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
  QuantNbr(p) = n & q is_subformula_of p implies QuantNbr(q) <= n
proof
  set L =the  PATH of q,p;
  set m = len L;
  assume that
A1: QuantNbr(p) = n and
A2: q is_subformula_of p;
  1 <= m by A2,Def5;
  then ex r st r = L.1 & QuantNbr(r) <= n by A1,A2,Th29;
  hence thesis by A2,Def5;
end;
