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 Th6:
  P is QC-pred_symbol of k,Al2
proof
  the_arity_of P = k by QC_LANG1:11;
  then
A1: P`1 = 7+k by QC_LANG1:def 8;
  reconsider P as QC-pred_symbol of Al2 by Th3,TARSKI:def 3;
  the_arity_of P = k by QC_LANG1:def 8, A1;
  hence thesis by QC_LANG3:1;
end;
