Let $A, B$ be well ordered sets. If $A$ is order isomorphic to a subset of $B$, and $B$ is order isomorphic to a subset of $A$, prove that $A, B$ are order isomorphic.

I know that two well ordered set is order isomorphic iff there exists a increasing function(i.e. $x<\_A y \Rightarrow f(x)<\_B f(y)$) from $A$ to $B$ which is bijective. The first thing I thought was the well ordering isomorphic theorem(if $(A, \le\_A), (B, \le\_B )$ are well orderd, only one of $A \cong B, A \cong B[b] \ \exists!b \in B, \ A[a] \cong B \ \exists!a \in A$) must hold, and the isomorphism is unique.)

Where do I have to start the proof?