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<b>let </b><font color="Maroon" title="c1">X</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="compts_1.html#V6" title="COMPTS_1:attr.6">compact</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>;<br/>

<a NAME="E1:148"/><i><font color="Green" title="E90">A1</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K19" title="PSCOMP_1:func.19">SW-corner</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K10" title="PSCOMP_1:func.10">inf</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">proj1</a>  <a href="pscomp_1.html#K12" title="PSCOMP_1:func.12">||</a> <font color="Maroon" title="c1">X</font></span>)</span>
 
<b>by </b><i><a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/>
<b>set </b><font color="Maroon" title="c2">LX</font> =  <a href="pscomp_1.html#K26" title="PSCOMP_1:func.26">S-most</a> <font color="Maroon" title="c1">X</font>;<br/>
<a NAME="E2:148"/><i><font color="Green" title="E91">A2</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K34" title="PSCOMP_1:func.34">S-min</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K10" title="PSCOMP_1:func.10">inf</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">proj1</a>  <a href="pscomp_1.html#K12" title="PSCOMP_1:func.12">||</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K26" title="PSCOMP_1:func.26">S-most</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/>
<a NAME="E3:148"/><i><font color="Green" title="E92">A3</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K33" title="PSCOMP_1:func.33">S-max</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K11" title="PSCOMP_1:func.11">sup</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">proj1</a>  <a href="pscomp_1.html#K12" title="PSCOMP_1:func.12">||</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K26" title="PSCOMP_1:func.26">S-most</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/>
<a NAME="E4:148"/><i><font color="Green" title="E93">A4</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">SE-corner</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K11" title="PSCOMP_1:func.11">sup</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">proj1</a>  <a href="pscomp_1.html#K12" title="PSCOMP_1:func.12">||</a> <font color="Maroon" title="c1">X</font></span>)</span>
 
<b>by </b><i><a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/>
<a NAME="E5:148"/><i><font color="Green" title="E94">A5</font></i>: 
 <a href="pscomp_1.html#K26" title="PSCOMP_1:func.26">S-most</a> <font color="Maroon" title="c1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E6:148"/><b>then </b><i><font color="Green" title="E95">A6</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K19" title="PSCOMP_1:func.19">SW-corner</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K34" title="PSCOMP_1:func.34">S-min</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
 
<b>by </b><i><a class="txt" href="pscomp_1.html#E1:148"><i><font color="Green" title="E90">A1</font></i></a>, <a class="txt" href="pscomp_1.html#E2:148"><i><font color="Green" title="E91">A2</font></i></a>, <a class="ref" href="pscomp_1.html#T62" target="_self" title="PSCOMP_1:th.62">Th62</a></i>;<br/>
<a NAME="E7:148"/><i><font color="Green" title="E96">A7</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K34" title="PSCOMP_1:func.34">S-min</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K33" title="PSCOMP_1:func.33">S-max</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
 
<b>by </b><i><a class="txt" href="pscomp_1.html#E2:148"><i><font color="Green" title="E91">A2</font></i></a>, <a class="txt" href="pscomp_1.html#E3:148"><i><font color="Green" title="E92">A3</font></i></a>, <a class="ref" href="pscomp_1.html#T53" target="_self" title="PSCOMP_1:th.53">Th53</a></i>;<br/>
<a NAME="E8:148"/><i><font color="Green" title="E97">A8</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K33" title="PSCOMP_1:func.33">S-max</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">SE-corner</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
 
<b>by </b><i><a class="txt" href="pscomp_1.html#E3:148"><i><font color="Green" title="E92">A3</font></i></a>, <a class="txt" href="pscomp_1.html#E4:148"><i><font color="Green" title="E93">A4</font></i></a>, <a class="txt" href="pscomp_1.html#E5:148"><i><font color="Green" title="E94">A5</font></i></a>, <a class="ref" href="pscomp_1.html#T63" target="_self" title="PSCOMP_1:th.63">Th63</a></i>;<br/>
<b>thus </b><a NAME="E9:148"/>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K19" title="PSCOMP_1:func.19">SW-corner</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K34" title="PSCOMP_1:func.34">S-min</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
 <b>by </b><i><a class="txt" href="pscomp_1.html#E1:148"><i><font color="Green" title="E90">A1</font></i></a>, <a class="txt" href="pscomp_1.html#E2:148"><i><font color="Green" title="E91">A2</font></i></a>, <a class="txt" href="pscomp_1.html#E5:148"><i><font color="Green" title="E94">A5</font></i></a>, <a class="ref" href="pscomp_1.html#T62" target="_self" title="PSCOMP_1:th.62">Th62</a></i>;<br/>

<b>thus </b><a NAME="E10:148"/>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K19" title="PSCOMP_1:func.19">SW-corner</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K33" title="PSCOMP_1:func.33">S-max</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
 <b>by </b><i><a class="txt" href="pscomp_1.html#E6:148"><i><font color="Green" title="E95">A6</font></i></a>, <a class="txt" href="pscomp_1.html#E7:148"><i><font color="Green" title="E96">A7</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>

<b>hence </b><a NAME="E11:148"/>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K19" title="PSCOMP_1:func.19">SW-corner</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">SE-corner</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
 <b>by </b><i><a class="txt" href="pscomp_1.html#E8:148"><i><font color="Green" title="E97">A8</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>

<b>thus </b><a NAME="E12:148"/>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K34" title="PSCOMP_1:func.34">S-min</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K33" title="PSCOMP_1:func.33">S-max</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
 <b>by </b><i><a class="txt" href="pscomp_1.html#E2:148"><i><font color="Green" title="E91">A2</font></i></a>, <a class="txt" href="pscomp_1.html#E3:148"><i><font color="Green" title="E92">A3</font></i></a>, <a class="ref" href="pscomp_1.html#T53" target="_self" title="PSCOMP_1:th.53">Th53</a></i>;<br/>

<b>thus </b><a NAME="E13:148"/>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K34" title="PSCOMP_1:func.34">S-min</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">SE-corner</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
 <b>by </b><i><a class="txt" href="pscomp_1.html#E7:148"><i><font color="Green" title="E96">A7</font></i></a>, <a class="txt" href="pscomp_1.html#E8:148"><i><font color="Green" title="E97">A8</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>

<b>thus </b><a NAME="E14:148"/>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K33" title="PSCOMP_1:func.33">S-max</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">SE-corner</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
 <b>by </b><i><a class="txt" href="pscomp_1.html#E3:148"><i><font color="Green" title="E92">A3</font></i></a>, <a class="txt" href="pscomp_1.html#E4:148"><i><font color="Green" title="E93">A4</font></i></a>, <a class="txt" href="pscomp_1.html#E5:148"><i><font color="Green" title="E94">A5</font></i></a>, <a class="ref" href="pscomp_1.html#T63" target="_self" title="PSCOMP_1:th.63">Th63</a></i>;<br/>


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