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There really isn't a way for you to prove that. As an employer, you don't have the right to subpoena records or demand responses to discovery. That being said, that isn't something that you really have to prove. You have sufficient evidence, based on what you've already stated, to legally justify a termination and to put the question to state for blocking unemployment.
Even if her mother is dead, and was having a funeral, the information she gave you was not correct and so the time taken off is not legally protected in any way (in fact, legally there is no requirement for bereavement leave in the first place). That would still justify a termination and the blocking of unemployment and then the burden would shift to her to prove that she was dead, the date was correct and the funeral home is lying.
Only if the matter were to go to court (if she sued you) would you then have the power of subpoena and discovery to obtain information proving life.
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