I don't understand when you say maps in are injective and preserve order...

If then an element of is any function from to I'll do as if we just have maybe there is something I misunderstood

As you said, since is well-ordered we can see an element of B as a sequence were for all in(from now on, means )

You want to order using the orders of and Intuitively, you can think something like, for any and in if then

But of course, could be greater than in some points, greater than in others, and that definition of an order of wouldn't be valid.

So let's use the fact that is a well order, and define:

This means that, if you take two map in the (strictly) greatest is the "first" (for the well order of ) to be strictly greater than the other.

Finally define , for any

Prove that this really is a linear order of