Tropical fish should not be kept with coldwater for a variety of reasons, and regardless of whether a fish is tropical, sub-tropical or coldwater etc, there is a lot more to keeping fish successfully than just keeping them in the right temps. You need to properly and thoroughly research fish before you buy them in the future rather than going on a quick bit of petshop advice, most petshop staff as you have found out do not know much on fish and tend to spread myths more than anything else rather than facts/correct info on fishkeeping. But either way petshops are not obliged to tell you how to be a good fish keeper, which is why most shops don't bother to train their staff much at all (if at all) on fishkeeping.
But anyway, for now you need help with your siamese fighter (also known as a betta).
The siamese fighter will heal if you separate it imediately- get a bucket of at least 10litres or more and fill it up with some water of the right temp (around 24-27 degree's C) and dechlorinate the water/put a dose of water conditioner in it and put a heater in the bucket if you can as it will help to keep the temp stable (which should be around 24-28 degree's C temp wise).
Acclimatise the betta to the bucket of water over a 30min period of time and then put a lid on the bucket or something similar to prevent the betta from jumping out. Medicate the betta with a med called Melafix, it is a good med for treating physical injuries on fish and helping prevent secondary infections from setting into their wounds. You should change the water in the bucket once a day, doing a 100% water change with dechlorinator in the process described above and put a new dose of melafix in each time-if you do this, the betta/siamese fighting fish should heal within a few weeks depending on how severe the fin injuries are.
If you have a spare aquarium of at least 10litres for the betta instead of a bucket then this is better as it will be easier to keep an eye on the betta's health in a glass or acrylic aquarium.
Fish are cold blooded animals and their metabolism is controlled by the temp of the water that surrounds them. If you put a coldwater fish in tropical temps, or vice versa, this will have a negative impact on its health as its metabolism is made too fast or too slow, and at the very least the fish will not live as long as it should do and has a very much increased chance of health problems/complications etc. So its very important to keep fish in the right temps if you want them to lead healthy lives with a good quality of life
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