  We rigorously verify that in the spatially homogeneous spacetimes (Bianchi
VIII and IX), the presence of matter does not affect the oscillatory behavior
of the solutions to the Einstein field equations as first conjectured by
Belinski, Khalatnikov, and Lifshitz (BKL) in 1970. This paper is an extension
of the paper by Reiterer and Trubowitz (arXiv 1005.4908) and uses the same
formalism. We use the ultrarelativistic equation of state.
