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The Boolean theorem $$AB + \overline{A} C + BC = AB + \overline{A} C$$ crresponds to
$$(A + B).(\overline{A} + C).(B + C) = (A + B).(\overline{A} + C)$$
$$AB + \overline{A} C + BC = AB + BC$$
$$AB + \overline{A} C + BC = (A + B).(\overline{A} + C).(B + C)$$
$$ (A + B).(\overline{A} + C).(B + C) = AB + \overline{A} C$$
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