How do you ask for proof? Just ask. Get a doctor's note on the doctor's stationery. This alleged condition essentially means that this employee is only intending to work a four-day week for the next . . . one to two years. Can your business afford to pay her full-time for less than full-time work? As an employer with only 10 employees, you do not meet the thresholds where the federal employment laws would apply.
You may have a number of options at your disposal. You may be able to fire her, reduce her salary or work out a telecommuting arrangement where she works from home on the weekend (to make up for the Friday she'll be missing. Speak to a local employment attorney to determine your options under federal and state law.