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Directory:Jon Awbrey/Papers/Futures Of Logical Graphs (view source)

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By way of gaining a minimal experience with how equational proofs look in the present forms of syntax, I will lay out the proofs of a few essential theorems in the primary algebra.
 
By way of gaining a minimal experience with how equational proofs look in the present forms of syntax, I will lay out the proofs of a few essential theorems in the primary algebra.
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The first theorem is known as the ''Double Negation Theorem'' (DNT).
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The first theorem goes under the names of ''Consequence&nbsp;1'' <math>(C_1)\!</math>, the ''double negation theorem'' (DNT), or ''Reflection''.
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<pre>
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{| align="center" cellpadding="10"
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o-----------------------------------------------------------o
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| [[Image:Logical_Graph_Figure_24.jpg|500px]]
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| C_1.` Double Negation Theorem ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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|}
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o-----------------------------------------------------------o
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| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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| ` ` ` ` ` ` ` ` ` a ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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| ` ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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| ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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| ` ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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| ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` a ` ` ` ` ` ` ` ` ` |
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| ` ` ` ` ` ` ` ` ` @ ` ` ` ` = ` ` ` ` @ ` ` ` ` ` ` ` ` ` |
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| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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o-----------------------------------------------------------o
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| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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| ` ` ` ` ` ` ` ` ((a)) ` ` ` = ` ` ` ` a ` ` ` ` ` ` ` ` ` |
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| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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o-----------------------------------------------------------o
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| ` ` ` ` ` ` ` Reflect <---- | ----> Reflect ` ` ` ` ` ` ` |
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o-----------------------------------------------------------o
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</pre>
     −
The proof that follows it is derived from the one that was given by George Spencer Brown in his book ''Laws of Form'', and credited to two of his students, John Dawes and D.A. Utting.  This result is annotated as ''Consequence&nbsp;1'' (''C''<sub>1</sub>) or as ''Reflection'' in LOF.
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The proof that follows is adapted from the one that was given by [[George Spencer Brown]] in his book ''Laws of Form'' (LOF) and credited to two of his students, John Dawes and D.A. Utting.
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<pre>
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{| align="center" cellpadding="10"
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o-----------------------------------------------------------o
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| [[Image:Logical_Graph_Figure_25.jpg|500px]]
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| C_1.` Double Negation Theorem.` Proof.` ` ` ` ` ` ` ` ` ` |
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|}
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o-----------------------------------------------------------o
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| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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{| align="center" cellpadding="10"
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| ` ` ` ` ` a o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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| [[Image:Logical_Graph_Figure_26.jpg|500px]]
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| ` ` ` ` ` ` `\` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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|}
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| ` ` ` ` ` ` ` \ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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| ` ` ` ` ` ` ` `o` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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| ` ` ` ` ` ` ` ` \ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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| ` ` ` ` ` ` ` ` `\` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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| ` ` ` ` ` ` ` ` ` @ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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o=============================< I2. Unfold "(())" >=========o
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| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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| ` ` ` ` ` a o ` ` ` ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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| ` ` ` ` ` ` `\` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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| ` ` ` ` ` ` ` \ ` ` ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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| ` ` ` ` ` ` ` `o` ` `o` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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| ` ` ` ` ` ` ` ` \ ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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| ` ` ` ` ` ` ` ` `\`/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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| ` ` ` ` ` ` ` ` ` @ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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o=============================< J1. Insert "(a)" >==========o
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| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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| ` ` ` ` ` ` ` ` ` ` ` ` `a o` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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| ` ` ` ` ` ` ` ` ` ` ` ` ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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| ` ` ` ` ` ` ` ` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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| ` ` ` ` ` a o ` a o ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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| ` ` ` ` ` ` `\` ` `\` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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| ` ` ` ` ` ` ` \ ` ` \ / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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| ` ` ` ` ` ` ` `o` ` `o` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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| ` ` ` ` ` ` ` ` \ ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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| ` ` ` ` ` ` ` ` `\`/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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| ` ` ` ` ` ` ` ` ` @ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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o=============================< J2. Distribute "((a))" >====o
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| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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| ` ` ` ` ` a o ` a o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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| ` ` ` ` ` ` `\` ` `\` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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| ` ` ` ` ` ` ` \ ` ` \ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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| ` ` ` ` ` ` ` `o` ` `o` `a o` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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| ` ` ` ` ` ` ` ` \ ` ` \ ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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| ` ` ` ` ` ` ` ` `\` ` `\`/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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| ` ` ` ` ` ` ` ` a o ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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| ` ` ` ` ` ` ` ` ` `\` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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| ` ` ` ` ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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| ` ` ` ` ` ` ` ` ` ` `o` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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| ` ` ` ` ` ` ` ` ` ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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| ` ` ` ` ` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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| ` ` ` ` ` ` ` ` ` @ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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o=============================< J1. Delete "(a)" >==========o
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| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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| ` ` ` ` ` a o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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| ` ` ` ` ` ` `\` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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| ` ` ` ` ` ` ` `o` ` `o` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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| ` ` ` ` ` ` ` ` a o ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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| ` ` ` ` ` ` ` ` ` ` `o` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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| ` ` ` ` ` ` ` ` ` @ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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o=============================< J1. Insert "a" >============o
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| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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| ` ` ` ` ` a o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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| ` ` ` ` ` ` `\` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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| ` ` ` ` ` ` ` \ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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| ` ` ` ` ` ` ` `o` ` `o a` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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| ` ` ` ` ` ` ` ` \ ` ` \ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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| ` ` ` ` ` ` ` ` `\` ` `\` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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| ` ` ` ` ` ` ` ` a o ` ` o a ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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| ` ` ` ` ` ` ` ` ` `\` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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| ` ` ` ` ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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| ` ` ` ` ` ` ` ` ` ` `o` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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| ` ` ` ` ` ` ` ` ` ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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| ` ` ` ` ` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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| ` ` ` ` ` ` ` ` ` @ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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o=============================< J2. Collect "a" >===========o
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| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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| ` ` ` ` ` a o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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| ` ` ` ` ` ` `\` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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| ` ` ` ` ` ` ` \ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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| ` ` ` ` ` ` ` `o` ` `o a` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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| ` ` ` ` ` ` ` ` ` o ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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| ` ` ` ` ` ` ` ` ` `\` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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| ` ` ` ` ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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| ` ` ` ` ` ` ` ` ` ` `o` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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| ` ` ` ` ` ` ` ` ` ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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| ` ` ` ` ` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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| ` ` ` ` ` ` ` ` a @ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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o=============================< J1. Delete "((a))" >========o
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| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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| ` ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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| ` ` ` ` ` ` ` ` ` `\` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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| ` ` ` ` ` ` ` ` ` ` \ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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| ` ` ` ` ` ` ` ` ` ` `o` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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| ` ` ` ` ` ` ` ` ` ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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| ` ` ` ` ` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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| ` ` ` ` ` ` ` ` a @ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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o=============================< I2. Refold "(())" >=========o
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| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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| ` ` ` ` ` ` ` ` ` a ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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| ` ` ` ` ` ` ` ` ` @ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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o=============================< QED >=======================o
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</pre>
      
==Weed and seed theorem==
 
==Weed and seed theorem==
Jon Awbrey
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