Picard's existence and uniqueness theorem states that if a function satisfies some conditions on a particular interval then it has a solution in that interval. The function being f in the IVP:
$$ \begin{cases} x'=f(x(t),t) \\ x(t_0)=x_0 \end{cases}$$
But then there exists a theorem that states that these solutions can be continued further by repeating the argument with the old final time as the new initial time. I am looking for a detailed explanation and formulation of the proof for this theorem. Could somebody help me out? as I've been searching for quite a while now and can't seem to find anything.