Uniqueness of solution for linear first-order partial differential equations

Question

In Polyanin's Handbook of First Order Partial Differential Equations (2002), in Section 10.1.2, it is stated that the non-homogeneous linear, first-order partial differential equation: $$\sum_{i=1}^nf_i(x_1,\dots ,x_n)\frac{\partial w}{\partial x_i}=g(x_1,\dots ,x_n)w+h(x_1,\dots ,x_n)$$ with initial condition $$w (0,x_2,\dots ,x_n)=\psi (x_2,\dots ,x_n)$$ where $\psi (x_2,\dots ,x_n)$ is a given function, has unique, continuously differentiable solution in domain $G=\{0<x_1<a,-\infty<x_i<\infty,i=2,\dots,n\}$, if $f_1=1$, $f_2,\dots f_n,g,h,\psi$ are continuously differentiable functions of $x_1,\dots,x_n$ in $G$ and the following inequality holds in $G$: $$\sqrt{f_2^2(x_1,\dots ,x_n)+\dots+f^2_n(x_1,\dots ,x_n)}\leq k\Big(1+\sqrt{x_2^2+\dots+x_n^2}\Big),k=const.$$ However, I couldn't find a proof for this uniqueness theorem. Can anyone give a hint?