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;
reserve x,y for set;
reserve t for Element of dom tree_of_subformulae(F),
  s for Element of dom tree_of_subformulae(G);
reserve t for Element of dom tree_of_subformulae(F),
  s for FinSequence;
reserve C for Chain of dom tree_of_subformulae(F);

theorem
  for F,G,H,C st G in { (tree_of_subformulae(F)).t where t is Element of
dom tree_of_subformulae(F) : t in C } & H in { (tree_of_subformulae(F)).t where
t is Element of dom tree_of_subformulae(F) : t in C } holds G is_subformula_of
  H or H is_subformula_of G
proof
  let F,G,H,C;
  assume that
A1: G in { (tree_of_subformulae(F)).t where t is Element of dom
  tree_of_subformulae(F) : t in C } and
A2: H in { (tree_of_subformulae(F)).t where t is Element of dom
  tree_of_subformulae(F) : t in C };
  consider t9 such that
A3: G = (tree_of_subformulae(F)).t9 and
A4: t9 in C by A1;
  consider t99 such that
A5: H = (tree_of_subformulae(F)).t99 and
A6: t99 in C by A2;
A7: t9,t99 are_c=-comparable by A4,A6,TREES_2:def 3;
  per cases by A7;
  suppose
    t9 is_a_prefix_of t99;
    hence thesis by A3,A5,Th13;
  end;
  suppose
    t99 is_a_prefix_of t9;
    hence thesis by A3,A5,Th13;
  end;
end;
