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<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="E1:150"/><i><font color="Green">E30</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K19" target="_self">SW-corner</a> <font color="Maroon">X</font></span>)</span> <a href="euclid.html#K21">`1</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#K13" target="_self">proj1</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#K26" target="_self">S-most</a> <font color="Maroon">X</font>;<br/>
<a NAME="E2:150"/><i><font color="Green">E31</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K34" target="_self">S-min</a> <font color="Maroon">X</font></span>)</span> <a href="euclid.html#K21">`1</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#K13" target="_self">proj1</a>  <a href="pscomp_1.html#K12" target="_self">||</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K26" target="_self">S-most</a> <font color="Maroon">X</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="pscomp_1.html#T18" target="_self">Th18</a></i>;<br/>
<a NAME="E3:150"/><i><font color="Green">E32</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K33" target="_self">S-max</a> <font color="Maroon">X</font></span>)</span> <a href="euclid.html#K21">`1</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#K13" target="_self">proj1</a>  <a href="pscomp_1.html#K12" target="_self">||</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K26" target="_self">S-most</a> <font color="Maroon">X</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="pscomp_1.html#T18" target="_self">Th18</a></i>;<br/>
<a NAME="E4:150"/><i><font color="Green">E34</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#K21">`1</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#K13" target="_self">proj1</a>  <a href="pscomp_1.html#K12" target="_self">||</a> <font color="Maroon">X</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E5:150"/><i><font color="Green">E35</font></i>: 
 <a href="pscomp_1.html#K26" target="_self">S-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="E6:150"/><b>then </b><i><font color="Green">E36</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K19" target="_self">SW-corner</a> <font color="Maroon">X</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K34" target="_self">S-min</a> <font color="Maroon">X</font></span>)</span> <a href="euclid.html#K21">`1</a> 
 
<b>by </b><i>, <a class="ref" href="pscomp_1.html#T19" target="_self">Th19</a>, <a class="ref" href="pscomp_1.html#D2" target="_self">Def2</a></i>;<br/>
<a NAME="E7:150"/><i><font color="Green">E40</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K34" target="_self">S-min</a> <font color="Maroon">X</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K33" target="_self">S-max</a> <font color="Maroon">X</font></span>)</span> <a href="euclid.html#K21">`1</a> 
 
<b>by </b><i><a class="ref" href="pscomp_1.html#T19" target="_self">Th19</a>, <a class="ref" href="pscomp_1.html#T20" target="_self">Th20</a>, </i>;<br/>
<a NAME="E8:150"/><i><font color="Green">E41</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K33" target="_self">S-max</a> <font color="Maroon">X</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <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#K21">`1</a> 
 
<b>by </b><i><a class="ref" href="pscomp_1.html#T20" target="_self">Th20</a>, <a class="ref" href="pscomp_1.html#T21" target="_self">Th21</a>, <a class="ref" href="pscomp_1.html#T22" target="_self">Th22</a>, </i>;<br/>
<b>thus </b><a NAME="E9:150"/>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K19" target="_self">SW-corner</a> <font color="Maroon">X</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K34" target="_self">S-min</a> <font color="Maroon">X</font></span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i>, <a class="ref" href="pscomp_1.html#T19" target="_self">Th19</a>, <a class="ref" href="pscomp_1.html#T22" target="_self">Th22</a>, <a class="ref" href="pscomp_1.html#D2" target="_self">Def2</a></i>;<br/>

<b>thus </b><a NAME="E10:150"/>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K19" target="_self">SW-corner</a> <font color="Maroon">X</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K33" target="_self">S-max</a> <font color="Maroon">X</font></span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i><a class="txt" href="pscomp_1.html#E34"><i><font color="Green">Lemma113</font></i></a>, <a class="txt" href="pscomp_1.html#E36"><i><font color="Green">Lemma114</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>

<b>hence </b><a NAME="E11:150"/>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K19" target="_self">SW-corner</a> <font color="Maroon">X</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <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#K21">`1</a> 
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>

<b>thus </b><a NAME="E12:150"/>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K34" target="_self">S-min</a> <font color="Maroon">X</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K33" target="_self">S-max</a> <font color="Maroon">X</font></span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i><a class="ref" href="pscomp_1.html#T19" target="_self">Th19</a>, <a class="ref" href="pscomp_1.html#T20" target="_self">Th20</a>, </i>;<br/>

<b>thus </b><a NAME="E13:150"/>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K34" target="_self">S-min</a> <font color="Maroon">X</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <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#K21">`1</a> 
 <b>by </b><i><a class="txt" href="pscomp_1.html#E36"><i><font color="Green">Lemma114</font></i></a>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>

<b>thus </b><a NAME="E14:150"/>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K33" target="_self">S-max</a> <font color="Maroon">X</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <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#K21">`1</a> 
 <b>by </b><i><a class="ref" href="pscomp_1.html#T20" target="_self">Th20</a>, <a class="ref" href="pscomp_1.html#T21" target="_self">Th21</a>, <a class="ref" href="pscomp_1.html#T22" target="_self">Th22</a>, </i>;<br/>


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