Authorship;first,second,co I am finishing my first paper. I am unaware of how I should treat a certain situation, and have come  here for advice.

The paper is about 30 pages long. I have written 26 of those pages. On the pages I wrote, the other person helped with how thoughts are to be expressed, but all the theorems and proofs were done by me. The ideas of the paper are mine, and their execution.

How is the authorship to be divided in such an example? Should I be the first author or the corresponding author? Both? Should he be the second author, or the co-author?

I read this answer here, saying in math and computer science all authors are considered to have contributed equally. How should the ranking be notated than?

EDIT:

Question in partially answered in the linking answer (duplicate), at the comments bellow.
Simply, you should be the first author. 

A corresponding author will be responsible for replying to the queries come from the research community and help them to a better understanding as well as gain more citations. There are other things can you find them in the link below.

Finally, If the other author can do this task, make him the corresponding author and you can help him when needs.

This post talk about The difference between Corresponding author and First author.