Look up “Theory of Mind.” It may not fully answer your question but it may help.
Most commonly we hear the word “theory” in connection with models (e.g., models of the physical world) which are shared by many people. Theory of Mind is not like that. It is not shared with other people. Theory of mind is your own internal model of the mind of others.
You observe their behavior. You try to understand and predict their behavior. You try to engage in behavior with them. Perhaps you try to control their behavior. You do all this by imagining their mind.
In the area of evolutionary psychology, it is speculated that the brain developed at first only to control the movement of the organism in a sort of mindless way. A primitive animal uses sensory information to move in such as way as to obtain food, avoid danger, reproduce, and so on.
This primitive brain evolve larger and more complex neural networks that facilitated interaction with other animals. A predator needs to predict the movement of prey, and vice versa. Animals need to engage with others of their own species at least occasionally in order to reproduce.
Cooperation among social animals requires even more sophisticated brain functions. Perhaps the only reason we evolved mind at all is so that we can use it to model the higher-level brain function of others of our species. In other words, we have a mind so that we can imagine others’ minds and work out how to interact with them.
I suggest you turn the original question around. Is there a way to prove that your mind exists?