Confusion about the proof

[Coder-Yu]

Hi,

Thanks for the good work on graph condensation. I am recently working on this topic. When I read the proof of proposition 1, I had the following confusions.
1). In Eq. (12), do the subscript i and j both start from 0 or 1? In the current form, i starts from 0 while j starts from 1.
2). In Eq. (14), the outer integral is combined. Howover, the form of Eq. (14) does't make sense as the two matrices in the bracket have different sizes. Although I do understand the implication here, it seems the form of Eq. (14) needs to be revised? Once the form is revised, the transition from Eq. (14) to Eq. (15) could be more smooth.