The Mishnah had taught: ...by giving a document. What is the source for this method of Kidushin?

The pasuk states (Devarim 24,1-2), "...and he wrote for her a document of divorce and put it in her hand and sent her out from his house. And she left his house and she went and married another man'' - the Torah compares marriage to divorce, to allow the use of a document to marry.

R. Avin/Chizkiyah's Braisa: (Devarim 22, 13) "When (a man) will acquire (a woman)'' - (this teaches that) she is acquired through money. (This opinion disagrees with R. Yochanan 4-2) (j) - who quoted "and she shall leave for free'' as the source.)

Question: Why is a pasuk needed - we could learn it through a Kal Vechomer - if a Jewish maidservant who cannot be acquired through relations, can be acquired with money; a woman who can be acquired through relations should certainly be acquired with money?!

Answer: Yevamah proves it, as she can be acquired through relations but not with money; so too a regular wife, even though she can be acquired through relations, she cannot be acquired through money...therefore, we need "uboaloh - and he will have relations with her'' to teach that she is acquired with relations.

Question: Surely we have a Kal Vechomer to prove this law - why use a pasuk? If a yevamah who cannot be acquired with money, can be acquired through relations; a woman who can be acquired with money should certainly be acquired though relations?!

Answer: The Jewish maidservant disproves this, as she is acquired through money but not relations; so too a regular wife, even though she can be acquired with money, she cannot be acquired through relations...therefore, we need "ki yikach - when a man will acquire'' to teach that she is acquired with money and "uboaloh - and he will have relations with her'' to teach that she is acquired through relations.