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 Th5:
  for k,l holds l is CQC-variable_list of k,Al2
proof
  let k,l;
  rng l c= bound_QC-variables(Al) &
     bound_QC-variables(Al) c= bound_QC-variables(Al2) by Th4;
  then rng l c= bound_QC-variables(Al2);
  hence thesis by FINSEQ_1:def 4, XBOOLE_1:1;
end;
