reserve Al for QC-alphabet;
reserve PHI for Consistent 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 Nat,
        l for CQC-variable_list of k,Al,
        P for QC-pred_symbol of k,Al,
        x,y,z for bound_QC-variable of Al,
        b for QC-symbol of Al,
        PR for FinSequence of [:set_of_CQC-WFF-seq(Al),Proof_Step_Kinds:];
reserve Al2 for Al-expanding QC-alphabet,
        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
  PHI is Al2-Consistent
proof
    let PSI be Subset of CQC-WFF(Al2) such that
A1:  PHI = PSI;
    for CHI being Subset of CQC-WFF(Al2) st CHI c= PSI & CHI is finite
     holds CHI is Consistent
    proof
      let CHI be Subset of CQC-WFF(Al2) such that
A2:    CHI c= PSI & CHI is finite;
      reconsider CHI as finite Subset of CQC-WFF(Al) by A1,A2,XBOOLE_1:1;
      consider Al1 being countable QC-alphabet such that
A3:    CHI is finite Subset of CQC-WFF(Al1) & Al is Al1-expanding by Th20;
      reconsider Al as Al1-expanding QC-alphabet by A3;
      reconsider CHI as finite Subset of CQC-WFF(Al);
      reconsider PHI as Consistent Subset of CQC-WFF(Al);
      reconsider CHI as Consistent Subset of CQC-WFF(Al) by A1,A2,Th22;
      CHI is Al1-Consistent by Th18;
      then reconsider CHI as Consistent Subset of CQC-WFF(Al1) by A3;
      still_not-bound_in CHI is finite;
      then consider CZ being Consistent Subset of CQC-WFF(Al1), JH being
       Henkin_interpretation of CZ such that
A4:    JH,valH(Al1) |= CHI by GOEDELCP:34;
      Al1 c= Al & Al c= Al2 by Def1;
      then reconsider Al2 as Al1-expanding QC-alphabet by Def1,XBOOLE_1:1;
      consider CHI2 being Subset of CQC-WFF(Al2) such that
A5:    CHI = CHI2;
      ex A being non empty set, J2 being interpretation of Al2,A, v2
       being Element of Valuations_in(Al2,A) st J2,v2 |= CHI2 by A4,A5,Th21;
      hence thesis by A5,HENMODEL:12;
    end;
    hence thesis by HENMODEL:7;
  end;
