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 Th20:
  G is_subformula_of F iff F-entry_points_in_subformula_tree_of G <> {}
proof
  now
    assume G is_subformula_of F;
    then G in rng tree_of_subformulae(F) by Th10;
    then ex x being object st x in dom tree_of_subformulae(F) & G = (
    tree_of_subformulae(F)).x by FUNCT_1:def 3;
    hence F-entry_points_in_subformula_tree_of G <> {} by Def3;
  end;
  hence G is_subformula_of F implies F-entry_points_in_subformula_tree_of G <>
  {};
  assume F-entry_points_in_subformula_tree_of G <> {};
  then consider x being object such that
A1: x in F-entry_points_in_subformula_tree_of G by XBOOLE_0:def 1;
  x in { t where t is Element of dom tree_of_subformulae(F) : (
  tree_of_subformulae(F)).t = G } by A1,Th19;
  then ex t st x = t & (tree_of_subformulae(F)).t = G;
  then G in rng tree_of_subformulae(F) by FUNCT_1:def 3;
  hence thesis by Th10;
end;
