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 Th22:
  for CHI being Subset of CQC-WFF(Al) st CHI c= PHI holds CHI is Consistent
proof
 let CHI be Subset of CQC-WFF(Al) such that
A1: CHI c= PHI;
 assume CHI is Inconsistent;
 then ex f being FinSequence of CQC-WFF(Al) st rng f c= CHI &
  |- f^<*'not' VERUM(Al)*> by HENMODEL:def 1, GOEDELCP:24;
 then PHI |- ('not' VERUM(Al)) by A1,HENMODEL:def 1,XBOOLE_1:1;
 hence contradiction by GOEDELCP:24;
end;
