Is anyone else struggling to understand theorem 2 on pages 15-16? It sounds like what they are saying is equivalent to the following: If S1 intersect S2 has a

Is anyone else struggling to understand theorem 2 on pages 15-16? It sounds like what they are saying is equivalent to the following: If S1 intersect S2 has a solution, and S1 intersect S3 has a solution, then the system S1 intersect S2 intersect S3 has a solution. But this is evidently false. Consider the case where the CTS included each of the following rows, and were empty everywhere else (after re-ordering the columns so the same-name columns were in the same final column): (1) 000 (2) 001 (3) 0 00 In this case, (1) and (2) are consistent, and (1) and (3) are consistent, but (2) and (3) are inconsistent. I suspect the problem they set up the induction for might not perfectly align with the theorem, but it needs more careful examination.