This is exactly what I've been saying from the start of this thread.
I have shown quotes from the quran which say they should cover up. This isn't a pure choice on their part, especially given how literally the book is taken, this would be taught as 'correct' from birth. So ending up believing it should be done and it is a willing choice on their part is going to happen.
Let's be honest, at some point someone said "women should cover up and not show themselves to other men". We know this because it is written in the quran (and in some countries the laws), which means someone has said it (let's ignore whether it was a damn good fiction writer or a god). So this is women being told, you should cover up.
It is an oppressive device, which someone has decided women should wear because they say so (again, leave the 'who said this' out of it for now). I have seen no evidence that at some point in time a woman went "you know what, I think I'll wear a bed sheet today" and somehow started a fashion trend which was then written into their holy book (perhaps some sort of old school fashion advert or catalogue page?

) and has grown to what it is today.
EDIT: lisab, I think there is definitely an edge of what you have said above in this law despite my views above. But then, is it such a bad thing? I don't see a problem with wanting people to integrate into the current population better. Otherwise you face potential segregation of immigrant groups.