I am stuck understanding the proof of existence of vector-valued Integration in S. Kesavan's book "Functional Analysis" . This older post has the image of proof and the version of Hahn-Banach theorem that author uses in proving the result. The problem I am having is that I don't understand how does the author infer $ \{m_1,m_2, \cdots m_k\} \in K $ from $ \{t_1,t_2, \cdots t_k\} \notin K , \{t_1,t_2, \cdots t_k\} \neq\{m_1,m_2, \cdots m_k\} $.
I don't understand how is he arriving at this conclusion from the non-equality of these 2 tuples. If the proof is incorrect please provide a correct proof or a reference for it.
The key to the conclusion is that $\ \big(t_1,t_2,\dots,t_k\big)\ $ is an arbitrary member of $\ K^c\ $, so the inequality $\ \big(m_1,m_2,\dots,m_k\big)\ne\big(t_1,t_2,\dots,t_k\big)\ $ holds not just for some particular $\ \big(t_1,t_2,\dots,t_k\big)\ $ in $\ K^c\ $, but for all $\ \big(t_1,t_2,\dots,t_k\big)\ $ in $\ K^c\ $. But if $\ \big(m_1,m_2,\dots,m_k\big)\ $ is not equal to any member of $\ K^c\ $ then it cannot be a member of that set, and so must lie in its complement $\ \big(K^c\big)^c=K\ $.