%  file is s10.tex
{\bf Proof}
If $B-\nu C + \omega C > 0$, then $\phi(\nu,\omega)$  is
finite. Moreover, $L(y,\nu,\omega) = \phi(\nu,\omega)$ for $y=(B-\nu C +
\omega C)^{-1}\psi$. Substituting for $y$ in $L$ yields
\[
 h(\nu,\omega)=\phi(\nu,\omega).
\]
Now if $z$ is feasible for (P), then for all
nonpositive $\nu,\omega$ we have
$\phi(\nu,\omega) \leq  L(z,\nu,\omega) \leq \mu(z)$.
We now have
\[
\begin{array}{rcl}
\mu^* &=& \min_{z~feasible} \mu(z) \\
&\geq& \sup_{\nu \leq 0,\omega \leq 0} \phi(\nu,\omega),\\
&=& \sup_{\stackrel{B - \nu C + \omega C \geq 0}{\nu \leq 0,\omega \leq
0}} \phi(\nu,\omega)\\
&\geq& \sup_{\stackrel{B - \nu C + \omega C > 0}{\nu \leq 0,\omega \leq
0}} h(\nu,\omega).
\end{array}
\]
