one of those parts: By “composite,” do you mean those parts, their coming together, both, or a fourth possibility? If you mean the first, this would mean that those parts depend upon those parts. It would amount to saying that the composite thing depends upon the composite, that a thing depends upon itself, and that what is necessary in itself depends upon what is necessary in itself.
It is evident that what is necessary in itself cannot be independent of itself. Indeed, its being necessary in itself entails that it cannot be independent of itself.
Thus, the dependence you mention establishes its being necessary in itself, rather than precluding it.
If it is said that the composite is the conjunction—that is, the coming together and combination of the parts—the reply is: This conjunction is an attribute and an accident of the parts. No rational person says that it is necessary in itself independently of the parts. Rather, it is said only to be an inseparable concomitant of the parts. What is necessary in itself is the self-subsisting essence, namely the parts, not merely the attribute that constitutes a relation between the parts. Since this conjunction is not itself the essence that is necessary in itself, but only an attribute of it, the same reasoning applies to it as to the other things you have called parts. At most, some of the parts would be