“…Then U ∩ A x is a nonempty relatively w-open subset of B A x with diam(U ∩ A x ) < 2, so that, by Lemma 2.2, Ω x is finite, and hence A x is finite-dimensional. Since x is arbitrary in U, and U has nonempty n-interior in A, it follows from either [7,Theorem 1] or [8,Theorem D] that in fact A x is finite-dimensional for every x in A. Now, for every x in A, the set σ(x) := {λ ∈ R : x − λ1 is not invertible in the unital hull of A x } is finite.…”