UPDATE: I was able to prove AssumeS1(C)&AssumeS2(C)≃AssumeS1∪S2(C) in general whenever S1 and S2 are disjoint and both in Obs(C), with help from Rohin Shah, following the “restrict attention to world S1∪S2” approach I hinted at earlier.
UPDATE: I was able to prove AssumeS1(C)&AssumeS2(C)≃AssumeS1∪S2(C) in general whenever S1 and S2 are disjoint and both in Obs(C), with help from Rohin Shah, following the “restrict attention to world S1∪S2” approach I hinted at earlier.