Strict Ordering can be Expanded to Compare Additional Pair

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

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