I work in a hospital. I want to know what we should call the proof of appointment. And I want to know the name of the place where we pay the bill after treatment.
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In English, you'd call the proof that you have an appointment to see a doctor at a hospital clinic an appointment slip. A "slip" is a piece of paper. The slip in the link is an official US Army document labeled Appointment Slip. That term should work in Korea.
A case paper is something the patient may provide as proof of having an appointment to see the doctor.