Yikes yeah that is fin rot. Not surprising given the care he had before... Shame on that mother, it was solely up to her to care for the pet, and she COMPLETELY failed. *dismounts high horse*
To treat fin rot (and this is almost advanced to body rot on that bottom/anal fin) you will want to add aquarium salt, gradually increasing the salinity to 1tsp/gallon. I mix some up in some heated aq water, so I'd try to get 2 or 3 cups heated up, and add mix 10 tsp of aquarium salt into that. Then add maybe 1/4 of a cup at a time, giving 30 mins in between for him to get used to it.
That will purge the infection from his fins. If it continues to advance, you can increase to 2 tsp/gal, but do not leave the salt in for more than 10 days, it's pretty hard on their kidneys. I imagine with that and the clean/warm water, he'll heal up pretty quickly :)