To show sets are equal, you show they are subsets of each other.
Let x be an element of interior of (A?B). That means that there is a ball around x which is entirely contained in (A?B). That means it is entirely contained in A so x is in the interior of A. And same for B.
That establishes that int(A?B) is a subset of (interior of A) ? (interior of B)
Now the other direction. Let x be an element of (interior of A) ? (interior of B). That means it's an element of (interior of A) and it's an element of (interior of B).
That means there's a ball around X which is entirely contained in A, and there's also a ball around X which is entirely contained within B. Pick the smaller of those two balls.
