Bell's Theorem is a consequence of the violation of Bell's Inequality.
Loosely stated the theorem says ... "No physical theory of local hidden variables can ever reproduce all of the predictions of quantum mechanics." (from wikipedia).
"According to the quantum entanglement theory of quantum mechanics, on the other hand, distant events may under some circumstances have instantaneous correlations with local ones." (from wikipedia).
This is a fairly readable account of Bell's Theorem.
Even so, it takes some time and effort to wade through.
http://www.upscale.utoronto.ca/PVB/H...lsTheorem.html
"Believe me when I tell you that I'm not that honest" (see document text for relevance).
Regards, Rob