Let $n$ be an integer such that $n \geqslant 2$. We denote by $E' = \operatorname{Vect}(e_{1}, \ldots, e_{n-1})$ and $\pi$ the orthogonal projection onto $E'$ $$\pi : \left\lvert \, \begin{aligned} E & \rightarrow E' \\ \sum_{i=1}^{n} x_{i} e_{i} & \mapsto \sum_{i=1}^{n-1} x_{i} e_{i} \end{aligned} \right.$$ We set $X' = \pi \circ X = \sum_{i=1}^{n-1} \varepsilon_{i} e_{i}$. For $t$ in $\{-1, 1\}$ we denote $H_{t}$ the affine hyperplane $E' + te_{n}$ and $C_{t} = \pi(C \cap H_{t})$. Show, for $x' \in E'$ and $t \in \{-1, 1\}$, that $x' \in C_{t} \Longleftrightarrow x' + te_{n} \in C$.
Let $n$ be an integer such that $n \geqslant 2$. We denote by $E' = \operatorname{Vect}(e_{1}, \ldots, e_{n-1})$ and $\pi$ the orthogonal projection onto $E'$
$$\pi : \left\lvert \, \begin{aligned} E & \rightarrow E' \\ \sum_{i=1}^{n} x_{i} e_{i} & \mapsto \sum_{i=1}^{n-1} x_{i} e_{i} \end{aligned} \right.$$
We set $X' = \pi \circ X = \sum_{i=1}^{n-1} \varepsilon_{i} e_{i}$. For $t$ in $\{-1, 1\}$ we denote $H_{t}$ the affine hyperplane $E' + te_{n}$ and $C_{t} = \pi(C \cap H_{t})$.
Show, for $x' \in E'$ and $t \in \{-1, 1\}$, that $x' \in C_{t} \Longleftrightarrow x' + te_{n} \in C$.