I am reading through this book: st.openlogicproject.org.
Currently stuck on Lemma 13.6. on page 178.
I want to apologize for using an image instead of typing it down, for some users can't read it then. I'll try to give the details soon if there is any need, and hope that access through the link is sufficient for you now.
For the proof of the first statement it seems as if $0 \times \{1\} = \{0\}$. Or why is that equal, then? (Zero is the empty set, I suppose.)
Similarly, in the second statement, I can follow only the first line.
For the last case, the limit case, my understanding is also totally unclear. At the moment only very few aspects of it make a little bit of sense to me.
As always, also short hints and suggestions are appreciated.. Thank you for reading!
:)
EDIT: Following definitions are applied:
Hint: Isomorphic well-orderings have the same order type. So what you have to show in the base case is that the well-ordering $(\alpha \times \{0\} \cup 0 \times \{ 1\}, <)$ is isomorphic to the well ordering $(\alpha \times \{0\} \cup \{0\}, <)$. Analogously for the successor case.