  In this paper, the easier methods of my thesis are applied to give a simple
proof of a theorem of Goussarov. The theorem relates two possible notions of
finite type equivalence of knots, links or string links, showing that the
resulting filtrations are the same up to a degree shift by a factor of two.
This is then applied to the situation of rooted claspers to show that rooted
claspers of sufficiently high degree must preserve type k invariants. As a
consequence, grope cobordisms of sufficiently high class must preserve type k
invariants. This result is applied in math.GT/0012118, to show Theorem 2 of
that paper.
