reserve x, y, z for object,
  i, j, n for Nat,
  p, q, r for FinSequence,
  v for FinSequence of NAT;

theorem Th3:
  dom root-tree x = elementary_tree 0 & (root-tree x).{} = x
proof
 {} in elementary_tree 0 by TARSKI:def 1,TREES_1:29;
  hence thesis by FUNCOP_1:7;
end;
