A couple of remarks. For each claim it should be clear whether you are assuming it, trying to prove it or something else. Your proof is just a series of statements, and the role of those statements is not clear. Write comments when necessary, such as "We assume ...," "We need to prove ...," "It follows from ... that ..."

Mathematics is case-sensitive, so K and k may denote different variables.

Edit: Don't forget the base case.