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);

theorem
  for p, Sub holds QuantNbr(p) = QuantNbr(CQC_Sub([p,Sub]))
proof
  defpred P[Element of CQC-WFF(Al)] means for Sub holds
QuantNbr($1) = QuantNbr(CQC_Sub([$1,Sub]));
A1: 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 Th15,Th18,Th21,Th23,Th24;
  thus for r holds P[r] from CQC_LANG:sch 1(A1);
end;
