Iterated antiderivative extensions

Let F be a characteristic zero differential field with an algebraically closed field of constants and let E be a no new constants extension of F. We say that E is an iterated antiderivative extension of F if E is a liouvillian extension of F obtained by adjoining antiderivatives alone. In this article, we will show that if E is an iterated antiderivative extension of F and K is a differential subfield of E that contains F then K is an iterated antiderivative extension of F.