I haven't really understood how the proposed simplification should work... It's clearly exponential without it, but I am not sure about the supposed added label

I haven't really understood how the proposed simplification should work... It's clearly exponential without it, but I am not sure about the supposed added labels. What should those be, conjunction and group subsets? Does that make them exponential? The proofs and even explanations in this paper aren't very clear to me...