Part I, Chapter 1: Fundamentals, Section 1.3

§ 1.3) Uniqueness Theorem

The Uniqueness Theorem states that any given element et must absolutely
choose to belong to one panel and no others. Therefore, the set
0f elements et belonging to Pn is the set of those objects with the property
of not belonging to any prior or subsequent panels.

1.3.1) Plain language proof:

Given the Axiom of Panels and its corollary, let us define a panel ϕ(x) with the property of being the set of all elements contained in both Pn and Pn+1. Now every x that is an element of ϕ(x) will have the following properties:

1. It belongs to neither Pn-1, nor Pn+1 .
   
2. It belongs to neither Pn, nor Pn+2.

In terms of predicate logic for a two-valued function ϕ(x,y) where x=y.

(x,y)ϕ(x,y): ~x ^ x
(x,y)ϕ(x,y): y ^ ~y

Using a truth table we can compute the value for these statements.

~x ^ x
F F T
F F T
T F F
T F T

y ^ ~y
T F F
T F F
F F T
F F T

And, since both statements are equally false, the statement ~x^x = y^~y is true for all cases. Substituting y=x for y, we get ~x^x = x^~x. Being true in all cases, then, means that we can say without a doubt that the statement (x, ...)ϕ(x,...):{x NOT in Pn-1, x NOT in Pn, x NOT in Pn+1, x NOT in Pn+2, ...} is always false for any given set of x's, since by mathematical induction, we can see that the proof works for the n+1 case as well. Therefore, by induction, we conclude that because ϕ(x) contains all the elements x that do not belong to any union of spaces Pn U Pn+1, it is therefore equal to the empty set. Therefore, any element et must belong to one and only one Pn. ▮.

1.3.2) Proof in Symbolic Logic.

A proof in symbolic logic will follow.

NB: Bear in mind that these assertions make no statement on the movement of the eye over the design. At this stage, we are merely trying to build up precise definitions for concepts such as "belonging to" or "exists".

1.3.3) Implications of the Uniqueness Theorem.

The Uniqueness Theorem implies that, whatever their position in space related to the layout, an element will belong to that panel where it was first read. Imagine if each panel were on its own unique layer, and whatever arrangement of elements was free to exist either within or outside the panel boundaries. By the Uniqueness Theorem, all elements first encountered within those panels will belong strictly to those panels and those alone, since each dwells within its own unique space.