  We introduce a sequent calculus for the temporal-over-topological fragment
$\textbf{DTL}_{0}^{\circ * \slash \Box}$ of dynamic topological logic
$\textbf{DTL}$, prove soundness semantically, and prove completeness
syntactically using the axiomatization of $\textbf{DTL}_{0}^{\circ * \slash
\Box}$ given in \cite{paper3}. A cut-free sequent calculus for
$\textbf{DTL}_{0}^{\circ * \slash \Box}$ is obtained as the union of the
propositional fragment of Gentzen's classical sequent calculus, two $\Box$
structural rules for the modal extension, and nine $\circ$ (next) and $*$
(henceforth) structural rules for the temporal extension. Future research will
focus on the construction of a hypersequent calculus for dynamic topological
$\textbf{S5}$ logic in order to prove Kremer's Next Removal Conjecture for the
logic of homeomorphisms on almost discrete spaces $\textbf{S5H}$.
