I wonder if there is a mistake in the proof of Theorem IV in section VIII.4.2 in Cassel's book on geometry of numbers. I refer to the claim in equation (12) on page 217 and its proof three equations later (see the highlighted section in the screenshot below).
Here $F$ is a norm on $\mathbb{R}^n$, $J$ is an integer between $1$ and $n$ and for $w\in \mathbb{R}^{n-J}$, $$ S_J(t,w)=\left\lbrace [x] \in \mathbb{R}^J/\mathbb{Z}^J : F(x,w) < t \right\rbrace. $$ $m_J$ denotes the Haar measure on $\mathbb{R}^J/\mathbb{Z}^J$ induced by the Lebesgue measure on $\mathbb{R}^J$.
Question 1: Does the highlighted portion above make sense? It seems to be implying that $$ F(y-y_0,sz) = F(y-y_0,z) $$ which I cannot justify.
Question 2: If there is some mistake here, how do I prove the claim in equation (12) of the screenshot above?
Hope you can help me with this difficulty.
I'm not sure what is going on in Cassel's book but when I looked at the book of Gruber-Lekkerkerker, the following argument was suggested. We let $\pi_J: \mathbb{R}^J \to \mathbb{R}^J/\mathbb{Z}^J$ be the projection. Recall that, for $t\in \mathbb{R}$ and $z \in \mathbb{R}^{n-J}$, $$ S_J(t,z) := \pi_J \{x \in \mathbb{R}^J: F(x,z)<t\}. $$ Assume $S_J(t,z) \neq \varnothing$ and choose $w \in \mathbb{R}^J$ so that we have \begin{equation}\label{0}\tag{1} 0 \in w +\{x\in \mathbb{R}^J: F(x,z) < t\}. \end{equation} Using this, that $s\geq 1$, and the fact that $w+\{x \in \mathbb{R}^J: F(x,z)<t\}$ is convex, we see that \begin{equation} \begin{split} m_J(S_J(st,sz)) &= m_J(\pi_J \{x \in \mathbb{R}^J: F(x,sz) < st\})\\ &= m_J(\pi_J (s\cdot\{y \in \mathbb{R}^J: F(y,z) < t\}))\\ &= m_J\left(\pi_J (s\cdot \{y \in \mathbb{R}^J: F(y,z) < t\} + \pi_J(s\cdot w)\right)\\ &= m_J\left(\pi_J (s\left( \{y \in \mathbb{R}^J: F(y,z) < t\} + w \right))\right) \\ &\geq m_J\left(\pi_J ( \{y \in \mathbb{R}^J: F(y,z) < t\} + w )\right)\ \text{ (Use the facts here)}\\ &=m_J(S_J(t,z) + \pi_J(w)) = m_J(S_J(t,z)). \end{split} \end{equation}