A
document is drafted by the man which proclaims that a woman is hereby
betrothed to him. The Kiddushin is effected when the man gives the
document to the woman.
Why? Because “And she will leave the first husband's house, and she will go and be a wife unto another man.” Since “leave” can be effected through a document, so too “be” can be effected through a document.
But
if so, can divorce be effected with money? - No, because a defender
cannot become an accuser. And while the documents would be different
and distinct, money would be the same.