Ordering can be Expanded to compare Additional Pair

Theorem
Let $(S, \preceq)$ be an ordered set.

Let $a$ and $b$ be non-comparable elements of $S$. That is, let:
 * $a \not\preceq b$ and $b \not\preceq a$.

Let ${\preceq'} = {\preceq} \cup \left\{ {\left({a,b}\right)} \right\}$.

Let $\preceq'^+$ be the transitive closure of $\preceq'$.

Then:

$\preceq'^+$ is an ordering.

$\preceq'^+$ can be defined by letting $p \preceq'^+ q$ iff:
 * $p \preceq q$ or
 * $p \preceq a$ and $b \preceq q$.