= (d-1)^n$.

\[c:mu23\] If $P$ is a general-at-infinity deformation then $$\mu_{II}(s) + \mu_{III}(s) = \lambda(0) + \sum_{p_i \in W_0}\beta_{p_i}(0),$$ where $\beta_{p_i}(s) := \mu_{p_i,gen}(s) + \mu_{p_i}^{{\infty}}(s)$.

Since $f_s$ is GI for $s\not= 0$, we have $\lambda(s)=0$ and $W_s =\emptyset$, hence $\beta_{p_i}(s) = 0$. From the formula (\[eq:mulambda\]) for $f_0$ and $f_s$, we get $b_{n-1}(G_s) - b_{n-1}(G_0) = \mu_{II}(s) + \mu_{III}(s) - \lambda(0)$, whereas from (\[eq:chi\]), see also (\[eq:gammas\]), we get $b_{n-1}(G_s) - b_{n-1}(G_0) = \Delta \beta := 
\sum_{p_i \in W_0}\beta_{p_i}(0)$.

Related to the semi-continuity result Proposition \[l:semicont\], a question which arises next would be: what is the behaviour of $\mu(s)$ and of $\lambda(s)$ when $s\to {{\infty}}$? It is clear that the total Milnor number $\mu(s)$ cannot increase. It can either be constant (e.g. in homogeneous deformations of polynomials) or decrease, like in Example \[e:broughton\]. In case $\mu(s)$ decreases, as $s\to 0$, we say that there is [*loss of $\mu$ at infinity*]{}; this happens if and only if $\Gamma_{II}\cup \Gamma_{III} \not= 0$.

In contrast, $\lambda(s)$ may either increase or decrease (or be constant). Examples where it increases may be found in Siersma-Smeltink paper [@SS], see also Example \[e:mixed\]. Examples of $\lambda$-constant deformations are all those with $\lambda(s)=0$. Example \[e:decrease\] (of a non-[[$\mathcal 