I am going through Bernard F. Schutz's book on "Geometrical Methods of Mathematical Physics". Currently I am in section 2.15 about "When is a basis a coordinate basis". I don't understand the logic in that section exactly. If we have two linear independent vector fields: $$\vec{A}=d/d\lambda$$ and $$\vec{B}=d/d\mu$$ Then one can compute the coordinates at a different point by acting on $x^i$ with the exponential of the scaled vector field: $$x^i(\alpha,\beta)=e^{\beta d/d\mu}e^{\alpha d/d\lambda}x^i$$ Then he shows that since the vector fields $\vec{A}=d/d\lambda$ and $\vec{B}=d/d\mu$ are assumed commute: $$\frac{d(x^i)}{\partial \alpha}=e^{\beta d/d\mu}e^{\alpha d/d\lambda}\frac{d(x^i)}{\partial \lambda}$$ $$\frac{d(x^i)}{\partial \beta}=e^{\beta d/d\mu}e^{\alpha d/d\lambda}\frac{d(x^i)}{\partial \mu}$$
However, the RHS of both equations are just the transported vector components of the vector fields $d/\lambda$ and $d/d\mu$ So he concludes that: $$\partial/\partial\alpha=d/d\lambda$$ and $$\partial/\partial\beta=d/d\mu$$ He then writes that this is a proof that the vector fields $\vec{A}=d/d\lambda$ and $\vec{B}=d/d\mu$ are therefore coordinate basis vectors. I don't understand this last assertion. What is the reasoning behind this?