Understanding proof detail (Gauss theorem form small geodesic triangles)

Question

I am reading the following proof:

But I don’t understand how Proposition 4.13 implies the highlighted part .

I know that since $E=1 $ in geodesic coordinates then $E_v=0$ but I can’t see how did $G_u=G_r$ becomes $(\sqrt{G})_r$ .

I think it must be a simple detail I am missing. Thanks in advance