  We prove the cosmic no-hair conjecture for all orthogonal Bianchi cosmologies
with matter in the $R+\beta R^2$ theory using the conformally equivalent
Einstein field equations, with the scalar field having the full
self-interacting potential, in the presence of the conformally related matter
fields. We show, in particular, that the Bianchi IX universe asymptotically
approaches de Sitter space provided that initially the scalar three-curvature
does not exceed the potential of the scalar field associated with the conformal
transformation. Our proof relies on rigorous estimates of the possible bounds
of the so-called Moss-Sahni function which obeys certain differential
inequalities and a non-trivial argument which connects the behaviour of that
function to evolution of the spatial part of the scalar curvature.
