Great question — answering as the UphillTech team:
Purchase the replacement, it comes with a tool you will need to remove the cartridge. Before you start turn the water off both hot and cold. The tricky part is finding the set screw that holds the handle on the stem. They all have a set screw. Remove the screw and put it in a safe place, I generally use an old prescription bottle. Work the handle loose, it may take a bit of jiggling. It is possible there is an adapter plate below that, remove that as well. Next use the tool and remove the ring, by unscrewing the big nut behind it, the tool pulls the cartridge out. Here is a link showing how to remove the cartridge Just reverse the process to install the new one. There should be instructions...
Full answer by Gil: https://diy.stackexchange.com/a/231340 (CC BY-SA 4.0).
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