Come to think of it, my proof of c) needs more elaboration. If |(H \cap K)| is bigger than 1 it means that it's impossible each element of HK to have only one representation of type hk. But it does not prove that it's the case when |(H \cap K)|=1.

It's intuitive, but I doubt how well I can formalize it. By definition of HK, each sequence hk represents element of HK so there is "map" between the set of representations and the set of distinct elements. This map is surjective by def. of HK and the two sets have equal cardinality, hence the map is bijective. Injectivity follows...