reserve i,k for Nat;
reserve A for QC-alphabet;
reserve x for bound_QC-variable of A;
reserve a for free_QC-variable of A;
reserve p,q for Element of QC-WFF(A);
reserve l for FinSequence of QC-variables(A);
reserve P,Q for QC-pred_symbol of A;
reserve V for non empty Subset of QC-variables(A);
reserve s,t for QC-symbol of A;

theorem
  p is existential implies Vars(p,V) = Vars(the_argument_of the_scope_of
  the_argument_of p, V)
proof
  set p1 = the_argument_of the_scope_of the_argument_of p;
  set x = bound_in the_argument_of p;
  assume p is existential;
  then p = Ex(x,p1) by QC_LANG2:40;
  then p = 'not' All(x,'not' p1) by QC_LANG2:def 5;
  then Vars(p,V) = Vars(All(x,'not' p1), V) by Th39
    .= Vars('not' p1, V) by Th44
    .= Vars(p1, V) by Th39;
  hence thesis;
end;
