  A rigorous demonstration that given appropriate data on our past light cone
leads to the determination of the metric functions and all their time
derivatives on our past light cone is presented, thus showing how to evolve the
solution we obtain from data on the light cone off it in a well-defined and
straightforward way. It also automatically gives a procedure for constructing
the solution for all spherically symmetric, inhomogeneous cosmological
Lema\^itre-Tolman-Bondi models in observational coordinates as a Taylor series
in time of however many terms we need. Our procedure takes into account the
essential data giving the maximum of the observer area (angular-diameter)
distance, and the redshift $z_{max}$ at which that occurs. This enables the
determination of the vacuum-energy density $\mu_{\Lambda}$, which would
otherwise remain undetermined.
