Let $($A$,$ R$)$ and $($B$,$ S$)$ be linear orderings. Define a relation L on the set A $\backslash$times B by

$($a$1,$ b$1)$L$($a$2,$ b$2)($a$1<$ a$2)$ or $($a$1=$ a$2$ and b$1<$ b$2).$

Show that $($A $\backslash$times B$,$ L$)$ is a linear order. This is sometimes called lexicographic ordering $($or dictionary ordering$).$