apostrophe possession question

I am a medical transcriptionist and they have their own rules of grammar, spelling, Β and punctuation (drives me crazy). However, one thing I am not certain on is when the doctor dictates, " John Doe is a patient of Dr. Smith's." I want to put, "John Doe is a patient of Dr. Smith." Does the inference of Dr. Smith's practice make the Dr. Smith's spelling correct OR should it be changed to Dr. Smith?
I'm sure I've taken part in discussions before about the "double possessive". If I recall the outcome correctly, it's used when the thing possessed is a representative of a whole group of similar things that belong to the possessor. So in your example, Dr Smith has a bunch of patients, as a group they're "Dr Smith's patients", and John Doe is one member of that group.
'Patient of Dr Smith' is not wrong, except for being too rare compared to the other form. It's a clue of foreignism.
Some discussions here and here.
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