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?