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
  PSI |= p implies PSI |- p
proof
  set CHI = PSI \/ {'not' p};
  assume A1: PSI |= p;
  assume not PSI |- p;
  then CHI is Consistent by HENMODEL:9;
  then ex A,J,v st J,v |= CHI by Def1;
  hence contradiction by GOEDELCP:37,A1;
end;
