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
  (P!l).(x,y) = P!CQC_Subst(l,Sbst(x,y)) & QuantNbr(P!l) = QuantNbr((P!l
  ).(x,y))
proof
  set S = [P!l,Sbst(x,y)];
  S = Sub_P(P,l,Sbst(x,y)) by SUBSTUT1:9;
  then
A1: (P!l).(x,y) = P!CQC_Subst(l,Sbst(x,y)) by SUBLEMMA:8;
  QuantNbr(P!CQC_Subst(l,Sbst(x,y))) = 0 by CQC_SIM1:15;
  hence thesis by A1,CQC_SIM1:15;
end;
