proofs.

Some general results
====================

Let ${\widetilde }{\mathcal{P}}=({\widetilde }{F},{\widetilde }{G},{\widetilde }{S},{\widetilde }{K},{\widetilde }{\Lambda})$ be another numerical problem. Suppose we have an algorithm for problem ${\widetilde }{\mathcal{P}}$, and we want to construct one for problem $\mathcal{P}$. Furthermore, for each input $f\in F$ of problem $\mathcal{P}$ we can produce an input $R(f)$ for problem ${\widetilde }{\mathcal{P}}$ such that $S(f)=\Psi\circ{\widetilde }{S}\circ R(f)$ with a certain mapping $\Psi: {\widetilde }{G}\to G$. Finally, each information about $R(f)$ can be obtained from $\kappa$ suitable informations about $f$. Then we say that problem $\mathcal{P}$ reduces to ${\widetilde }{\mathcal{P}}$. Let us specify the assumptions.

Let $R:F\to{\widetilde }{F}$ be a mapping such that there exist a $\kappa \in {{\mathbb N}}$, mappings $\eta_j:{\widetilde }{\Lambda}\to\Lambda \quad (j=0, \dots, \kappa-1)$ and ${\varrho }:{\widetilde }{\Lambda}\times K^\kappa\to {\widetilde }{K}$ with $$\label{XG3}
(R(f))({\widetilde }{\lambda})={\varrho }({\widetilde }{\lambda}, f(\eta_0({\widetilde }{\lambda})), \dots,f(\eta_{\kappa-1}({\widetilde }{\lambda})))$$ for all $f\in F$ and ${\widetilde }{\lambda}\in{\widetilde }{\Lambda}$. Furthermore, let $\Psi:{\widetilde }{G}\to G$ be a Lipschitz mapping and assume that $$\label{XG4}
S=\Psi\circ{\widetilde }{S}\circ R.$$ Note that (\[XG3\]) defines also a mapping $$R:\mathcal{F}(\Lambda,K) \to \mathcal{F}({\widetilde }{\Lambda},{\widetilde }{K})$$ (we use the same notation $R$), where $\mathcal{F}(\Lambda,K)$ stands for the set of all mappings from $\Lambda$ to $K$.

\[lem:1\] Let $F_0\subseteq F$ be any nonempty subset. Suppose that 