  Given a sequence of pairs of spin-one half particles in the singlet state,
assume that Alice measures the normalized projections along some vector of the
spins of one vector per pair along that vector while Bob measures the
normalized projections along some vector of the spins of the other member of
each pair. Then Quantum Mechanics, or QM, lets one evaluate the correlation of
the projections along these two vectors as minus the cosinus of the angle
between said vectors; we assume that all vectors are chosen in a fixed plane.
Assuming Classical Microscopic Realism, or CMR, there exist also normalized
projection pairs of the spins of the pairs of particles along some other pair
of vectors. Assuming QM and MR, we also have that the correlations of the
projections along the other vectors as minus the cosinus of the angle between
the extra vectors. Assuming Locality,i.e., the impossibility of any effect of
an event on another event when said events are spatially separated, beside QM
and MR, the theory of Bell lets one deduce various violations of some
inequalities at some choices of quadruplets of the vectors that have been
chosen. Our main result is the existence of quadruplets where at least one of
the said inequalities is violated if one only assumes QM, MR and some very mild
further hypotheses. These weak hypotheses only concern the behavior of
correlations that we use near special quadruplets. We thus get versions of
Bell's theorem that are strictly stronger than the original one and in
particular do not assume Locality.
