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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===