<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">X</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="compts_1.html#V6">compact</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/>

<a NAME="E1:142"/><i><font color="Green">E30</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K22" target="_self">SE-corner</a> <font color="Maroon">X</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K10" target="_self">inf</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K14" target="_self">proj2</a>  <a href="pscomp_1.html#K12" target="_self">||</a> <font color="Maroon">X</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">LX</font> =  <a href="pscomp_1.html#K25" target="_self">E-most</a> <font color="Maroon">X</font>;<br/>
<a NAME="E2:142"/><i><font color="Green">E31</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K32" target="_self">E-min</a> <font color="Maroon">X</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K10" target="_self">inf</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K14" target="_self">proj2</a>  <a href="pscomp_1.html#K12" target="_self">||</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K25" target="_self">E-most</a> <font color="Maroon">X</font></span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E3:142"/><i><font color="Green">E32</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K31" target="_self">E-max</a> <font color="Maroon">X</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K11" target="_self">sup</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K14" target="_self">proj2</a>  <a href="pscomp_1.html#K12" target="_self">||</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K25" target="_self">E-most</a> <font color="Maroon">X</font></span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E4:142"/><i><font color="Green">E34</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K21" target="_self">NE-corner</a> <font color="Maroon">X</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K11" target="_self">sup</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K14" target="_self">proj2</a>  <a href="pscomp_1.html#K12" target="_self">||</a> <font color="Maroon">X</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E5:142"/><i><font color="Green">E35</font></i>: 
 <a href="pscomp_1.html#K25" target="_self">E-most</a> <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E6:142"/><b>then </b><i><font color="Green">E36</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K22" target="_self">SE-corner</a> <font color="Maroon">X</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K32" target="_self">E-min</a> <font color="Maroon">X</font></span>)</span> <a href="euclid.html#K22">`2</a> 
 
<b>by </b><i>, <a class="txt" href="pscomp_1.html#E22"><i><font color="Green">Lemma102</font></i></a>, <a class="ref" href="pscomp_1.html#D2" target="_self">Def2</a></i>;<br/>
<a NAME="E7:142"/><i><font color="Green">E40</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K32" target="_self">E-min</a> <font color="Maroon">X</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K31" target="_self">E-max</a> <font color="Maroon">X</font></span>)</span> <a href="euclid.html#K22">`2</a> 
 
<b>by </b><i><a class="txt" href="pscomp_1.html#E22"><i><font color="Green">Lemma102</font></i></a>, <a class="txt" href="pscomp_1.html#E23"><i><font color="Green">Lemma103</font></i></a>, </i>;<br/>
<a NAME="E8:142"/><i><font color="Green">E41</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K31" target="_self">E-max</a> <font color="Maroon">X</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K21" target="_self">NE-corner</a> <font color="Maroon">X</font></span>)</span> <a href="euclid.html#K22">`2</a> 
 
<b>by </b><i><a class="txt" href="pscomp_1.html#E23"><i><font color="Green">Lemma103</font></i></a>, <a class="ref" href="pscomp_1.html#T15" target="_self">Th15</a>, <a class="ref" href="pscomp_1.html#T17" target="_self">Th17</a>, </i>;<br/>
<b>thus </b><a NAME="E9:142"/>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K22" target="_self">SE-corner</a> <font color="Maroon">X</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K32" target="_self">E-min</a> <font color="Maroon">X</font></span>)</span> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i>, <a class="txt" href="pscomp_1.html#E22"><i><font color="Green">Lemma102</font></i></a>, <a class="ref" href="pscomp_1.html#T17" target="_self">Th17</a>, <a class="ref" href="pscomp_1.html#D2" target="_self">Def2</a></i>;<br/>

<b>thus </b><a NAME="E10:142"/>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K22" target="_self">SE-corner</a> <font color="Maroon">X</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K31" target="_self">E-max</a> <font color="Maroon">X</font></span>)</span> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i><a class="ref" href="pscomp_1.html#T18" target="_self">Th18</a>, <a class="txt" href="pscomp_1.html#E28"><i><font color="Green">Lemma107</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>

<b>hence </b><a NAME="E11:142"/>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K22" target="_self">SE-corner</a> <font color="Maroon">X</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K21" target="_self">NE-corner</a> <font color="Maroon">X</font></span>)</span> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>

<b>thus </b><a NAME="E12:142"/>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K32" target="_self">E-min</a> <font color="Maroon">X</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K31" target="_self">E-max</a> <font color="Maroon">X</font></span>)</span> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i><a class="txt" href="pscomp_1.html#E22"><i><font color="Green">Lemma102</font></i></a>, <a class="txt" href="pscomp_1.html#E23"><i><font color="Green">Lemma103</font></i></a>, </i>;<br/>

<b>thus </b><a NAME="E13:142"/>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K32" target="_self">E-min</a> <font color="Maroon">X</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K21" target="_self">NE-corner</a> <font color="Maroon">X</font></span>)</span> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i><a class="txt" href="pscomp_1.html#E28"><i><font color="Green">Lemma107</font></i></a>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>

<b>thus </b><a NAME="E14:142"/>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K31" target="_self">E-max</a> <font color="Maroon">X</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K21" target="_self">NE-corner</a> <font color="Maroon">X</font></span>)</span> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i><a class="txt" href="pscomp_1.html#E23"><i><font color="Green">Lemma103</font></i></a>, <a class="ref" href="pscomp_1.html#T15" target="_self">Th15</a>, <a class="ref" href="pscomp_1.html#T17" target="_self">Th17</a>, </i>;<br/>


</div>
