S3 (i9300/i9305) Is it possible to install CM10 after installing the 4.3 OTA update? I heard you cannot downgrade to 4.3 to 4.1.2.
But, is it possible to change it down altogether to CM10?   

Reason: The new 4.3 update runs terrible on my phone. I sent it back to Samsung and my carrier both and tried to update with KIES, but it's not getting any better. Some minor lag disappeared after they performed a full reset and flash, but nothing huge change. It's still crap. The lock screen (which I see at least 50 times a day) is absolutely horrible. The battery life also gone crap, since the update, Google+ loves to eat battery. Deleted the updates, disabled the app, signed out, none of those helped. Google of course won't care since I'm just one sucker from the many people.  

Sidenote: I know I won't be able to use Triangle Away (since KNOX) and such. But the phone is not usable anymore, it's a brick with a few hours of battery life. It's useless.
Yes, the reason you can't downgrade official ROMs is not a technical one; it's just a decision Samsung have taken.  

There's no reason at all why you can't go backwards and forwards between versions of Cyanogenmod as long as you have the appropriate recovery. And CM10.1.3 and CM10.2 are currently much more stable than CM11 on the i9300.  

Consult the Cyanogenmod site (http://www.cyanogenmod.org/) or XDA-Developers (http://forum.xda-developers.com/showthread.php?t=2382573) for full details before actually flashing a new recovery.  (Personally I'm staying with CM10.1.3 (http://forum.xda-developers.com/showthread.php?t=2035141) as I found the camera unstable on CM10.2)