reserve Al for QC-alphabet;
reserve a,b,b1 for object,
  i,j,k,n for Nat,
  p,q,r,s for Element of CQC-WFF(Al),
  x,y,y1 for bound_QC-variable of Al,
  P for QC-pred_symbol of k,Al,
  l,ll for CQC-variable_list of k,Al,
  Sub,Sub1 for CQC_Substitution of Al,
  S,S1,S2 for Element of CQC-Sub-WFF(Al),
  P1,P2 for Element of QC-pred_symbols(Al);
reserve F1,F2,F3 for QC-formula of Al,
  L for FinSequence;

theorem
  for p st (for q st q is_subformula_of p holds for x,r holds q <> All(x
  ,r)) holds QuantNbr(p) = 0
proof
  let p such that
A1: for q st q is_subformula_of p holds for x,r holds q <> All(x,r);
  defpred P[Element of CQC-WFF(Al)] means $1 is_subformula_of p
implies QuantNbr($1) = 0;
A2: for r,s st P[r] & P[s] holds P[r '&' s]
  proof
    let r,s such that
A3: ( P[r])& P[s];
    assume
A4: r '&' s is_subformula_of p;
    s is_immediate_constituent_of r '&' s by QC_LANG2:45;
    then
A5: s is_proper_subformula_of p by A4,QC_LANG2:63;
    r is_immediate_constituent_of r '&' s by QC_LANG2:45;
    then r is_proper_subformula_of p by A4,QC_LANG2:63;
    then QuantNbr(r '&' s) = 0+0 by A3,A5,CQC_SIM1:17,QC_LANG2:def 21;
    hence thesis;
  end;
  for r st P[r] holds P['not' r]
  proof
    let r such that
A6: P[r];
A7: r is_immediate_constituent_of 'not' r by QC_LANG2:43;
    assume 'not' r is_subformula_of p;
    then r is_proper_subformula_of p by A7,QC_LANG2:63;
    hence thesis by A6,CQC_SIM1:16,QC_LANG2:def 21;
  end;
  then
A8: for r,s,x,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,CQC_SIM1:14,15;
  for r holds P[r] from CQC_LANG:sch 1(A8);
  hence thesis;
end;
