Existence of Solution to System of First Order ODEs

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

\dfrac {\d y} {\d x} = \map f {x, y, z} & : \map y {x_0} = y_0 \\ & \\ \dfrac {\d z} {\d x} = \map g {x, y, z} & : \map z {x_0} = z_0 \\ \end{cases}$

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

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