Detailed proof of Theorem 2.1 in Papadimitrou book (Multitape TM to SingleTape TM)

I want to know if anybody knows a detailed proof of Theorem 2.1 of Papadimitrou's book Computational Complexity. The theorem states "Given any $$k$$-string Turing machine $$M$$ operating within time $$O(f(n))$$, we can construct a Turing Machine $$M'$$ operating within time $$O(f(n)^2)$$ and such that, for any input x, $$M(x)=M'(x)$$."

When asking for a detailed proof, I am asking for a complete description of $$M'$$. For instance, in the proof of the book, there are statements (paragraph 4), like: "$$M'$$ must contain new states, each of which corresponds to a particular combination of a state of $$M$$ and of a $$k$$-tuple of symbols of $$M$$.", which are not detailed.

Do not misunderstand me, I do understand the proof, so I am not asking for an explanation. I want to know if anybody (maybe an old paper) make the effort to write a complete configuration for the new machine.