Theorem 1 in page 4 of the book Numerical Solution of Partial Differential Equations in Science and Engineering by L. Lapidus:
The general solution of the quasilinear PDE $$a(x,y,u)u_x + b(x, y,u)u_y = c(x, y,u),$$ is given by $$G(v,w)=0,$$ where $G$ is an arbitrary function and where $v(x, y, u) = c_1$, and $w(x,y,u)=c_2$ form a solution of the equations $$\frac{dx}{a}=\frac{dy}{b}=\frac{du}{c}.$$
My question is how can I prove this theorem?
In that book, it explains how can interpret quasilinear PDEs geometrically, but I can't understand how to obtain $\frac{dx}{a}=\frac{dy}{b}=\frac{du}{c}$.
Thanks in advance for any explanation or clarification about this theorem.