Cats

So a couple of weeks ago my neighbours cat decided it was a good idea to climb through our kitchen window in the middle of the night and our greys got a hold of it and killed it, I have apologised to the neighbour, obviously they were angry and upset but accepted the apology but the refuse to believe that there cat would climb through our window, her cats are real pain in the neighbourhood with pooping in everybody's garden and entering other homes any advice would be great