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);
reserve G for Subformula of F;
reserve t, t9 for Entry_Point_in_Subformula_Tree of G;
reserve G1, G2 for Subformula of F,
  t1 for Entry_Point_in_Subformula_Tree of G1,
  s for Element of dom tree_of_subformulae(G1);
reserve s for FinSequence;

theorem
  for F,G1,G2 holds { t^s where t is Entry_Point_in_Subformula_Tree of
  G1, s is Element of dom tree_of_subformulae(G1) : s in G1
-entry_points_in_subformula_tree_of G2 } c= entry_points_in_subformula_tree(G2)
proof
  let F,G1,G2;
  { t^s where t is Entry_Point_in_Subformula_Tree of G1, s is Element of
dom tree_of_subformulae(G1) : s in G1-entry_points_in_subformula_tree_of G2 } =
  { t^s where t is Element of dom tree_of_subformulae(F), s is Element of dom
  tree_of_subformulae(G1) : t in F-entry_points_in_subformula_tree_of G1 & s in
  G1-entry_points_in_subformula_tree_of G2 } by Th39;
  hence thesis by Th29;
end;
