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 $E'$. We set $X' = \pi \circ X = \sum_{i=1}^{n-1} \varepsilon_{i} e_{i}$. For $t$ in $\{-1, 1\}$ we denote $C_{t} = \pi(C \cap H_{t})$ where $H_t = E' + te_n$. For $t$ in $\{-1, 1\}$, we denote by $Y_{t}$ the projection of $X'$ onto the non-empty closed convex set $C_{t}$. Show that $$\mathbb{P}(X \in C) = \frac{1}{2}\mathbb{P}(X' \in C_{+1}) + \frac{1}{2}\mathbb{P}(X' \in C_{-1})$$
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 $E'$. We set $X' = \pi \circ X = \sum_{i=1}^{n-1} \varepsilon_{i} e_{i}$. For $t$ in $\{-1, 1\}$ we denote $C_{t} = \pi(C \cap H_{t})$ where $H_t = E' + te_n$. For $t$ in $\{-1, 1\}$, we denote by $Y_{t}$ the projection of $X'$ onto the non-empty closed convex set $C_{t}$.
Show that
$$\mathbb{P}(X \in C) = \frac{1}{2}\mathbb{P}(X' \in C_{+1}) + \frac{1}{2}\mathbb{P}(X' \in C_{-1})$$