If (A,<1) and (B,<2) are ordered sets such that is (A,<1) is isomorphic to an initial segment of (B,<2) and (B,<2) is isomorphic to an end segment of (A,<1), then (A,<1) and (B,<2) are isomorphic.
I tried to use Cantor-Bernstein Theorem, but I do not think isomorphism between the ordered sets follows from isomorphic embedding in both sides. How can I prove it?