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I don't think so, I think you have to just assume that when electrons are in the same orbital pattern, one is 1/2 and the other is -1/2. If they were going in the same direction, they would go against Hund's Law
There is a way to prove that an atom's electron can spin up or spin down through the beaming of silver atoms shown in one of Lavelle's lectures. The electrons spin dictated where the atom would go. However,I think you just need to know that paired electrons have the opposite spin.
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