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<b>let </b><font color="Maroon">c<sub>1</sub></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/>

<i><font color="Green">E112</font></i>: <span class="p1">(<span class="default"><a href="pscomp_1.html#K19" target="_self">SW-corner</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K21">`1</a>  = 
 <a href="pscomp_1.html#K15" target="_self">W-bound</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="ref" href="pscomp_1.html#T72" target="_self">Th72</a></i>
<br/>.= 
 <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">c<sub>1</sub></font></span>)</span>

;<br/>
<b>set </b><font color="Maroon">c<sub>2</sub></font> =  <a href="pscomp_1.html#K26" target="_self">S-most</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<a NAME="E2:150"/><i><font color="Green">E113</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K34" target="_self">S-min</a> <font color="Maroon">c<sub>1</sub></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">c<sub>1</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="pscomp_1.html#T116" target="_self">Th116</a></i>;<br/>
<a NAME="E3:150"/><i><font color="Green">E114</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K33" target="_self">S-max</a> <font color="Maroon">c<sub>1</sub></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">c<sub>1</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="pscomp_1.html#T116" target="_self">Th116</a></i>;<br/>
<i><font color="Green">E115</font></i>: <span class="p1">(<span class="default"><a href="pscomp_1.html#K22" target="_self">SE-corner</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K21">`1</a>  = 
 <a href="pscomp_1.html#K17" target="_self">E-bound</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="ref" href="pscomp_1.html#T78" target="_self">Th78</a></i>
<br/>.= 
 <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">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="pscomp_1.html#D32" target="_self">Def32</a></i>
;<br/>
<a NAME="E5:150"/><i><font color="Green">E116</font></i>: 
 <a href="pscomp_1.html#K26" target="_self">S-most</a> <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></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">E117</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K19" target="_self">SW-corner</a> <font color="Maroon">c<sub>1</sub></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">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K21">`1</a> 
 
<b>by </b><i><a class="txt" href="pscomp_1.html#E1:150"><i><font color="Green">E112</font></i></a>, <a class="txt" href="pscomp_1.html#E2:150"><i><font color="Green">E113</font></i></a>, <a class="ref" href="pscomp_1.html#T62" target="_self">Th62</a></i>;<br/>
<a NAME="E7:150"/><i><font color="Green">E118</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K34" target="_self">S-min</a> <font color="Maroon">c<sub>1</sub></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">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K21">`1</a> 
 
<b>by </b><i><a class="txt" href="pscomp_1.html#E2:150"><i><font color="Green">E113</font></i></a>, <a class="txt" href="pscomp_1.html#E3:150"><i><font color="Green">E114</font></i></a>, <a class="ref" href="pscomp_1.html#T53" target="_self">Th53</a></i>;<br/>
<a NAME="E8:150"/><i><font color="Green">E119</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K33" target="_self">S-max</a> <font color="Maroon">c<sub>1</sub></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">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K21">`1</a> 
 
<b>by </b><i><a class="txt" href="pscomp_1.html#E3:150"><i><font color="Green">E114</font></i></a>, <a class="txt" href="pscomp_1.html#E4:150"><i><font color="Green">E115</font></i></a>, <a class="txt" href="pscomp_1.html#E5:150"><i><font color="Green">E116</font></i></a>, <a class="ref" href="pscomp_1.html#T63" target="_self">Th63</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">c<sub>1</sub></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">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i><a class="txt" href="pscomp_1.html#E1:150"><i><font color="Green">E112</font></i></a>, <a class="txt" href="pscomp_1.html#E2:150"><i><font color="Green">E113</font></i></a>, <a class="txt" href="pscomp_1.html#E5:150"><i><font color="Green">E116</font></i></a>, <a class="ref" href="pscomp_1.html#T62" target="_self">Th62</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">c<sub>1</sub></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">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i><a class="txt" href="pscomp_1.html#E6:150"><i><font color="Green">E117</font></i></a>, <a class="txt" href="pscomp_1.html#E7:150"><i><font color="Green">E118</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">c<sub>1</sub></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">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i><a class="txt" href="pscomp_1.html#E8:150"><i><font color="Green">E119</font></i></a>, <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">c<sub>1</sub></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">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i><a class="txt" href="pscomp_1.html#E2:150"><i><font color="Green">E113</font></i></a>, <a class="txt" href="pscomp_1.html#E3:150"><i><font color="Green">E114</font></i></a>, <a class="ref" href="pscomp_1.html#T53" target="_self">Th53</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">c<sub>1</sub></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">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i><a class="txt" href="pscomp_1.html#E7:150"><i><font color="Green">E118</font></i></a>, <a class="txt" href="pscomp_1.html#E8:150"><i><font color="Green">E119</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">c<sub>1</sub></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">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i><a class="txt" href="pscomp_1.html#E3:150"><i><font color="Green">E114</font></i></a>, <a class="txt" href="pscomp_1.html#E4:150"><i><font color="Green">E115</font></i></a>, <a class="txt" href="pscomp_1.html#E5:150"><i><font color="Green">E116</font></i></a>, <a class="ref" href="pscomp_1.html#T63" target="_self">Th63</a></i>;<br/>


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