Part I, Chapter 1: Fundamentals, Section 1.5

§1.5) Definition of Panel as an Operator

In order to develop a comprehensive theory of the mechanics of
sequential art, we must necessarily come to a functionally robust,
yet adequately general definition for what it means for an object
to be "inside" or "outside" of a panel. This section develops the idea
of a panel as a binary relation that maps objects from the space of
elements E to the spaces defined by a set of borders G. We develop
notation for the operator as Pm{E,G,k}.

1.5.1) Logical Definition of Panel:

Let Pm(x) be a predicate consisting of the statement "is contained within a border G", and x be any element of E or any n-tuple of elements selected therefrom. This predicate is true for all x, and is false for no x, e.g. for things not included in the universe of discourse.

(∀x∈E).Pm(x)...............Df.
(∀~x∈E).Pm(x) ⊃ f........Df.

1.5.2) Functional Definition of Panel:

With the logical description in hand, we now expand the definition into its full functional form.

Define: Pm{E,G,k}: E→G under k.

Where E is the tuple consisting of all elements in the layout L, and G is the set of all finite, simply-connected borders in L, such that E=(e1, e2, ..., et) and G={G1, G2, ...Gm}. Here, k is an index containing entries {σ1, σ2,..., σm}, where σ is some positive integer satisfying Σσm = t.

Then Pm is defined as the mapping which associates a certain sequence of elements out of E, specified by k, to a specific G.

Thus:

Pm{E,G,k} :=

G1 = (e1, e2, ..., e(σ1)) : k= σ1
G2 = (e(σ1) +1,e(σ1) +2, ..., e(σ1)+(σ2)) : k= σ2
...
Gm = (e(σt-m), ..., et) : k= σm

1.5.3) Some Properties of the Panel Operator:

Pm{E,G,k} is useful insofar as several interesting properties can be derived from this definition. These properties are analogous to the basic operations of arithmetic. They are: Union (Adding panels to each other) and Disjunction (Breaking panels up).

Union of adjacent Borders: ... Gn-1 ∪ Gn ∪ Gn+1 ...

Which can be represented by the operator as the following, given u, v, w ∈ E; Gn, Gn-1, Gn+1 ∈ G, σn, σn-1, σn+1 ∈ k:

1. Gn-1 ∪ Gn: Pm{E, Gn-1 + Gn, k}= Pm{u+v, Gn-1+Gn, σn-1 + σn}
2. Gn ∪ Gn+1: Pm{E, Gn + Gn+1, k}= Pm{v+w, Gn+Gn+1, σn + σn+1}
3. Gn-1 ∪ Gn∪ Gn+1: Pm{E, Gn-1 + Gn + Gn+1, k}= Pm{u+v+w, Gn-1+Gn+Gn+1, σn-1 + σn + σn+1}

Since E is well-ordered, we can write u+v+w=x, Gn-1+Gn+Gn+1 = G', σn-1 + σn + σn+1 = σ'. Thus:

Pm{E, Gn-1 ∪ Gn ∪ Gn+1, k} = Pm{x, G', σ'} .

Corollary: L = Σ G. ∎

However, this only makes sense for consecutive panels. We can run this oppositely to get disjunctions.