I was reading MIT's reading notes on relations and partial orders. In doing so I came across a section which said that because posets have topological sort, it will helps us in executing tasks sequentially. Then it says that for any posets there is a legal parallel schedule that runs in t steps where t is the length of longest chain. To prove this claim I came across this theorem:
I don't understand the last sentence "for all a in Ai, we have that all b<_a appear in the set A U...UAi-1." what does it mean by " for all a in Ai". What does a refer to in a in Ai?