Strict Ordering can be Expanded to Compare Additional Pair

Theorem
Let $\struct {S, \prec}$ be an ordered set.

Let $a$ and $b$ be distinct, $\prec$-incomparable elements of $S$.

That is, let:
 * $a \not \prec b$ and $b \not \prec a$.

Let ${\prec'} = {\prec} \cup \set {\tuple {a, b} }$.

Define a relation $\prec'^+$ by letting $p \prec'^+ q$ :
 * $p \prec q$

or:
 * $p \preceq a$ and $b \preceq q$

where $\preceq$ is the reflexive closure of $\prec$.

Then:


 * $\prec'^+$ is a strict ordering
 * $\prec^+$ is the transitive closure of $\prec'$.