proof that $\frac d {dy}\left(G(x,y)G(y,z)\right)=0 \implies G(x,y)=c_2\frac{H(x)}{H(y)}$?

Question

Jaynes in probability theory the logic of science says that

$$\frac d {dy}\left(G(x,y)G(y,z)\right)=0 \implies G(x,y)=c_2\frac{H(x)}{H(y)}$$

Why is this true? I can see the implication going the other way of course.