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===Exemplary proofs===

===Exemplary proofs===

With the meagre utilities of the axioms and theorems set down so far, it is already possible to prove a multitude of much more complex theorems.  A couple of all-time favorites are given next.

Using no more than the axioms and theorems recorded so far, it is already possible to prove a multitude of much more complex theorems.  A couple of all-time favorites are given next.

====Peirce's law====

====Peirce's law====

This section presents a proof of Peirce's law, commonly written:

This section presents a proof of Peirce's law, commonly written:

: <math>((p \Rightarrow q) \Rightarrow p) \Rightarrow p</math>

<p><math>((p \Rightarrow q) \Rightarrow p) \Rightarrow p</math></p>

</center>

The first order of business is present the statement as it  appears in the so-called ''existential interpretation'' of Peirce's own ''logical graphs''.  Here is the statement of Peirce's law, as rendered under the existential interpretation into (the topological dual forms of) Peirce's logical graphs:

Figure&nbsp;31 shows the representation of Peirce's law as a tree-form logical graph, as it appears under the existential interpretation.

{| align="center" border="0" cellpadding="10" cellspacing="0"

| [[Image:Logical_Graph_Figure_31.jpg|500px]] || (31)

| Peirce's Law                                              |

|}

|          @                =                   @        |

|                                                           |

|                                                           |

|     (((p (q)) (p)) (p)))  =                            |

|                                                           |

Finally, here's the promised proof of Peirce's law:

Finally, here is the promised proof of Peirce's law:

<pre>

<pre>