reserve x,y,z,a,b,c,X,X1,X2,Y,Z for set,
  W,W1,W2 for Tree,
  w,w9 for Element of W,
  f for Function,
  D,D9 for non empty set,
  i,k,k1,k2,l,m,n for Nat,
  v,v1,v2 for FinSequence,
  p,q,r,r1,r2 for FinSequence of NAT;

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;
