As for their third contention—namely, that the common factor in which the two agree might itself be an accidental attribute of the element by which they differ—although such an assumption may in principle be admissible, it cannot be admitted in the present case. The two entities in question must either differ with respect to necessity and existence and then have some other attribute predicated of them, or else they must differ by some other attribute and have necessity and existence predicated of them.
If they differ in necessity and existence, then there are not two Necessary Existents at all, whereas the hypothesis presupposes the opposite. On the other hand, if they agree in necessity and existence and some further attribute attaches to them, the cause of that adventitious attribute must be either necessity and existence themselves or something outside them. Necessity and existence cannot be the cause, for one and the same reality does not give rise to mutually divergent properties; that is conceivable only among individual members of a species because the individuals are composite. Nor may the cause be something other than necessity and existence, for that would entail that the Necessary Existent depends on something else—an absurdity, for necessary existence is by definition in no need of anything beyond itself.
We say to them: this matter may be addressed from several angles. First, your statement that if the two differ in existence and necessity there cannot be two Necessary Existents, whereas the contrary was the very point at issue, may well arise in this proof, yet it is invalid, superfluous, and ought not to be introduced here. When you laid out the fourfold division concerning two Necessary Beings distinguished from one another, it was inevitable that they share in one respect and differ in another, or coincide in something yet one of them be set apart from the other by something else—in short, that they come together in one aspect and separate in another. It therefore becomes impossible that they should fail to agree in necessary existence, which is the very subject of the division. To posit two necessary beings that do not share in necessity is like positing two existents that do not share in existence, or two black things that do not share in blackness, and the like.
On this assumption such a division is invalid and need not be mentioned. An instance parallel to their inserting into the division what does not in fact belong to what is being divided is their statement at the outset of the proof, “The things that…”