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 Th29:
  B is quantifiable implies CQCSub_the_scope_of(CQCSub_All(B,SQ)) = B`1
proof
  assume
A1: B is quantifiable;
  then
A2: CQCSub_All(B,SQ) = Sub_All(B,SQ) by Def5;
  then CQCSub_All(B,SQ) is Sub_universal by A1,SUBSTUT1:14;
  then
  CQCSub_the_scope_of(CQCSub_All(B,SQ)) = Sub_the_scope_of(Sub_All(B,SQ))
  by A2,Def6;
  hence thesis by A1,SUBSTUT1:21;
end;
