William wrote:

> John, your definition of \\(f^{\ast}(P)\\) looks dubious if \\(f\\) is not surjective, because it may have empty parts. It happens to work in the example given...

There are never empty parts in \\(f^{\ast}(P)\\), because the parts in \\(f^{\ast}(P)\\) are inverse images of parts in \\(P\\), which are nonempty, and the inverse image of a nonempty set is always nonempty.