Let $g$ be a simple Lie algebra and $\mathfrak{h}$ a Cartan subalgebra of $g$. Let $W$ be the Weyl group of $g$. Then $W$ is generated by simple reflections $s_1, \ldots, s_n$, $n$ is the rank of $g$. Let $\mathfrak{h}^*$ be the dual of $\mathfrak{h}$. Let $\alpha_1, \ldots, \alpha_n$ be the simple roots and $\alpha_1^{\vee}, \ldots, \alpha_n^{\vee}$ the simple coroots. We have actions: \begin{align} s_i(h)=h-\alpha_i(h)\alpha_i^{\vee}, \quad h \in \mathfrak{h}, \\ s_i(\lambda) = \lambda - \lambda(\alpha_i^{\vee})\alpha_i, \quad \lambda \in \mathfrak{h}^*. \end{align}
How to prove that $w\lambda(h)=\lambda(w^{-1}(h))$ using the formula of the above actions? Thank you very much.