You'd probably need to uninstall BI first. I think attempting to remove BI will automatically require RTW to be uninstalled. It's possible either option removes both parts, although I know from past experience with addons, patches and so forth you sometimes get start menu entries left behind etc when done out of order. I've uninstalled RTW and BI before though with no probs.


The order of everything else seems correct.


If you reinstall RTW and BI, after patching make a backup of the data folder for both RTW and BI, or the entire game. Then if you anything untoward occurs again, you have the backup on hand which you can just revert to, hopefully meaning no need to reinstall again.