  This is the last in a series of papers where we prove a conjecture of Deser
and Schwimmer regarding the algebraic structure of ``global conformal
invariants''; these are defined to be conformally invariant integrals of
geometric scalars. The conjecture asserts that the integrand of any such
integral can be expressed as a linear combination of a local conformal
invariant, a divergence and of the Chern-Gauss-Bonnet integrand.
  The present paper, jointly with [6,7] gives a proof of an algebraic
Proposition regarding local Riemannian invariants, which lies at the heart of
our resolution of the Deser-Schwimmer conjecture. This algebraic Propositon may
be of independent interest, applicable to related problems.

Differential Geometry