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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?
Eugene

### I'm going to step through

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 ...

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