During the proof of the product rule, Jayne's used the lemma without proof that, if G(x,y)G(y,z) is independent of y, then G can be written as rH(x)/H(y). This is easy to believe, but quite an important step, so it's a shame he skipped it.

Below is a proof of this lemma (credit goes to a friend; I found a similar but more cumbersome proof):

We know G(x,y)G(y,z)=f(x,z) for some function f. Setting z=1 gives G(x,y)=f(x,1)/G(y,1), while setting y=z=1 gives f(x,0)=G(x,1)G(1,1). Substituting the second into the first gives, G(x,y)=G(1,1)G(x,1)/G(y,1), which has the desired form.