHI2 c= PHI & PHI c= CQC-WFF(Al) by A5,XBOOLE_1:18;
              then PHI2 c= CQC-WFF(Al);
              then v|(still_not-bound_in q2) = vp|(still_not-bound_in q2)
               & J,v |= q2 by A13,A19,A23,A24,CALCUL_1:def 11;
              hence J,vp |= q2 by SUBLEMMA:68;
            end;
            suppose
A25:          q2 in B;
              B c= Example_Formulae_of(Al) by A12,XBOOLE_1:18;
              then q2 in Example_Formulae_of(Al) by A25;
              then consider r,y such that
A26:           q2 = Example_Formula_of(r,y);
              set fcr = Example_of(r,y);
              set y2 = Al2-Cast(y);
              set r2 = Al2-Cast(r);
              reconsider fcr,y2 as bound_QC-variable of Al2;
              reconsider r2 as Element of CQC-WFF(Al2);
              per cases;
              suppose fcr = fc;
                then [the free_symbol of Al,[x,p]] =
                 [the free_symbol of Al,[y,r]] by XTUPLE_0:1;
                then [x,p] = [y,r] by XTUPLE_0:1;
                then r = p & x = y by XTUPLE_0:1;
                hence thesis by A21,A26,Th8;
              end;
              suppose
A27:            not fcr = fc;
A28:            q2 in PHI2 \/ B by A25,XBOOLE_0:def 3;
                per cases;
                suppose
A29:              J,v |= 'not' Ex(y2,r2);
                  set fml = 'not' Ex(y2,r2);
                  'not' Ex(y,r) = Al2-Cast('not' Ex(y,r)) by QC_TRANS:def 3
                   .= 'not' Al2-Cast(Ex(y,r)) by QC_TRANS:8
                   .= fml by Th7;
                   then v|(still_not-bound_in fml) =
                   vp|(still_not-bound_in fml) & J,v |= fml by A19,A29;
                   then J,vp |= fml by SUBLEMMA:68;
                   hence J,vp |= q2 by A26,Th8;
                end;
                suppose not J,v |= 'not' Ex(y2,r2);
                  then J,v |= r2.(y2,fcr) by A13,A26,A28,Th8,CALCUL_1:def 11;
                  then consider a2 being Element of A such that
A30:               v.fcr = a2 & J,v.(y2|a2) |= r2 by CALCUL_1:24;
A31:              vp.fcr = a2 by A27,A30,SUBLEMMA:48;
                  not [the free_symbol of Al,[x,p]] in QC-symbols(Al)
                   by Def2;
                  then not [4,[the free_symbol of Al,[x,p]]] in
                   [:{4},QC-symbols(Al):] by ZFMISC_1:87;
                  then not fc in bound_QC-variables(Al) by QC_LANG1:def 4;
                  then not fc in still_not-bound_in r;
                  then not fc in still_not-bound_in r2 by QC_TRANS:12;
                  then v.(fc|a)|still_not-bound_in r2 =
                   v|still_not-bound_in r2 by CALCUL_1:26;
                  then vp.(y2|a2)|still_not-bound_in r2 =
                   v.(y2|a2)|still_not-bound_in r2 by SUBLEMMA:64;
                  then J,vp.(y2|a2) |= r2 by A30,SUBLEMMA:68;
                  then J,vp |= r2.(y2,fcr) by A31,CALCUL_1:24;
                  hence thesis by A26,Th8;
               end;
              end;
            end;
            suppose not q2 in PHI2 & not q2 in B;
              then q2 = b by A22,TARSKI:def 1,XBOOLE_0:def 3;
              hence thesis by A14,A21,Th8;
            end;
          end;
          hence thesis by CALCUL_1:def 11;
        end;
      end;
      P[Ex2] from FINSET_1:sch 2(A9,A10,A11);
      hence thesis;
    end;
    hence contradiction by A2,A8,HENMODEL:12;
  end;
  hence thesis;
end;
