reserve Al for QC-alphabet;
reserve a,b,c,d for object,
  i,k,n for Nat,
  p,q for Element of CQC-WFF(Al),
  x,y,y1 for bound_QC-variable of Al,
  A for non empty set,
  J for interpretation of Al,A,
  v,w for Element of Valuations_in(Al,A),
  f,g for Function,
  P,P9 for QC-pred_symbol of k,Al,
  ll,ll9 for CQC-variable_list of k,Al,
  l1 for FinSequence of QC-variables(Al),
  Sub,Sub9,Sub1 for CQC_Substitution of Al,
  S,S9,S1,S2 for Element of CQC-Sub-WFF(Al),
  s for QC-symbol of Al;
reserve vS,vS1,vS2 for Val_Sub of A,Al;
reserve B for Element of [:QC-Sub-WFF(Al),bound_QC-variables(Al):],
  SQ for second_Q_comp of B;
reserve B for CQC-WFF-like Element of [:QC-Sub-WFF(Al),
  bound_QC-variables(Al):],
  xSQ for second_Q_comp of [S,x],
  SQ for second_Q_comp of B;
reserve B1 for Element of [:QC-Sub-WFF(Al),bound_QC-variables(Al):];
reserve SQ1 for second_Q_comp of B1;

theorem Th41:
  [S,x] is quantifiable implies S`2 = ExpandSub(x,S`1,RestrictSub(
  x,All(x,S`1),xSQ))
proof
  set Z = [All(x,S`1),xSQ];
  set q = All(x,S`1);
  assume [S,x] is quantifiable;
  then
A1: ([S,x]`1)`2 = (QSub(Al)).[All([S,x]`2,([S,x]`1)`1),xSQ] by SUBSTUT1:def 23;
A2: Z in [:QC-WFF(Al),vSUB(Al):] by ZFMISC_1:def 2;
  [:QC-WFF(Al),vSUB(Al):] c= dom QSub(Al) by Th34;
  then [Z,([S,x]`1)`2] in QSub(Al) by A2,A1,FUNCT_1:1;
  then [Z,S`2] in QSub(Al);
  then consider p being QC-formula of Al,Sub1,b such that
A3: [Z,S`2] = [[p,Sub1],b] and
A4: p,Sub1 PQSub b by SUBSTUT1:def 15;
  Z = [p,Sub1] by A3,XTUPLE_0:1;
  then
A5: All(x,S`1) = p & xSQ = Sub1 by XTUPLE_0:1;
A6: q is universal by QC_LANG1:def 21;
  then
A7: bound_in q = x by QC_LANG1:def 27;
  S`2 = b by A3,XTUPLE_0:1;
  then
  S`2 = ExpandSub(bound_in q,the_scope_of q,RestrictSub(bound_in q,q, xSQ
  )) by A4,A5,A6,SUBSTUT1:def 14;
  hence thesis by A6,A7,QC_LANG1:def 28;
end;
