If $n$ order stats are iid from Uniform(0,1), why does dividing by the highest order stat give $n−1$ order stats iid from Uniform(0,1)?

If $n$ order stats are iid from Uniform(0,1), why does dividing by the
highest order stat give $n−1$ order stats iid from Uniform(0,1)?

As the title states:
If $P_{(1)}, ... ,P_{(n)}$ are order statistics of $n$ independent uniform
$(0,1)$ random variables, why are $P_{(1)}/P_{(n)} .....
P_{(n-1)}/P_{(n)}$ also order statistics of $n-1$ independent uniform
random variables on $(0,1)$, independent of $P_{(n)}$?
I'd prefer seeing an actual proof of this as I can't seem to locate one
anywhere. Any place I've found that mentions this theorem treats it as
"common knowledge" so-to-speak and doesn't give any theoretical
justification for the claim.
Any help is greatly appreciated!