Show that if (P,<) and (Q,$\displaystyle \prec$) are isomorphic strictly ordered sets and < is a linear ordering, then $\displaystyle \prec$ is a linear ordering.
we need to show that if $\displaystyle a,b\in Q, a \neq b$ then $\displaystyle a\prec b $ or $\displaystyle a\succ b$.
Let $\displaystyle \varphi : P\to Q$ be an order isomorphism.
As $\displaystyle a\neq b$ then $\displaystyle \varphi ^{-1}(a)\neq \varphi ^{-1}(b)$, so $\displaystyle \varphi ^{-1}(a)< \varphi ^{-1}(b)$ or $\displaystyle \varphi ^{-1}(a)> \varphi ^{-1}(b)$ ($\displaystyle (P,<)$ is linear).
Therefore $\displaystyle a=\varphi \varphi^{-1}(a)\prec \varphi \varphi^{-1}(b)=b$ or $\displaystyle a=\varphi \varphi^{-1}(a)\succ \varphi \varphi^{-1}(b)=b$, hence $\displaystyle (Q,\prec )$ is linear.