How does a health insurance company ensures someone else doesn't use my name to get a doctor appointment?

I mean he could falsify or steal my card and go see a physician saying he is me.
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Same as the waiter when you give him your credit card, and all the other instances where ID theft might happen. It is called Identity Theft and it is up to you to defend it with all tools available. You can download a booklet by the federal trade commission with all steps laid out: http://123automotive-insurance.solutions…
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