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

theorem
  root-tree x = y-flat_tree p implies x = y & p = {}
proof
  assume
A1: root-tree x = y-flat_tree p;
  thus x = (root-tree x).{} by Th3
    .= y by A1,Def3;
 dom (y-flat_tree p) = elementary_tree len p by Def3;
  hence thesis by A1,Th2;
end;
