On amalgamations of Heegaard splittings with high distance

Let $M$ be a compact, orientable 3-manifold and $F$ an essential closed surface which cuts $M$ into $M_{1}$ and $M_{2}$. Suppose that $M_{i}$ has a Heegaard splitting $V_{i}\cup_{S_{i}}W_{i}$ with distance $D{(S_{i})}\geqslant{2g(M_{i})+1}$, $i=1, 2$. Then $g(M)=g(M_1)+g(M_2)-g(F)$, and the amalgamation of $V_{1}\cup_{S_{1}}W_{1}$ and $V_{2}\cup_{S_{2}}W_{2}$ is the unique minimal Heegaard splitting of $M$ up to isotopy.