On countably saturated linear orders

We will say that a linear order L is countably saturated if for any two countable subsets A,B of L, such that any element of A is less than any element of B, we can find an element of L between them. This obvious generalization of density corresponds to ”countable saturation” from model theory. We’ll say, that a countably saturated linear order L is prime, if every countably saturated linear order contains an isomorphic copy of L. I would like to present, in a rather detailed way, a proof that all prime countably saturated linear orders are isomorphic.