Composite of Orderings is not necessarily Ordering

Theorem
Let $A$ be a set.

Let $\RR$ and $\SS$ be orderings on $A$.

Then their composite $\RR \circ \SS$ is not necessarily also an ordering on $A$.

Proof
Let $\RR$ and $\SS$ be orderings as asserted.

Both $\RR$ and $\SS$ are both antisymmetric and transitive.

But we have:


 * From Composite of Antisymmetric Relations is not necessarily Antisymmetric, it is not necessarily the case that $\RR \circ \SS$ is itself antisymmetric.


 * From Composite of Transitive Relations is not necessarily Transitive, it is not necessarily the case that $\RR \circ \SS$ is itself transitive.

Hence it is not necessarily the case that $\RR \circ \SS$ is an ordering.