and $F^{(\varepsilon)}\subset{\mathbb{R}^d_{+}}$, $F^{(\varepsilon)}$ $(1+\varepsilon)$-[stick]{}-covers $F$ if and only if: $$\begin{aligned}
\forall \bm{z}'\in F^{(\varepsilon)},~~
\exists \bm{z}\in F &:&
\bm{z}'{\succsim}\bm{z}\\
\text{and}~~~
\forall \bm{z}\in F,~~
\exists \bm{z}'\in F^{(\varepsilon)} &:&
(1+\varepsilon)\bm{z}{\succsim}\bm{z}'\end{aligned}$$

The first condition is easily satisfiable by $F^{(\varepsilon)}\subseteq F$. The second condition states that $F^{(\varepsilon)}$ sticks to $F$. Given $F$, one can implement Definition \[def:stick:covering\] as in Figure \[fig:mo:approx\]: Take one element of ${\mathcal{F}}$ per cell of the logarithmic grid, and then take ${\mbox{WST}}$ of this set of elements. Now we can state that with an approximate Phase 1, the precision transfers to Phase 2 in polynomial time, as follows.

![MO approximations, depictions of under and stick coverings[]{data-label="fig:mo:approx"}](art1/logGridphi.pdf)

 

$E^{(\varepsilon)}$ (the green dots below ${\mbox{WST}}[{\mathcal{E}}]$) is a $(1+\varepsilon)$-under-covering of set ${\mbox{WST}}[{\mathcal{E}}]$.\
$F^{(\varepsilon)}$ (the three red dots in ${\mathcal{F}}$) is a $(1+\varepsilon)$-stick-covering of the dark-red set ${\mathcal{F}}$.

\[lem:approx\] Given $\varepsilon_1,\varepsilon_2>0$ and approximations $E$ of ${\mathcal{E}}$ and $F$ of ${\mathcal{F}}$, if $$\begin{aligned}
\forall \bm{y}\in{\mathcal{E}}, \exists \bm{y}'\in E,\quad \bm{y}{\succsim}\bm{y}' &
\quad \text{and}\quad & 
\forall \bm{y}'\in E, \exists \bm{y}\in{\mathcal{E}},\quad (1+\varepsilon_1)\bm{y}'{\succsim}\bm{y}\label{eq:approx:E}\\
\forall \bm{z}'\in F, \exists \bm{z}\in{\mathcal{F}},\quad \bm{z}'{\succsim}\bm{z} &
\quad \text{and}\quad & 
\forall \bm{z}\in{\mathcal{F}}, \exists \bm{z}'\in F,\quad (1+\varepsilon_2)\bm{z}{\succsim}\bm{z}'\label{eq:approx:F}\end{aligned}$$ holds, then it follows that $R[E,F]\subseteq R[{\mathcal{E}},{\mathcal{F}}]$ and: $$\begin{aligned}
\forall \bm{\rho}\in R[{\mathcal{E}},{\mathcal{F}}],\quad \exists \bm{\rho}'\in R[E,F],\quad (1+\varepsilon_1)(1+\varepsilon_2)\bm{\rho}'{\succsim}\bm{\rho}\label{eq:approx:R}\end{aligned}$$

Equations (\[eq:approx:E\]) and (\[eq:approx:F\]) state approximation 