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 Th7:
  for Al2 being Al-expanding QC-alphabet
  for p holds p is Element of CQC-WFF(Al2)
proof
  let Al2 be Al-expanding QC-alphabet;
  defpred P[Element of CQC-WFF(Al)] means $1 is Element of CQC-WFF(Al2);
A1: P[VERUM(Al)]
  proof
    VERUM(Al) = VERUM(Al2);
    hence thesis;
  end;
A2: for k,P,l holds P[P!l]
  proof
    let k,P,l;
A3: the_arity_of P = len l by Th1;
    P is QC-pred_symbol of k,Al2 & l is CQC-variable_list of k,Al2 by Th5,Th6;
    then consider P2 being QC-pred_symbol of k,Al2,
    l2 being CQC-variable_list of k,Al2 such that
A4: P=P2 & l=l2;
    the_arity_of P2 = len l2 by Th1;
    then P2!l2 = <*P2*>^l2 by QC_LANG1:def 12;
    hence thesis by A3,A4,QC_LANG1:def 12;
  end;
A5: P[p] implies P['not' p]
  proof
    assume P[p];
    then consider q being Element of CQC-WFF(Al2) such that
A6: p = q;
    'not' p = 'not' q by A6;
    hence thesis;
  end;
A7: P[p] & P[q] implies P[p '&' q]
  proof
    assume P[p] & P[q];
    then consider t,u being Element of CQC-WFF(Al2) such that
A8: p = t & q = u;
    p '&' q = t '&' u by A8;
    hence thesis;
  end;
A9: for x holds P[p] implies P[All(x,p)]
  proof
    let x;
    assume P[p];
    then consider q being Element of CQC-WFF(Al2) such that
A10: p = q;
    x is bound_QC-variable of Al2 by Th4,TARSKI:def 3;
    then consider y being bound_QC-variable of Al2 such that
A11: x = y;
    All(x,p) = All(y,q) by A10,A11;
    hence thesis;
  end;
A12: for r,s being Element of CQC-WFF(Al)
  for x being bound_QC-variable of Al for k
  for l being CQC-variable_list of k, Al for P being
  QC-pred_symbol of k,Al holds P[VERUM(Al)] & P[P!l] &
  (P[r] implies P['not' r]) & (P[r] & P[s] implies P[r '&' s]) &
  (P[r] implies P[All(x, r)]) by A1,A2,A5,A7,A9;
  for p holds P[p] from CQC_LANG:sch 1(A12);
  hence for p holds p is Element of CQC-WFF(Al2);
end;
