+Moving away from considering order, note that creating Definition
+\autoref{defi:similar} imposed order on our universe at the cost of some
+uniqueness. Let's examine that loss of uniqueness a bit further.
+
+The twenty numbers from generations 0-2 break down into seven similarity-based
+equivalence classes, as shown below.
+
+$$\surreal{}{-1} \similar \surreal{}{-1,0} \similar \surreal{}{-1,1} \similar \surreal{}{-1,0,1}$$
+$$\surreal{-1}{0} \similar \surreal{-1}{0,1} \similar \surreal{-1}{}$$
+$$\surreal{}{0} \similar \surreal{}{0,1}$$
+$$\surreal{}{} \similar \surreal{-1}{1}$$
+$$\surreal{0}{} \similar \surreal{-1,0}{} \similar \surreal{}{1}$$
+$$\surreal{0}{1} \similar \surreal{-1,0}{1}$$
+$$\surreal{1}{} \similar \surreal{0,1}{} \similar \surreal{-1,1}{} \similar \surreal{-1,0,1}{}$$
+
+Thus, whereas Knuth's universe has 20 numbers by generation-2, our universe
+only has seven numbers. For convenience, we introduce the following two
+definitions (and theorems).
+
+% ==================================================================================================
+% ==================================================================================================
+% ==================================================================================================
+
+\begin{theorem} \label{thm:reduced-form}
+Given an arbitrary number $x$, there exists a number $x' =
+\surreal{X'_L}{X'_R}$ which is similar to $x$ and has the property that the
+cardinality of both $X'_L$ and $X'_R$ are one or zero.
+\end{theorem}
+
+\begin{proof}
+TODO
+% Since a finite, total ordered set contains a maximum and minimum element, it
+% follows that we can define $X'_L$ as containing only the maximum element of
+% $X_L$ and define $X'_R$ as containing only the minimum element of $X_R$. If
+% either $X_L$ or $X_R$ are the empty set, the corresponding $X'_L$ or $X'_R$ is
+% also the empty set.
+%
+% Then, to show that $x \similar x'$, we apply Axiom \autoref{ax:leq-relation} in
+% both directions.
+%
+% In the forward direction, $x \leq x'$ means that, $\forall x_L \in X_L$, it
+% must hold that $x_L \leq x'$. Applying \autoref{thm:strong-self-relation}, we
+% conclude that $x_L \leq x'_L < x'$. (this isn't quite right. reconsider my
+% approach)
+%
+% ...
+%
+% By Axiom \autoref{ax:leq-relation} and \autoref{thm:weak-self-relation}, for
+% $x'_l \in X'_L$,
+%
+% ==================
+%
+% The current ten equivalence classes were created by noting that every number
+% with a left or right set containing more than one element is similar to another
+% number whose sets do NOT contain more than one element. Since similarity is
+% built atop the LEQ relation from Axiom \autoref{ax:leq-relation} which itself
+% is based upon element-wise comparisons, we see that the only elements of
+% importance to these relations are the largest element of the left set and the
+% smallest element of the right set. Of course, for such a statement to be
+% sensical, we must first prove that Axiom \autoref{ax:leq-relation} imposes a
+% total ordering, otherwise there might not be a largest or smallest element in a
+% set.
+%
+% From this we see that, since Axiom \autoref{ax:leq-relation} makes its
+% comparison element-wise, every surreal number generated by our current, finite
+% methods must be similar to a surreal number containing one or zero elements in
+% its left and right sets since only the largest member of the left set and
+% smallest member of the right set are important. This motivates the following
+% definition.
+\end{proof}