Thankfully you are not teaching in an elementary school because I would expect an elementary school teacher to know the correct spelling of "bureaucracy".

The point you are flailing at is that Taiwan, Province of China is de jure part of China (and the vast majority of the world's nations recognizes the People's Republic of China as the sole legitimate government of China. De facto there is another entity occupying Taiwan Province.

Contrary to you claim that they are "not tied whatsoever", the ROC splinter government on Taiwan maintains a de jure claim over all of China (plus Mongolia) as blatantly illustrated by the official emblem of the ROC Marine Corps:

No country or international organization recognizes de facto control as the primary determing factor of independence or sovereignty. This is why terrorist organizations like ISIS or criminal gangs like certain cartels aren't considered sovereign states simply because they occupy and administer territory.

It's bureaucracy and bureaucratically

I'm not even trying to be a dick, I just had to look up what you wrote to make sure that it wasn't something else.

Furthermore, your point is correct regardless of whether or not people agree with the PRC or ROC being the leader of China overall, Taiwan province is an independently functioning government.