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 Th18:
  (for Sub holds QuantNbr(p) = QuantNbr(CQC_Sub([p,Sub]))) implies
  for Sub holds QuantNbr('not' p) = QuantNbr(CQC_Sub(['not' p,Sub]))
proof
  assume
A1: for Sub holds QuantNbr(p) = QuantNbr(CQC_Sub([p,Sub]));
  let Sub;
  set S = ['not' p,Sub];
  S = Sub_not [p,Sub] by Th16;
  then QuantNbr(CQC_Sub(S)) = QuantNbr('not' CQC_Sub([p,Sub])) by SUBSTUT1:29
    .= QuantNbr(CQC_Sub([p,Sub])) by CQC_SIM1:16
    .= QuantNbr(p) by A1;
  hence thesis by CQC_SIM1:16;
end;
