I'm reading "Differential Topology" by Guillemin, V and Pollack, A. While reading the chapter about transversality, I got through this theorem :https://i.stack.imgur.com/5wYBo.jpg (I'm not allowed to put pictures yet).
After the proof he says that this theorem easily implies that transverse maps are general when the target manifold ($Y$) in the euclidean space $\mathbb{R}^m$. If you take $S$ to be an open ball, $F(x,s)=f(x)+s$, and you fix $x\in X$ you get that $F$ is a traslation of the ball, and therefore a submersion. Everything ok until now. But then he says "So, of course, $F$ is a submersion of $X\times S$ and therefore transversal to any submanifold $Z\subset\mathbb{R}^m$". Can you help me with that step? Thank you very much.