for each $\delta>0$ and each finite subset $\Lambda_0\subseteq \Lambda$ there are mappings $$\theta:K\to K,\quad \Theta:F_0\to F$$ such that $\theta(K)$ is a finite set, $$\label{U4}
(\Theta(f))({{\lambda }})=\theta(f({{\lambda }}))\quad(f\in F_0,\, {{\lambda }}\in \Lambda_0),$$ and $$\label{U2}
\sup_{f\in F_0}\|S(f)-S(\Theta(f))\|\le \delta.$$ Then for all $n\in {{\mathbb N}}_0$, $$\label{U3}
e_{2\kappa n}^{{\rm q }}(S,F_0)\le \|\Psi\|_{\rm Lip}\, e_n^{{\rm q }}({\widetilde }{S},{\widetilde }{F}).$$

Let $\delta>0$, $n\in{{\mathbb N}}_0$ and let ${\widetilde }{A}$ be any quantum algorithm from ${\widetilde }{F}$ to ${\widetilde }{G}$ with $n_q({\widetilde }{A})\le n$ and $$e({\widetilde }{S},{\widetilde }{A},{\widetilde }{F})\le e_n^{{\rm q }}({\widetilde }{S},{\widetilde }{F})+\delta.$$ Let $${\widetilde }{A}=(({\widetilde }{A}_l)_{l=0}^{k-1},({\widetilde }{b}_l)_{l=0}^{k-1},{\widetilde }{{\varphi }}),\quad 
{\widetilde }{A}_l=({\widetilde }{Q}_l,({\widetilde }{U}_{l,j})_{j=0}^{n_l}),$$ $${\widetilde }{Q}_l=({\widetilde }{m}_l,{\widetilde }{m}_l',{\widetilde }{m}''_l,{\widetilde }{Z}_l,{\widetilde }{\tau}_l,{\widetilde }{\beta}_l),$$ where ${\widetilde }{Z}_l\subseteq {{\mathbb Z}}[0,2^{{\widetilde }{m}_l'})$ and $${\widetilde }{\tau}_l:{\widetilde }{Z}_l\to {\widetilde }{\Lambda}, \quad
{\widetilde }{\beta}_l:{\widetilde }{K}\to{{\mathbb Z}}[0,2^{{\widetilde }{m}''_l}).$$ Denote $${\widetilde }{\Lambda}_0=\{{\widetilde }{\tau}_l(i)\,:\, i\in {\widetilde }{Z}_l,\, l=0,\dots, k-1\}$$ and $$\Lambda_0=\{\eta_j({\widetilde }{{{\lambda }}})\,:\,{\widetilde }{{{\lambda }}}\in {\widetilde }{\Lambda}_0\}.$$ Now let $\theta$ and $\Theta$ be according to the assumptions, and choose $m^{*}$ so that $|\theta(K)|\le 2^{m^{*}}$. It is easily checked that one can find $$\beta: K\to {{\mathbb Z}}[0,2^{m^{*}})$$ and $${\gamma }:{{\mathbb Z}}[0,2^{m^{*}})\to K$$ such that ${\gamma }\circ \beta=\theta$. Define $$\bar{{\varrho }}:{\widetilde }{\Lambda}\times {{\mathbb Z}}[0,2^{m^{*}})^\kappa \to {\widetilde }{K}$$ for ${\widetilde }{{{\lambda }}}\in {\widetilde }{\Lambda}$, $t_0,\dots,t_{\kappa-1}\in {{\mathbb Z}}[0,2^{m^{*}})^\kappa$ by $$\bar{{\varrho }}({\widetilde 