I think your question might not be very clear. Unless you find a bill of sale on the system I don't know if actual ownership can be determined except perhaps demonstrated by possession of the system in question.
If you mean you want to demonstrate if the computer in question logged into a particular Facebook account I believe you should be able to establish that by the presence of a cookie associated with the Facebook account. If the cookie is active it could even be used to access the Facebook account though there may be other ethical and legal implications to that.
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