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 Th23:
  (for Sub holds QuantNbr(p) = QuantNbr(CQC_Sub([p,Sub]))) implies
  for Sub holds QuantNbr(All(x,p)) = QuantNbr(CQC_Sub([All(x,p),Sub]))
proof
  assume
A1: for Sub holds QuantNbr(p) = QuantNbr(CQC_Sub([p,Sub]));
  let Sub;
  set S1 = [All(x,p),Sub];
  set S = [p,(CFQ(Al)).[All(x,p),Sub]];
  set y = S_Bound(@CQCSub_All(QScope(p,x,Sub),Qsc(p,x,Sub)));
A2: QScope(p,x,Sub) is quantifiable by Th22;
A3: Sub_All(QScope(p,x,Sub),Qsc(p,x,Sub)) = CQCSub_All(QScope(p,x,Sub),Qsc(p
  ,x,Sub)) by Th22,SUBLEMMA:def 5
    .= S1 by Th22;
  then
A4: S1 is Sub_universal by A2,SUBSTUT1:def 28;
  then
A5: CQC_Sub(S1) = CQCQuant(S1,CQC_Sub(CQCSub_the_scope_of S1)) by SUBLEMMA:28;
  CQCSub_the_scope_of S1 = Sub_the_scope_of Sub_All(QScope(p,x,Sub),Qsc(p,
  x,Sub)) by A3,A4,SUBLEMMA:def 6
    .= [S,x]`1 by A2,SUBSTUT1:21
    .= S;
  then
  CQC_Sub(S1) = CQCQuant(CQCSub_All(QScope(p,x,Sub), Qsc(p,x,Sub)),CQC_Sub
  (S)) by A5,Th22;
  then QuantNbr(CQC_Sub(S1)) = QuantNbr(All(y,CQC_Sub(S))) by Th22,SUBLEMMA:31
    .= QuantNbr(CQC_Sub(S))+1 by CQC_SIM1:18
    .= QuantNbr(p)+1 by A1;
  hence thesis by CQC_SIM1:18;
end;
