Your fact check does a good job of identifying the main issue with the claim and referencing a reliable source to support your point. You correctly pointed out that Illinois does not require party registration and clarified that voting in Republican primaries does not mean someone is a registered Republican. The structure could be improved to make your reasoning clearer. For example, starting with Illinois' election laws, then discussing Prevost's voting history, and finishing with why the claim is misleading. Some of the phrasing is a little unclear, like “newly Chose Pope,” and could be made smoother to help the reader better understand your point.