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I have an smooth structure on $M$ given by the atlas $\{(U_\alpha,\varphi_\alpha)\}$

And on another manifold $N$ we define $\{(V_\alpha,\mu_\alpha)\}$ with $V_\alpha = \pi(U_\alpha)$ and $\mu_\alpha = \pi_2\circ\varphi_\alpha\circ \sigma$

where $\pi:M\rightarrow N$ is a smooth submersion, $\pi_2:\mathbb{R}^{n+k}\rightarrow \mathbb{R}^n$ the projection on the $n$ last coordinates, and $\sigma:N\rightarrow M$ a homeomorphism.

What I want to show is that $\{(V_\alpha,\mu_\alpha)\}$ is indeed a smooth structure. In the proof https://www.mathi.uni-heidelberg.de/~lee/StephanSS16.pdf, I don't understand the reasoning so decided to make my own version and would like to know if there are any mistakes in it.

So here's what I do:

I suppose $V_\alpha\cap V_\beta \neq \varnothing$ and so need to show that $\mu_\alpha\circ\mu_\beta^{-1}:\mu_\beta(V_\alpha\cap V_\beta)\rightarrow \mu_\alpha(V_\alpha\cap V_\beta)$ is smooth.

$$\mu_\alpha\circ\mu_\beta^{-1} = \pi_2\circ\varphi_\alpha\circ\sigma\circ\sigma^{-1}\circ\varphi_\beta^{-1}\circ\pi_2^{-1} = \pi_2\circ\varphi_\alpha\circ\varphi_\beta^{-1}\circ\pi_2^{-1}$$

but $\varphi_\alpha\circ\varphi_\beta^{-1}$ is smooth and $\pi_2$ and $\pi_2^{-1}$ are also smooth so the whole thing is smooth.

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