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;

theorem Th31:
  [S,x] is quantifiable implies CQCQuant(CQCSub_All([S,x],xSQ),
  CQC_Sub(S)) = All(S_Bound(@CQCSub_All([S,x],xSQ)),CQC_Sub(S))
proof
  set S1 = CQCSub_All([S,x],xSQ);
  set p = CQC_Sub(CQCSub_the_scope_of S1);
A1: Quant(S1,p) = All(S_Bound(@S1),p) by SUBSTUT1:def 37;
  assume
A2: [S,x] is quantifiable;
  then CQCSub_All([S,x],xSQ) = Sub_All([S,x],xSQ) by Def5;
  then CQCSub_All([S,x],xSQ) is Sub_universal by A2,SUBSTUT1:14;
  then
A3: CQCQuant(S1,p) = Quant(S1,p) by Def7;
  CQCQuant(S1,CQC_Sub(S)) = CQCQuant(S1,p) by A2,Th30;
  hence thesis by A2,A3,A1,Th30;
end;
