& \bar{X}\ar@{^{(}->}[rr]^{\Phi_{|\pi^*(K_X+\Lambda)|}}\ar@{-->}[rrd]^{\Phi'}\ar[d]_{\pi}& & \mathbb{P}\ar@{-->}[d]^{pr_c}\\
  C\ar@{^{(}->}[r] &X\ar@{-->}[rr]_{\Phi_{|K_X+\Lambda|}} &&\mathbb{P}',
  }$$ where $\Phi'$ is a map defined by the partial linear system $\pi^*H^0(X,K_X+\Lambda)$. We regard $\Phi_{|\pi^*(K_X+\Lambda)|}$ as an inclusion. By Lemma \[restsecions\], $r_{X\to C}(K_X+\Lambda)\ge1$. If ’=’ holds, $\Phi_{|K_X+\Lambda|}(C)$ is a point. Hence $pr_c(\bar{C})$ is a point, which implies $\bar{C}$ is a line passing $c$. Hence $C$ is a line passing through $\pi(c)$, which is the base point of $|K_X+\Lambda|$.

\[2comps\] Let $(X,\Lambda)$ be a connected Gorenstein stable log surface. Assume $X=X_1\cup X_2$, where $X_1$ is connected and $X_2$ is irreducible. Denote $C:=X_1\cap X_2$ as the connecting curve of $X_1$ and $X_2$.

Then:

-   $\Delta(X,K_X+\Lambda)\ge \Delta(X_1,(K_X+\Lambda)|_{X_1})$.

-   if ’=’ holds and $\Delta(X_2,(K_X+\Lambda)|_{X_2})\le 1$, then

    -   either $X_2$ is non-normal and $\Delta(X_2,(K_X+\Lambda)|_{X_2})=1$. $r_{X_2\to C}((K_X+\Lambda)|_{X_2})=1$. $C$ is a line on $X_2$ passing through the base point of $|(K_X+\Lambda)|_{X_2}|$. Moreover, $r_{X_1\to C}((K_X+\Lambda)|_{X_1})\le 1$.

    -   $X_2$ is normal and $\Delta(X_2,(K_X+\Lambda)|_{X_2})=0$, $|(K_X+\Lambda)|_{X_2}|$ is very ample. $r_{X_2\to C}((K_X+\Lambda)|_{X_2})=2$. $C$ is a line on $X_2$. Moreover, $r_{X_1\to C}((K_X+\Lambda)|_{X_1})\le 2$;

    -   or $X_2$ is normal and $\Delta(X_2,(K_X+\Lambda)|_{X_2})=1$, $|(K_X+\Lambda)|_{X_2}|$ is composed with a pencil of elliptic curves. $r_{X_2\to C}((K_X+\Lambda)|_{X_2})=1$. $C$ is a fiber on $X_2$. Moreover, $r_{X_1\to C}((K_X+\Lambda)|_{X_1})\le 1$.

$(K_X+\Lambda)^2=(K_X+\Lambda)|_{X_1}^2+(K_X+\Lambda)|_{X_2}^2$ together with Cor \[fibersection\] gives $$\label{deltaIneq}
\begin{split}
\Delta(X,K_X+\Lambda) & \ge \Delta(X_1,(K_X+\Lambda)|_{X_1})+\Delta(X_2,(K_X+\Lambda)|_{X_2})\\
&+\max\{r_{X_1\to C}((K_X+\Lambda)|_{X_1}),r_{X_2\to 