I am currently self-studying differential geometry from O'Neill's Semi-Riemannian Geometry With Applications to Relativity. In the proof of Lemma 6 (page 40 of my edition, first Lemma in the section regarding contractions in chapter 2), he uses that, if $\xi = (x^1,\dots,x^n),\ \eta = (y^1, \dots, y^n)$ are two overlapping coordinate systems on a manifold $M$, then
$$ \sum_m \frac{\partial y^m}{\partial x^j}\frac{\partial x^i}{\partial y^m} = \delta^i_j $$
Where $m$ ranges from 1 to the dimension of the manifold. I do understand that the notation is thought to be reminiscent of the case where we simply have derivatives in $\mathbb{R}^n$, in which case the result is trivial, but following the definitions given in said book about derivatives in a manifold, shouldn't this last sum be equal to
$$ \sum_m \left(\frac{\partial \left(y^m\circ\xi^{-1}\right)}{\partial u^j}\circ\xi\right)\cdot\left(\frac{\partial \left(x^i\circ\eta^{-1}\right)}{\partial u^m}\circ\eta\right) $$
Where (following the conventions of the book), $u^1,\dots, u^n$ are the natural coordinate functions of $\mathbb{R}^n$. If this is the case, I wonder how one should go about proving that this last sum is actually equal to $\delta^i_j$.