  We prove versions of the Bell and the GHZ Theorems that do not assume
Locality but only the Effect After Cause Principle (EACP) according to which
for any Lorentz observer the value of an observable cannot change because of an
event that happens after the observable is measured. We show that the EACP is
strictly weaker than Locality. As a consequence of our results, Locality cannot
be considered as the common cause of the contradictions obtained in all
versions of Bell's Theory. All versions of Bell's Theorem assume Weak Realism
according to which the value of an observable is well defined whenever the
measurement could be made and some measurement is made. As a consequence of our
results, Weak Realism becomes the only hypothesis common to the contradictions
obtained in all versions of Bell's Theory. Usually, one avoids these
contradictions by assuming Non-Locality; this would not help in our case since
we do not assume Locality. This work indicates that it is Weak Realism, not
Locality, that needs to be negated to avoid contradictions in microscopic
physics, at least if one refuses as false the de Broglie-Bohm Hidden Variable
theory because of its essential violation of Lorentz invariance.
