You have heard the saying "Forgive and Forget" but there is something slightly wrong with it. You want to forget in a certain manner but not forget completely. What I mean is that you don't want to obsess over a mistake that you might have made but you also don't want to forget about the lesson that you learned from the situation.
When you fail or make a mistake their is always something to be learned. Failure is good, as long as you learn from it. If you are having some disagreement with a friend and you forgive and forget than you might not learn anything.
Just think to yourself is their anything that I can learn from the previous situation. Which there usually is. Than learn from the situation and move on. You just need to better yourself by learning and than forgive the other person. If you try to prove yourself later, than it will just cause more trouble and fret.
I find myself trying to prove myself more often than not. I would guess that many other people do the same thing as me. I usually find myself doing this to my parents when they make me mad about a certain topic. It never gets any better from there. I and probably the majority of people need to learn that proving yourself and living in the past never ends well. It just causes more stress and anger. If you apologize or show gratitude to the other person than they will most likely be fine and also drop the situation.
One thing you have to know is that you don't want to immediately apologize to the other person because if they are still mad than it also won't turn out well. This is probably common sense though.
If you found this article helpful you might also find the article stop trying to prove yourself helpful.