12. Show that, if $0 \leqslant j \leqslant n - 2$, for all $0 \leqslant i \leqslant n - j - 1$, the polynomial $p _ { i }$ is orthogonal to $\mathbb { R } _ { i - 1 } [ X ]$ for the inner product $\langle \cdot , \cdot \rangle _ { j + 1 }$.
12. Show that, if $0 \leqslant j \leqslant n - 2$, for all $0 \leqslant i \leqslant n - j - 1$, the polynomial $p _ { i }$ is orthogonal to $\mathbb { R } _ { i - 1 } [ X ]$ for the inner product $\langle \cdot , \cdot \rangle _ { j + 1 }$.\\