Delete other versions of iReal Pro application completely from your computer.
Use Spotlight to find other copies of iReal Pro that you might have installed in folders other than Applications.
Use Spotlight (magnifying glass on the top right corner) to find other copies of iReal b/Pro that you might have installed in folders other than Applications. You can Command-click from the Spotlight results to find the location of the other copies of app showing up in Spotlight and delete them. Then empty trash.
Then delete any related files using the Go to Folder command from the Go menu in the Finder:
and delete the folder called iReal b
and delete the file: com.massimobiolcati.irealbookmac.plist
After this clean up, restart your computer and download iReal Pro from the Mac App Store.