I'm trying to understand the relationship between Graver and Grobner basis, in particular how Graver basis can be computed via Grobner basis via Lawrence lifting. The key result appear to be Theorem 7.1 of "B. Sturmfels, Grobner bases and convex polytopes, University Lecture Series", however, I'm having trouble understanding the proof of this theorem. The theorem statement and part of the proof are as attached:
where I have highlighted the key parts which I don't understand and it has been defined that \begin{equation*} \Lambda(A) = \begin{pmatrix} A & 0\\ 1 & 1 \end{pmatrix} \end{equation*} Could someone explain to me how to see that $Gr_{\Lambda(A)}$ is a Grobner basis of $I_{\Lambda(A)}$?