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 Th19:
  F-entry_points_in_subformula_tree_of G = { t where t is Element
  of dom tree_of_subformulae(F) : (tree_of_subformulae(F)).t = G }
proof
  thus F-entry_points_in_subformula_tree_of G c= { t where t is Element of dom
  tree_of_subformulae(F) : (tree_of_subformulae(F)).t = G }
  proof
    let x be object;
    assume
A1: x in F-entry_points_in_subformula_tree_of G;
    F-entry_points_in_subformula_tree_of G c= dom tree_of_subformulae(F)
    by TREES_1:def 11;
    then reconsider t9 = x as Element of dom tree_of_subformulae(F) by A1;
    (tree_of_subformulae(F)).t9 = G by A1,Def3;
    hence thesis;
  end;
  thus { t where t is Element of dom tree_of_subformulae(F) : (
  tree_of_subformulae(F)).t = G } c= F-entry_points_in_subformula_tree_of G
  proof
    let x be object;
    assume x in { t where t is Element of dom tree_of_subformulae(F) : (
    tree_of_subformulae(F)).t = G };
    then ex t9 st t9 = x & (tree_of_subformulae(F)).t9 = G;
    hence thesis by Def3;
  end;
end;
