If prestige will be lost as sub city is occupied?

As you know you'll win prestige if you upgrade the buildings in the sub city.
If these prestige will be lost as this sub city is occupied by others?
or else as you occupy a sub city, if you will win all the prestige of that sub city including upgrading the buildings?

You won't lose the prestige you've obtained by upgrading buildings.

No prestige is lost/gained from stealing sub cities. If you built a sub and got prestige from building up the buildings you will keep that prestige. If you steal a sub that is already maxed you do not have to go through the hassle of building it up yourself but you do not get the prestige from building.

got it, thanks a lot!