Ordering can be Expanded to compare Additional Pair

Theorem
Let $\struct {S, \preceq}$ 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 ${\preceq'} = {\preceq} \cup \set {\tuple {a, b} }$.

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

Then:


 * $\preceq'^+$ is an ordering.

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