What happens next?

How to win an argument against someone who won't lose
Stay cool-headed.
When these people are losing an argument, they tend to do little more than cut you off and then insult you. If you become irritated or enraged, they can continue to insult you as they know you are weak at that point.
Your anger can fuel their anger as well. However, when you stay rational and calm, you can continue to prove their points wrong without getting into physical violence.