1. If you understand this first distinction, the much more difficult division of the intelligible world will make more sense. Think over this carefully: the visible world, that is, the world you see, has two kinds of visible objects in it. The first kind are shadows and reflections, that is, objects you see but aren't really there but derive from the second type of visible objects, that is, those that you see and are really there. The relation of the visible world to the intelligible world is identical to the relation of the world of reflections to the world of visible things that are real.
2. The lower region of the intelligible world corresponds to the upper region in the same way the lower region of the visible world corresponds to the upper region. Think of it this way: the lower region deals only with objects of thought (that are, in part, derived from visible objects), which is why it is part of the intelligible world. There have to be certain first principles (such as the existence of numbers or other mathematical postulates) that are just simply taken without question: these are hypotheses. These first principles, however, derive from other first principles; the higher region of the intelligible world encompasses these first principles. So you can see that the lower region derives from the higher region in that the thinking in the lower region derives from the first principles that make up the higher region, just as the mirror reflects a solid object. When one begins to think about first principles (such as, how can you prove that numbers exist at all?) and derives more first principles from them until you reach the one master, first principle upon which all thought is based, you are operating in this higher sphere of intellection ...