Jacobians in Tensor Notation

Question

I am teaching myself tensor calculus from Pavel Grinfields book, see this previous question for all the background: 1. I am stuck about half a page further down. In exercise 47 we are asked to prove that $$J^{i\prime}_{i}J^{i}_{j\prime} = \delta^{i\prime}_{j\prime}$$ by multiplication of $$J^{i}_{i\prime}J^{i\prime}_{j} = \delta^{i}_{j}$$ by $J^{j}_{j\prime}$. By definition we have that this is the same as $$\frac{\partial Z^i(Z^\prime)}{\partial Z^{i\prime}}\frac{\partial Z^{i\prime}(Z)}{\partial Z^{j}} = \frac{\partial Z^i}{\partial Z^j}$$ and a naive interpretation of multiplication by $J^{j}_{j\prime}$ from the right yields $$\frac{\partial Z^i(Z^\prime)}{\partial Z^{i\prime}}\frac{\partial Z^{i\prime}(Z)}{\partial Z^{j}} \frac{\partial Z^j(Z^\prime)}{\partial Z^{j\prime}} = \frac{\partial Z^i}{\partial Z^j}\frac{\partial Z^j(Z^\prime)}{\partial Z^{j\prime}}$$ but this does not seem right and I just get more confused the more I think about it. Can someone please point me in the right direction?