We are given a continuous function $\xi : \mathbb{R} \rightarrow \mathbb{R}$ satisfying condition (V.1) (with $d \geqslant 2$), where $$\forall A \in \mathcal{M}_d(\mathbb{R}), \quad A \text{ invertible} \Rightarrow f_\xi(A) = \left(\xi(A_{i,j})\right)_{1\leqslant i,j\leqslant d} \text{ invertible} \tag{V.1}$$ Let $\eta = \xi^{-1} : I \rightarrow \mathbb{R}$ be the inverse function of the bijection $\xi : \mathbb{R} \rightarrow I$. Show that where it is defined $$(\eta(xy))^2 = \eta\left(x^2\right)\eta\left(y^2\right)$$
We are given a continuous function $\xi : \mathbb{R} \rightarrow \mathbb{R}$ satisfying condition (V.1) (with $d \geqslant 2$), where
$$\forall A \in \mathcal{M}_d(\mathbb{R}), \quad A \text{ invertible} \Rightarrow f_\xi(A) = \left(\xi(A_{i,j})\right)_{1\leqslant i,j\leqslant d} \text{ invertible} \tag{V.1}$$
Let $\eta = \xi^{-1} : I \rightarrow \mathbb{R}$ be the inverse function of the bijection $\xi : \mathbb{R} \rightarrow I$. Show that where it is defined
$$(\eta(xy))^2 = \eta\left(x^2\right)\eta\left(y^2\right)$$