$$\hat{n}_2=-\hat{n}_1$$
$$\iiint_D \nabla \overrightarrow{F} dV=\iiint_{D_1} \nabla \overrightarrow{F} dV+\iiint_{D_2} \nabla \overrightarrow{F} dV$$
$$\\$$
$$\iint_S \overrightarrow{F} \cdot \hat{n} d \sigma=\iint_{S_1} \overrightarrow{F} \cdot \hat{n} d \sigma+\iint_{S_2} \overrightarrow{F} \cdot \hat{n} d \sigma+\iint_K \overrightarrow{F} \cdot \hat{n}_1 d \sigma+\iint_K \overrightarrow{F} \cdot \hat{n}_2 d \sigma$$ Since $\displaystyle{\iint_K \overrightarrow{F} \cdot \hat{n}_1 d \sigma+\iint_K \overrightarrow{F} \cdot \hat{n}_2 d \sigma=0}$, because $\hat{n}_2=-\hat{n}_1$, we have that:
$$\iint_S \overrightarrow{F} \cdot \hat{n} d \sigma=\iint_{S_1} \overrightarrow{F} \cdot \hat{n} d \sigma+\iint_{S_2} \overrightarrow{F} \cdot \hat{n} d \sigma$$
What are $S$, $S_1$ and $S_2$??
$S$ is the sphere, which is the boundary of the ball $D$.
The ball $D$ has been divided into 2 semi-balls $D_1$ and $D_2$.
The semi-ball $D_1$ is bounded by a semi-sphere $S_1$ and circle disk $K$. Similarly $D_2$ is bounded by $S_2$ and the same circle disk $K$.