hyperlinks in RTE textarea

Maybe I have drawn a wrong conclusion about not working hyperlinks in the FREE  RTE V3 B1, implemented in my own pages.
Looking again at the live demo, I saw that they don't work there either in the way I expected !
I mean, while you are editing the RTE-textarea, the inserted hyperlinks don't act as a hyperlink on a mouse-move-over.
After submiting the form, they come back as real usable hyperlinks,
but then it isn't still an editable RTE-textarea, but just a paragraph!!

 

My users edit the textarea, submit the form, which is updating the database, and then the same page comes back again, but the hyperlink in the RTE-textarea is still not clickable.
Questions:
1. Is it meant to work this way?
2. Could there possibly be a way to have 'active' hyperlinks in textareas?

 

(To all forumreaders, implementing V3 is very simple, compared with a lot of other applications, at least when you carefully read the instructions. The more we test, the sooner the beta version will come to the definitive V3, I hope!!  )