\begin{theorem}
Let $f(x)\in H_0^{12}[0; \pi],$ $h(x)\in H_0^{(12)}[0;\pi].$ Then the solution to the Dirichlet problem (\ref{eqn1-1}) exists and is unique.
\end{theorem}
\begin{proof}[Proof:]
The existence is proved in Lemma \ref{Existance}.

Let us prove that the solution is unique. Indeed, since solution (\ref{eqn1-1}) $u(x, t)$ exists under the conditions of the theorem, it can be found in the form of series (11). According to Lemma 3 and Lemma \ref{h10} this solution has a derivative $\displaystyle\frac{\partial u(x, t)}{\partial t}\in H_0^3[0; \pi] \forall t\in [0; 1].$ Having found that $\displaystyle\frac{\partial u(x, 1)}{\partial t}=g(x),$ $g(x)\in H_0^3[0; \pi],$ we can view the problem (\ref{eqn1-1}) as the Cauchy problem (\ref{eqn4-1}).

Thus, problem (\ref{eqn1-1}) is reduced to the Cauchy problem (\ref{eqn1-1b}), whose solution is unique.
\end{proof}