What do you do when one of you agrees with your point? Does it appease you, or do you just immediately find another point you need to make?

Sometimes I need to be heard, and I need to hear that I am right. But, even when it goes exactly as I want it to, then I back right back down. Okay, so what, I was right. Does that really matter in the long run? If I get stuck on a point like that, as soon as I hear "I was wrong" it all goes away.

If it continues even after the other person agrees with you, then you're fighting for the sake of fighting.