On Arutyunov theorem of coincidence point for two mapping in metric spaces

In the famous theorem of Arutyunov, it is asserted that the mappings $\psi,\varphi,$ acting from the complete metric space $(X, \rho_X)$ to the metric space $(Y, \rho_Y)$, one of which is $\alpha$-covering and the second is $\beta$-Lipschitz, $\alpha > \beta,$ have the coincidence point is the solution of the equation $\psi(x)=\varphi(x).$ We show that this assertion remains valid also in the case when the space $Y$ is not metric it is sufficient that the function $\rho_{Y}:Y^{2} \to \mathbb{R_{+}}$ satisfies only the axiom of identity. The function $\rho_{Y}$ may not be symmetric and does not correspond to the triangle inequality; moreover, it does not have to satisfy the $f$-triangle inequality (that is, it is possible that the space $Y$ is not even $f$-quasimetric).