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Two new derivations of Bell's theorem is given here. In the first derivation, we consider a definite set of two entangled photons wit honly two polarization directions, for which Bell's locality assumption is violated for the case of perfect correlation. Then, using a different approach, we derive an efficient Bell-Type inequality which is a CH-type inequality for real experiments. Quantum mechanics violates our proposed ineqality, independent of the detection-efficiency problem.
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