theorem Th15:
  t is_a_proper_prefix_of t9 implies (tree_of_subformulae(F)).t9
  <> (tree_of_subformulae(F)).t
proof
  set G = (tree_of_subformulae(F)).t;
  set H = (tree_of_subformulae(F)).t9;
  assume t is_a_proper_prefix_of t9;
  then len @H < len @G by Th14;
  hence thesis;
end;
