This is a question about pure logic.
There are two theries: Theory A and Theory B. Theory A assumes AssumptionA. Theory B assume AssumptionB.
The two assumptions are mutually exclusive: if AssumpionA then not AssumptionB and vice versa.
I believe that a philosophical result is that Theory A and Theory B cannot prove anything about each other. All you can do is preface each result with the assumption. For example, if Theory A proves X and Theory B proves Y, then we can say "If AssumptionA, then X" and "If AssumptionB then Y".
Who first proved this? Where is it documented?
I'm going to step through this carefully to make sure I follow. We have two theories: A and B . Theory A has an assumption: A and theory B has an assumption B . And A and B are mutually exclusive—can't both be true. Let's pause. To say that a theory has an assumption means that if the theory is true, then the assumption is true. It doesn't mean that if the assumption is true, then the theory is true. A silly example: the special theory of relativity assumes that objects can move in space. But from the assumption that objects can move in space, the special theory of relativity doesn't follow; you need a lot more than that. Otherwise, the "assumption" would be the real theory. You ask if it's true that neither theory can prove anything about the other. If I understand the question aright, it's not true. For one thing, trivially, if we take A as a premise, then by your own description, it follows that B is false. That seems like a case of proving something about B ...