In reference to this answer. I'm guessing it has something to do with being able to find some elements in I with the same initial terms as f and g, and then canceling out leading terms somehow, but I have no idea how to approach it, and haven't been able to find it in either of the textbooks I have access to.
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Your question seems to be far from what is proved in the mentioned answer, and somehow hardly understood. The answerer uses an induction on the leading monomials, while what you asked has nothing to do with this and in the context there is no reason for g to be in I. – user26857 Feb 13 '19 at 08:35