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 Th16:
  t is_a_proper_prefix_of t9 implies (tree_of_subformulae(F)).t9
  is_proper_subformula_of (tree_of_subformulae(F)).t
proof
  set G = (tree_of_subformulae(F)).t;
  set H = (tree_of_subformulae(F)).t9;
  assume
A1: t is_a_proper_prefix_of t9;
  then t is_a_prefix_of t9;
  then
A2: H is_subformula_of G by Th13;
  H <> G by A1,Th15;
  hence thesis by A2,QC_LANG2:def 21;
end;
