reserve A for QC-alphabet;
reserve i,j,k,l,m,n for Nat;
reserve a,b,e for set;
reserve t,u,v,w,z for QC-symbol of A;
reserve p,q,r,s for Element of CQC-WFF(A);
reserve x for Element of bound_QC-variables(A);
reserve ll for CQC-variable_list of k,A;
reserve P for QC-pred_symbol of k,A;
reserve f,h for Element of Funcs(bound_QC-variables(A),bound_QC-variables(A)),
  K,L for Element of Fin bound_QC-variables(A);

theorem
  QuantNbr(All(x,p)) = QuantNbr(p) + 1
proof
deffunc F(Element of CQC-WFF(A)) = QuantNbr($1);
A1: for p being Element of CQC-WFF(A), d being Element of NAT holds d = F(p)
iff ex F being Function of CQC-WFF(A), NAT st d = F.p & F.VERUM(A) = 0 &
for r,s,x,k
for l being CQC-variable_list of k,A for P being QC-pred_symbol of k,A holds
F.(P!l) = A(k,P,l) & F.('not' r) = N(F.r) & F.(r '&' s) = C(F.r,F.s) &
F.All(x,r) = Q(x,F.r) by Def6;
  thus F(All(x,p)) = Q(x,F(p)) from CQC_LANG:sch 9(A1);
end;
