Unfortunately after that I got the pop-up window with the message "A later vesrion of Apple Application Support is already installed on this computer".
That's a rarer one (in this particular context), Yuri. (Periodically over the past few months I've been musing over why that sort of thing doesn't hit AAS as often as it hits other components. I think it's because recent versions of AAS use slightly different installer technology than the other components.)
At any rate, unfortunately, this sort of trouble has gotten more complicated to deal with ever since Microsoft pulled the Windows Installer CleanUp utility from their Download Center on June 25 2010. First we have to find a copy of the utility.
Let's try Googling. (Best not to use Bing, I think.) Look for a working download site for at least version 3.0 of the Windows Installer CleanUp utility. (The results from mydigitallife and Major Geeks are worth checking.)
After downloading the utility installer file (msicuu2.exe), scan the file for malware, just in case. (I use the free version of Malwarebytes AntiMalware to do single-file scans for that.)
If the file is clean, to install the utility, doubleclick the msicuu2.exe file you've downloaded.
Now run the utility ("Start > All Programs > Windows Install Clean Up"). In the list of programs that appears in CleanUp, select any Apple Application Support entries and click "Remove".
Quit out of CleanUp. Restart the PC, and try another Apple Application Support install. Does it go through properly this time?