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;

theorem Th36:
  entry_points_in_subformula_tree(G) = { t where t is
  Entry_Point_in_Subformula_Tree of G : t = t }
proof
  thus entry_points_in_subformula_tree(G) c= { t where t is
  Entry_Point_in_Subformula_Tree of G : t = t }
  proof
    let x be object;
    assume x in entry_points_in_subformula_tree(G);
    then x in { t where t is Element of dom tree_of_subformulae(F) : (
    tree_of_subformulae(F)).t = G } by Th19;
    then consider t9 being Element of dom tree_of_subformulae(F) such that
A1: t9 = x and
A2: (tree_of_subformulae(F)).t9 = G;
    reconsider t9 as Entry_Point_in_Subformula_Tree of G by A2,Def5;
    t9 = t9;
    hence thesis by A1;
  end;
  thus { t where t is Entry_Point_in_Subformula_Tree of G : t = t } c=
  entry_points_in_subformula_tree(G)
  proof
    let x be object;
    assume x in { t where t is Entry_Point_in_Subformula_Tree of G : t = t };
    then ex t st t = x & t = t;
    hence thesis by Th35;
  end;
end;
