I am having trouble comprehending the proof for Theorem 11.1.1 (p.557) in Matrix Computations by Golub and Van Loan. The theorem is that after "reducing" a square matrix $A$ into Jordan canonical form the function $f$ acting on the Jordan block $J_i$ is given by: $$f(J_i)=\begin{pmatrix} f(\lambda_i) & f^{1}(\lambda_i) & ... & \frac{f^{(m_i-1)}(\lambda_i)}{(m_i-1)!}\\ 0 & f(\lambda_i)&...&...\\ ...&...&...&f^{(1)}(\lambda_i)\\ 0 $ ... & ... & ...& f(\lambda_i) \end{pmatrix}$$
This is a result that I've seen to be given as a definition in Functions of Matrices by Higham, and the proof in Golub and Van Loan introduces matrix $G$ such that $$G=\lambda I+E$$ where $$E=(\delta_{i,j-1})$$ which I can only assume is a different version of the Kronocker delta. I am leaving a screenshot of the proof in the book here. If anybody could help me understand this proof or give another one I would be grateful.
Thanks in advance.