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
  (tree_of_subformulae(F)).t = F iff t = {}
proof
  now
    reconsider t1 = {} as Element of dom tree_of_subformulae(F) by TREES_1:22;
    assume
A1: (tree_of_subformulae(F)).t = F;
A2: t1 is_a_prefix_of t;
    assume t <> {};
    then t1 is_a_proper_prefix_of t by A2;
    then (tree_of_subformulae(F)).t is_proper_subformula_of (
    tree_of_subformulae(F)).t1 by Th16;
    hence contradiction by A1,Def2;
  end;
  hence (tree_of_subformulae(F)).t = F implies t = {};
  assume t = {};
  hence thesis by Def2;
end;
