that $g$ admits a decomposition of the form $$g=h_1h_2h_3h_4,$$ where $$h_3=g\;\hbox{ on}\; \bigcup_{k=1}^{\infty} \p\t[ b_{2k},
a_{2k-1}],\quad \operatorname{supp}(h_3)\s U_3:=\bigcup_{k=1}^{\infty} \p\t(
\bar b_{2k}, \bar a_{2k-1}),$$ $$h_4=g\;\hbox{ on}
\;\bigcup_{k=0}^{\infty} \p\t[ b_{2k+1}, a_{2k}],\quad
\operatorname{supp}(h_4)\s U_4:=\bigcup_{n=0}^{\infty} \p\t( \bar b_{2k+1}, \bar
a_{2k}),$$ and where $a_0,\bar a_0$ satisfy $1> \bar a_0> a_0>
b_1$. Furthermore, $h_j$ satisfy conditions analogous to (3.4) for $j=1,2,3,4$.

In view of Prop. 6.3 there exist $\bar f_{j}\in  \H_c^{\p}(\p\t\R_+)$ of the form $\bar f_j=\operatorname{id}\t
f_j$ such that $\H_{U_j}(M)$ is $m$-displaceable by $\bar f_{j}$ for $j=1,2,3,4$ and for all $m\geq 1$.

Let $\nu$ be a conjugation-invariant norm on $\hdm$. In view of Theorem 5.6(2) and the invariance of $\nu$ we have $$\nu(g)\leq\nu(h_1)+\cdots+\nu(h_4)\leq 14(\nu(\bar
f_1)+\cdots\nu(\bar f_4)).$$

Observe that the sets $U_1\ld U_4$ depend on $g$. Nevertheless, in view of the second assertion of Prop. 6.3 and the invariance of $\nu$, the norms $\nu(\bar f_j)$ are independent of $g$. It follows that $\nu(g)$ is bounded, as required. $\square$

A connected open manifold $M$ is called *portable (in the wider sense)* if there are disjoint open subsets $U$, $V$ of $M$ such that there is $f\in \H_c(M)$ with $\overline{f(U\cup V)}$ contained in $V$. Furthermore, for every compact subset $K\s M$ there is $h\in \H_c(M)$ satisfying $h(K)\s U$.

The 