Existence of Solution to System of First Order ODEs

Theorem
Consider the system of initial value problems:
 * $$\begin{cases}

\dfrac {dy}{dx} = f \left({x, y, z}\right) & : y \left({x_0}\right) = y_0 \\ & \\ \dfrac {dz}{dx} = g \left({x, y, z}\right) & : z \left({x_0}\right) = z_0 \\ \end{cases}$$

where $$f \left({x, y, z}\right)$$ and $$g \left({x, y, z}\right)$$ are continuous real functions in some region of space $$xyz$$ that contains the point $$\left({x_0, y_0, z_0}\right)$$.

Then this system of equations has a unique solution which exists on some interval $$\left|{x - x_0}\right| \le h$$.