Inverse of a Lipschitz function

Question

If it is given that $\hat{\rho}(s)=\tilde{\rho}(s)s$. How do I prove that $s=(\tilde{\rho} \circ \hat{\rho}^{-1}(s))\hat{\rho}^{-1}(s)$ where $s \in \mathbb{R}$ and $\tilde{\rho} : \mathbb{R} \to \mathbb{R}$? All I can understand is that if $\hat{\rho}=(\tilde{\rho} \circ \hat{\rho}^{-1}(s))$, then the equality will be satisfied. But I don't know how to prove this.