  While initial versions of Bell's theorem captured the notion of locality with
the assumption of factorizability, in later presentations, Bell argued that
factorizability could be derived from the more fundamental principle of local
causality. Here we show that, contrary to what is commonly assumed, in order to
derive factorizability from the principle of local causality, a non-trivial
assumption, similar but strictly independent of settings independence, is
required. Loosely speaking, such an extra assumption demands independence
between the states of the measurement apparatuses. We conclude that it is
possible to construct a model, satisfying both the principle of local causality
and settings independence, but that, in virtue of violating this additional
assumption--and thus factorizability--is able to break Bell's inequality.
