From Exercise 4.4.9 of How To Prove It:
Suppose $R$ is a partial order on $A$ and $S$ is a partial order on $B$. Define a relation $L$ on $A \times B$ as follows: $L = \{((a, b), (a', b')) \in (A \times B) \times (A \times B)) | a R a', {\rm and\ if\ } a = a' {\rm then\ } b S b'\}$. Show that $L$ is a partial order on $A \times B$. If both $R$ and $S$ are total orders, will $L$ also be a total order?
I managed to show that $L$ was a partial order. For the second part I can show that $L$ is a total order if $R$ and $S$ are total orders by showing that for arbitrary $(a, b), (a', b') \in A \times B$, if $((a, b), (a', b')) \notin L$ then $((a', b'), (a, b)) \in L$. Is this correct way to do it and is it true that $L$ is a total order?