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;

theorem Th31:
  t in F-entry_points_in_subformula_tree_of G iff (
  tree_of_subformulae(F))|t = tree_of_subformulae(G)
proof
  now
    assume t in F-entry_points_in_subformula_tree_of G;
    then (tree_of_subformulae(F)).t = G by Def3;
    hence (tree_of_subformulae(F))|t = tree_of_subformulae(G) by Th30;
  end;
  hence t in F-entry_points_in_subformula_tree_of G implies (
  tree_of_subformulae(F))|t = tree_of_subformulae(G);
  now
    assume (tree_of_subformulae(F))|t = tree_of_subformulae(G);
    then
A1: (tree_of_subformulae(F)).t = (tree_of_subformulae(G)).{} by TREES_9:35;
    (tree_of_subformulae(G)).{} = G by Def2;
    hence t in F-entry_points_in_subformula_tree_of G by A1,Def3;
  end;
  hence thesis;
end;
