A Proof in for (y)((x)Ax É Ay)

We shall display this proof in "ordinary" drawings of Existential Graphs (executed in Adobe Illustrator and converted to GIF) on the left and with an equivalent display executed in the EGControlLibrary application () and saved from that application as SVG files. Since SVG is not quite, as yet, a universal Web Graphics format, these are shown as converted to GIF as well.

The first stage is, as usual, to set a Biclosure on the Sheet of Assertion as permitted by the rule of Positive Biclosure

 Figure 1. Biclosure inserted on SA

The next step involves insertion on Verso (i.e., in an Oddly Enclosed Area) of a Spot with attached Line of Identity (Dot). At this point in , the facility to show connections between different LI's in a given Ligature is turned off; the Graph inserted in is a Spot with its attached LI "Stub" and another segment of LI belonging to the same Ligature. There are a number of ways to effect this in , all of which would be legal according to the Beta Rules.

 Figure 2. Insert a Graph on Verso

Next, the Spot is Reiterated into the "inner close," maintaining its connection to the original Ligature.

 Figure 3. Reiterate Beta Graph

Now another Biclosure is inserted, this time around the original Spot and its LI Stub. In this is best shown as taking two steps: the first is the actual biclosure insertion, which here is shown to involve nothing resembling the placement of a Graph in two distinct Areas - no Graph crosses a Cut. The second phase of the operation here involves a Reiteration of the LI segment in the outer Cut into the outer, Recto, area of the new Biclosure; this is completely legal and brings into play the continuity of the Ligature across this Biclosure as shown in the original "traditional" EG diagram on the left.

 Figure 4. A Beta Biclosure

The next step is to break the Ligature (as represented in the generated diagram) by "Permission to break a LI on Recto (an Evenly-enclosed Line of Identity); is able to do this and to appropriately assign the components of the original Ligature to either Ligature as appropriate; note the use of color (in a manner similar to the way Peirce did it in manuscripts).

 Figure 5. Break Line of Identity on Recto

Now, to complete the proof, withdraw the new Ligature from the outer close of the second Biclosure; in this is done by Deiterating the inner chunk of Green LI back to the Green LI in the area enclosed by one Cut. The proof is formally finished at this point, but it may be brought to a more Peircean iconic state (emphasizing the continuity of identity) by making explicit visible connections between certain Hooks within Ligatures, as in the second diagram of this Figure.

 Figure 6. Withdraw Line of Identity end