Delete other versions of iReal Pro application completely from your computer.

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:
Go to:
~/Library/Application Support/
and delete the folder called iReal b

Go to:
~/Library/Preferences
and delete the file: com.massimobiolcati.irealbookmac.plist

After this clean up, restart your computer and re-install iReal Pro.

If you still have issues, let us know.