If my father died, mother is still living…and a sister is claiming she has power of attorney, how do I make her PROVE she IS in fact POA?

If sister does not have an original POA document nor can a copy be found, check with your mother's bank(s) or other financial institutions. Check with the attorney who may have drafted the POA. Check in your mother's papers. It's a long shot, but talk to your mother's bank(s) to see if they can help you determine whether one is in a safety deposit box. Without authority, they can't let you in, but they may have some ideas,

Also determine if your mother has in fact been diagnosed with Alzheimers or some form of dementia and is not COMPETENT. Even Alzheimers patients can be considered competent by the courts if they have lucid days. If she is assuredly not competent she cannot sign a new medical or financial POA. Unfortunately, the next step is to petition the court for conservatorship/guardianship. This can be complicated and expensive. Your best first step is to consult with an elder-issues attorney. Do not go to any attorney. A consultation should not cost a lot.

I am assuming that your Mother is incapable of speaking to you about this? Because the simple answer would be to ask your Mom if your Sis has POA.
If your Mom is suffering from dementia, what concerns do you have about POA? Do you suspect your Sister of fraudulent behavior? Of embezzling funds? Of doing an improper job?
When your Father died was your Sister ALSO the executor of his will and estate?
If you have reasons for concern (you haven't told us any), then do open a case with APS for wellness check on your Mom, let them know you have concerns, and let them know what those concerns ARE. Then provide any and all proof you are able to APS. While they will not come back to you with ANSWERS they will come back to the reporter to say they either did or did not find reasons for concern.

If it turns out she isn't the PoA, then what? Is she suspected of theft or fraud? You don't really give much detail. If I were in your situation and you actually have some suspicion or proof of theft/fraud, then I'd consult with an elder law attorney (but do not tell sister this) to see what is advised. Laws can differ by state, so that's why you have to talk to a professional for your mother's state.

FYI even if she is the PoA it usually isn't in effect until there is one or more medical diagnosis of incapacity (depending on what is stipulated in the PoA document).