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

<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/>

<b>set </b><font color="Maroon">c<sub>2</sub></font> = <a href="pscomp_1.html#K14" target="_self">proj2</a>  <a href="pscomp_1.html#K12" target="_self">||</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K25" target="_self">E-most</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>3</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K25" target="_self">E-most</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>;<br/>
<a NAME="E1:146"/><i><font color="Green">E108</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K25" target="_self">E-most</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K25" target="_self">E-most</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/>
<a NAME="E2:146"/>
<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">c<sub>1</sub></font></span>)</span></span>)</span> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K25" target="_self">E-most</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> is <a href="pscomp_1.html#V2" target="_self">with_min</a>
 
<b>by </b><i><a class="ref" href="pscomp_1.html#D15" target="_self">Def15</a></i>;<br/>
<a NAME="E3:146"/><b>then </b>
 <a href="pscomp_1.html#K3" target="_self">inf</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K14" target="_self">proj2</a>  <a href="pscomp_1.html#K12" target="_self">||</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K25" target="_self">E-most</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K25" target="_self">E-most</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2">in</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">c<sub>1</sub></font></span>)</span></span>)</span> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K25" target="_self">E-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#D4" target="_self">Def4</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>4</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:146"/><i><font color="Green">E109</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K25" target="_self">E-most</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K25" target="_self">E-most</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> &amp;  <a href="pscomp_1.html#K3" target="_self">inf</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K14" target="_self">proj2</a>  <a href="pscomp_1.html#K12" target="_self">||</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K25" target="_self">E-most</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K25" target="_self">E-most</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</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">c<sub>1</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="ref" href="funct_2.html#T115">FUNCT_2:115</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <b>by </b><i><a class="txt" href="pscomp_1.html#E1:146"><i><font color="Green">E108</font></i></a>, <a class="txt" href="pscomp_1.html#E5:146"><i><font color="Green">E109</font></i></a></i>;<br/>
<a NAME="E7:146"/><i><font color="Green">E110</font></i>: 
<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">c<sub>1</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K22">`2</a> 
 
<b>by </b><i><a class="txt" href="pscomp_1.html#E1:146"><i><font color="Green">E108</font></i></a>, <a class="txt" href="pscomp_1.html#E5:146"><i><font color="Green">E109</font></i></a>, <a class="ref" href="pscomp_1.html#T70" target="_self">Th70</a></i>;<br/>
<a NAME="E8:146"/><i><font color="Green">E111</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</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>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K21" target="_self">NE-corner</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="pscomp_1.html#E1:146"><i><font color="Green">E108</font></i></a>, <a class="txt" href="pscomp_1.html#E5:146"><i><font color="Green">E109</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E9:146"/>
( <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="hidden.html#R1">=</a>  <a href="pscomp_1.html#K17" target="_self">E-bound</a> <font color="Maroon">c<sub>1</sub></font> &amp; <span class="p1">(<span class="default"><a href="pscomp_1.html#K21" target="_self">NE-corner</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#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#T76" target="_self">Th76</a>, <a class="ref" href="pscomp_1.html#T78" target="_self">Th78</a></i>;<br/>
<a NAME="E10:146"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</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="txt" href="pscomp_1.html#E8:146"><i><font color="Green">E111</font></i></a>, <a class="ref" href="goboard7.html#T5">GOBOARD7:5</a></i>;<br/>
<b>hence </b><a NAME="E11:146"/>
 <a href="pscomp_1.html#K32" target="_self">E-min</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="pscomp_1.html#K25" target="_self">E-most</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="pscomp_1.html#E1:146"><i><font color="Green">E108</font></i></a>, <a class="txt" href="pscomp_1.html#E5:146"><i><font color="Green">E109</font></i></a>, <a class="txt" href="pscomp_1.html#E7:146"><i><font color="Green">E110</font></i></a>, <a class="ref" href="euclid.html#T57">EUCLID:57</a></i>;<br/>

<a NAME="E12:146"/>
<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">c<sub>1</sub></font></span>)</span></span>)</span> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K25" target="_self">E-most</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> is <a href="pscomp_1.html#V1" target="_self">with_max</a>
 
<b>by </b><i><a class="ref" href="pscomp_1.html#D14" target="_self">Def14</a></i>;<br/>
<a NAME="E13:146"/><b>then </b>
 <a href="pscomp_1.html#K2" target="_self">sup</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K14" target="_self">proj2</a>  <a href="pscomp_1.html#K12" target="_self">||</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K25" target="_self">E-most</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K25" target="_self">E-most</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2">in</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">c<sub>1</sub></font></span>)</span></span>)</span> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K25" target="_self">E-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#D3" target="_self">Def3</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>6</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E15:146"/><i><font color="Green">E112</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K25" target="_self">E-most</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K25" target="_self">E-most</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> &amp;  <a href="pscomp_1.html#K2" target="_self">sup</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K14" target="_self">proj2</a>  <a href="pscomp_1.html#K12" target="_self">||</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K25" target="_self">E-most</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K25" target="_self">E-most</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</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">c<sub>1</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="ref" href="funct_2.html#T115">FUNCT_2:115</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <b>by </b><i><a class="txt" href="pscomp_1.html#E1:146"><i><font color="Green">E108</font></i></a>, <a class="txt" href="pscomp_1.html#E15:146"><i><font color="Green">E112</font></i></a></i>;<br/>
<a NAME="E17:146"/><i><font color="Green">E113</font></i>: 
<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">c<sub>1</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="euclid.html#K22">`2</a> 
 
<b>by </b><i><a class="txt" href="pscomp_1.html#E1:146"><i><font color="Green">E108</font></i></a>, <a class="txt" href="pscomp_1.html#E15:146"><i><font color="Green">E112</font></i></a>, <a class="ref" href="pscomp_1.html#T70" target="_self">Th70</a></i>;<br/>
<a NAME="E18:146"/><i><font color="Green">E114</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</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>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K21" target="_self">NE-corner</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="pscomp_1.html#E1:146"><i><font color="Green">E108</font></i></a>, <a class="txt" href="pscomp_1.html#E15:146"><i><font color="Green">E112</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E19:146"/>
( <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="hidden.html#R1">=</a>  <a href="pscomp_1.html#K17" target="_self">E-bound</a> <font color="Maroon">c<sub>1</sub></font> &amp; <span class="p1">(<span class="default"><a href="pscomp_1.html#K21" target="_self">NE-corner</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#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#T76" target="_self">Th76</a>, <a class="ref" href="pscomp_1.html#T78" target="_self">Th78</a></i>;<br/>
<a NAME="E20:146"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</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="txt" href="pscomp_1.html#E18:146"><i><font color="Green">E114</font></i></a>, <a class="ref" href="goboard7.html#T5">GOBOARD7:5</a></i>;<br/>
<b>hence </b><a NAME="E21:146"/>
 <a href="pscomp_1.html#K31" target="_self">E-max</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="pscomp_1.html#K25" target="_self">E-most</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="pscomp_1.html#E1:146"><i><font color="Green">E108</font></i></a>, <a class="txt" href="pscomp_1.html#E15:146"><i><font color="Green">E112</font></i></a>, <a class="txt" href="pscomp_1.html#E17:146"><i><font color="Green">E113</font></i></a>, <a class="ref" href="euclid.html#T57">EUCLID:57</a></i>;<br/>


</div>
