,\mathbb{P}^0\times\{\rho\})\\     &=&I_{(c,\infty)}(\Delta_0,\mathbb{P}^0\times\{\infty\})     \end{array}$$  -   We want to prove that $\chi(F^{-0}(D^{\ast}(c,\eta))\cap G^{-0}(\rho)\cap D)=I_{(c,\infty)}(\Delta_0,\mathbb{P}^0\times\{\infty\})$.      As $G^{-0}(\rho)$ is smooth and cut transversally $D$, $$\begin{array}{lll}     \chi(F^{-0}(D^{\ast}(c,\eta))\cap G^{-0}(\rho)\cap D)&=&     