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
  'not' p.(x,y) = 'not' (p.(x,y)) & (QuantNbr(p) = QuantNbr(p.(x,y))
  implies QuantNbr('not' p) = QuantNbr('not' p.(x,y)))
proof
  set S = ['not' p,Sbst(x,y)];
A1: S = Sub_not [p,Sbst(x,y)] by Th16;
  then
A2: ('not' p).(x,y) = 'not' CQC_Sub([p,Sbst(x,y)]) by SUBSTUT1:29;
  QuantNbr(p) = QuantNbr(p.(x,y)) implies QuantNbr('not' p) = QuantNbr((
  'not' p).(x,y))
  proof
    assume
A3: QuantNbr(p) = QuantNbr(p.(x,y));
    QuantNbr(('not' p).(x,y)) = QuantNbr(p.(x,y)) by A2,CQC_SIM1:16;
    hence thesis by A3,CQC_SIM1:16;
  end;
  hence thesis by A1,SUBSTUT1:29;
end;
