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 Th3:
  QC-pred_symbols(Al) c= QC-pred_symbols(Al2)
proof
  for Q being object st Q in QC-pred_symbols(Al)
       holds Q in QC-pred_symbols(Al2)
  proof
    let Q be object such that
A1: Q in QC-pred_symbols(Al);
    set preds = { [k,b] : 7 <= k };
    set preds2 = { [k,b2] where b2 is QC-symbol of Al2 : 7 <= k };
    consider k,b such that
A2: Q=[k,b] & 7 <= k by A1;
    b in QC-symbols(Al2) by Th2,TARSKI:def 3;
    hence Q in QC-pred_symbols(Al2) by A2;
  end;
  hence thesis;
end;
