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This can be written inline as <math>{}^{\backprime\backprime} \texttt{(~(~)~)} = \quad {}^{\prime\prime}\!</math> or set off in a text display as follows:

This can be written inline as <math>{}^{\backprime\backprime} \texttt{( ( ) )} = \quad {}^{\prime\prime}\!</math> or set off in a text display as follows:

{| align="center" cellpadding="10"

{| align="center" cellpadding="10"

| width="33%" | <math>\texttt{(~(~)~)}\!</math>

| width="33%" | <math>\texttt{( ( ) )}\!</math>

| width="34%" | <math>=\!</math>

| width="34%" | <math>=\!</math>

| width="33%" | &nbsp;

| width="33%" | &nbsp;

This ritual is called ''traversing'' the tree, and the string read off is called the ''traversal string'' of the tree.  The reverse ritual, that passes from the string to the tree, is called ''parsing'' the string, and the tree constructed is called the ''parse graph'' of the string.  The speakers thereof tend to be a bit loose in this language, often using ''parse string'' to mean the string that gets parsed into the associated graph.

This ritual is called ''traversing'' the tree, and the string read off is called the ''traversal string'' of the tree.  The reverse ritual, that passes from the string to the tree, is called ''parsing'' the string, and the tree constructed is called the ''parse graph'' of the string.  The speakers thereof tend to be a bit loose in this language, often using ''parse string'' to mean the string that gets parsed into the associated graph.

We have treated in some detail various forms of the initial equation or logical axiom that is formulated in string form as <math>{}^{\backprime\backprime} \texttt{(~(~)~)} = \quad {}^{\prime\prime}.~\!</math>  For the sake of comparison, let's record the plane-embedded and topological dual forms of the axiom that is formulated in string form as <math>{}^{\backprime\backprime} \texttt{(~)(~)} = \texttt{(~)} {}^{\prime\prime}.\!</math>

We have treated in some detail various forms of the initial equation or logical axiom that is formulated in string form as <math>{}^{\backprime\backprime} \texttt{( ( ) )} = \quad {}^{\prime\prime}.~\!</math>  For the sake of comparison, let's record the plane-embedded and topological dual forms of the axiom that is formulated in string form as <math>{}^{\backprime\backprime} \texttt{( )( )} = \texttt{( )} {}^{\prime\prime}.\!</math>

First the plane-embedded maps:

First the plane-embedded maps:

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The choice of axioms for any formal system is to some degree a matter of aesthetics, as it is commonly the case that many different selections of formal rules will serve as axioms to derive all the rest as theorems.  As it happens, the example of an algebraic law that we noticed first, <math>a(~) = (~),\!</math> as simple as it appears, proves to be provable as a theorem on the grounds of the foregoing axioms.

The choice of axioms for any formal system is to some degree a matter of aesthetics, as it is commonly the case that many different selections of formal rules will serve as axioms to derive all the rest as theorems.  As it happens, the example of an algebraic law that we noticed first, <math>a \texttt{( )} = \texttt{( )},~\!</math> as simple as it appears, proves to be provable as a theorem on the grounds of the foregoing axioms.

We might also notice at this point a subtle difference between the primary arithmetic and the primary algebra with respect to the grounds of justification that we have naturally if tacitly adopted for their respective sets of axioms.

We might also notice at this point a subtle difference between the primary arithmetic and the primary algebra with respect to the grounds of justification that we have naturally if tacitly adopted for their respective sets of axioms.

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All of the axioms in this set have the form of equations.  This means that all of the inference steps that they allow are reversible.  The proof annotation scheme employed below makes use of a double bar <math>\overline{\underline{13:56, 6 November 2015 (UTC)~}}\!</math> to mark this fact, although it will often be left to the reader to decide which of the two possible directions is the one required for applying the indicated axiom.

All of the axioms in this set have the form of equations.  This means that all of the inference steps that they allow are reversible.  The proof annotation scheme employed below makes use of a double bar <math>=\!=\!=\!=\!=\!=</math> to mark this fact, although it will often be left to the reader to decide which of the two possible directions is the one required for applying the indicated axiom.

===Frequently used theorems===

===Frequently used theorems===