  As a generalization of a vector field on a manifold, the notion of an arc
field on a locally complete metric space was introduced in \cite{BC}. In that
paper, the authors proved an analogue of the Cauchy-Lipschitz Theorem i.e they
showed the existence and uniqueness of solution curves for a time independent
arc field. In this paper, we extend the result to the time dependent case,
namely we show the existence and uniqueness of solution curves for a time
dependent arc field. We also introduce the notion of the sum of two time
dependent arc fields and show existence and uniqueness of solution curves for
this sum.
