  We generalize Picard-Lindelof theorem/ the method of characteristics to the
following system of PDE: $C_{il}(x,y) {\partial y_i / \partial x_l} + {\partial
y_i / \partial x_m} = D_i(x,y)$. With a Lipschitz or $C^r$ $C_{il},D_i: [-a,
a]^{m} \times [-b, b]^{n} \rightarrow \mathbb{R}$ and initial condition $I_i:
[-\bar{a}, \bar{a}]^{m-1} \rightarrow (-b,b)$, $\bar{a} \leq a$, we obtain a
local unique Lipschitz or $C^r$ solution $f$, respectively that satisfies the
initial condition, $f_i (v, 0 ) = I_i(v)$, $v \in [-\bar{a}, \bar{a}]^{m-1}$.
To construct the solution we set bounds on the value of the solution by
discretizing the domain of the solution along the direction perpendicular to
the initial condition hyperplane. As the number of discretization hyperplanes
is taken to infinity the upper and lower bounds of the solution approach each
other, hence this gives a unique function for the solution ($Ufs$). A locality
condition is derived based on the constants of the problem. The dependence of
$C_{il}$, $D_i$ and $I_i$ on parameters, the generalization to nonlinear
systems of PDE and the application to hyperbolic quasilinear systems of first
order PDE in two independent variables is discussed.
