induces a locally trivial fibration $\bar\Psi_| : (B\cap {{\mathbb Y}}) \setminus (\Gamma_P \cup 
\{\sigma =0\} ) \to (D\times D') \setminus \Delta_P$. Let $Y_{U,V} := B\cap {{\mathbb Y}}_{U,V}$, for $U\subset D$ and $V\subset D'$. Since $\tau$ is a linear function, we have that $Y_{D,t}$ is contractible, for any $t\in D'$. Taking into account the singularities of the fibers of the projection $\sigma_| : Y_{D,t} \to D$ and using Lê’s attaching results [@Le-c], we deduce the following homotopy equivalences: $$Y_{D,t_0} \stackrel{{{\rm{ht}}}}{\simeq} Y_{s,t_0} \bigcup \mu_p(t_0) \mbox{ cells of dimension } n-1,$$ $$Y_{D,t} \stackrel{{{\rm{ht}}}}{\simeq} Y_{s,t} \bigcup \{\mu_p(t) \mbox{ cells of dimension } n-1 \}\bigcup 
\{ \alpha \mbox{ cells of dimension } n-1 \},$$ where $$\mu_p(t) := \mu({{\mathbb Y}}_{0,t},p), \hspace{1cm} \mu_p(t_0):= \mu({{\mathbb Y}}_{0,t_0},p),$$ $$\alpha := \sum_{(a,s,t) \in \Gamma_P \cap Y_{D,t}} \mu ({{\mathbb Y}}_{s,t}, a).$$

Since $Y_{s,t_0}\stackrel{{{\rm{ht}}}}{\simeq} Y_{s,t}$ we have the equality $\mu_p(t_0)=  \mu_p(t) + \alpha$ and therefore $\alpha = \mu_p(t_0)- \mu_p(t) = \lambda_{p,t_0}(0)$, by (\[eq:lambda\]). Since $\Gamma_P$ consists of trajectories of affine singular points, $\alpha$ is an intersection number, namely: $$\alpha = {{\rm{int}}}_{(p,t_0)}(\Gamma_P, \{ \tau = t_0\} ) = {{\rm{int}}}_{(0,t_0)}(\Delta_P, \{ \tau 
= t_0\} ),$$ where in the first intersection number, each point is counted with multiplicity equal to the Milnor number of the singularity 