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 Th22:
  [All(x,p),Sub] = CQCSub_All(QScope(p,x,Sub),Qsc(p,x,Sub)) &
  QScope(p,x,Sub) is quantifiable
proof
  set S = [p,(CFQ(Al)).[All(x,p),Sub]];
  set B = [[p,(CFQ(Al)).[All(x,p),Sub]],x];
A1: B`2 = x & (B`1)`1 = p;
  [All(x,p),Sub] in CQC-Sub-WFF(Al);
  then
A2: [All(x,p),Sub] in dom CFQ(Al) by FUNCT_2:def 1;
  (B`1)`2 = (QSub(Al)).[All(B`2,(B`1)`1),Sub] by A2,FUNCT_1:47;
  then
A3: B is quantifiable by SUBSTUT1:def 22;
  then
  CQCSub_All(QScope(p,x,Sub),Qsc(p,x,Sub)) = Sub_All(QScope(p,x,Sub),Qsc(p
  ,x,Sub)) by SUBLEMMA:def 5;
  hence thesis by A1,A3,SUBSTUT1:def 24;
end;
