reserve A for QC-alphabet;
reserve n,k,m for Nat;
reserve F,G,G9,H,H9 for Element of QC-WFF(A);
reserve t, t9, t99 for Element of dom tree_of_subformulae(F);
reserve x for set;

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;
