theorem
  for W being finite Tree holds W = union { W-level n: n <= height W }
proof
  let W be finite Tree;
  thus W c= union { W-level n: n <= height W }
  proof
    let x be object;
    assume x in W;
    then reconsider w = x as Element of W;
A1: len w <= height W by TREES_1:def 12;
A2: w in W-level len w;
 W-level len w in { W-level n: n <= height W } by A1;
    hence thesis by A2,TARSKI:def 4;
  end;
  let x be object;
  assume x in union { W-level n: n <= height W };
  then consider X such that
A3: x in X & X in { W-level n: n <= height W } by TARSKI:def 4;
 ex n st X = W-level n & n <= height W by A3;
  hence thesis by A3;
end;
