<?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/>

<b>set </b><font color="Maroon">p2W</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#K23" target="_self">W-most</a> <font color="Maroon">X</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">c</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#K23" target="_self">W-most</a> <font color="Maroon">X</font></span>)</span></span>)</span>;<br/>
<a NAME="E1:130"/><i><font color="Green">E30</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#K23" target="_self">W-most</a> <font color="Maroon">X</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K23" target="_self">W-most</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/>
<a NAME="E2:130"/>
<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#K23" target="_self">W-most</a> <font color="Maroon">X</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#K23" target="_self">W-most</a> <font color="Maroon">X</font></span>)</span></span>)</span> is <a href="pscomp_1.html#V2" target="_self">with_min</a>
 
<b>by </b><i/>;<br/>
<a NAME="E3:130"/><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#K23" target="_self">W-most</a> <font color="Maroon">X</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#K23" target="_self">W-most</a> <font color="Maroon">X</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#K23" target="_self">W-most</a> <font color="Maroon">X</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#K23" target="_self">W-most</a> <font color="Maroon">X</font></span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">p</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:130"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">p</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#K23" target="_self">W-most</a> <font color="Maroon">X</font></span>)</span></span>)</span> &amp; <font color="Maroon">p</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#K23" target="_self">W-most</a> <font color="Maroon">X</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#K23" target="_self">W-most</a> <font color="Maroon">X</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#K23" target="_self">W-most</a> <font color="Maroon">X</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#K23" target="_self">W-most</a> <font color="Maroon">X</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">p</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">p</font> = <font color="Maroon">p</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>, </i>;<br/>
<a NAME="E7:130"/><i><font color="Green">E32</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#K23" target="_self">W-most</a> <font color="Maroon">X</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> 
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E8:130"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">p</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#K19" target="_self">SW-corner</a> <font color="Maroon">X</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K20" target="_self">NW-corner</a> <font color="Maroon">X</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E9:130"/>
( <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#K15" target="_self">W-bound</a> <font color="Maroon">X</font> &amp; <span class="p1">(<span class="default"><a href="pscomp_1.html#K20" target="_self">NW-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#K15" target="_self">W-bound</a> <font color="Maroon">X</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E10:130"/><b>then </b>
<font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K15" target="_self">W-bound</a> <font color="Maroon">X</font>
 
<b>by </b><i>, <a class="ref" href="goboard7.html#T5">GOBOARD7:5</a></i>;<br/>
<b>hence </b><a NAME="E11:130"/>
 <a href="pscomp_1.html#K27" target="_self">W-min</a> <font color="Maroon">X</font> <a href="hidden.html#R2">in</a>  <a href="pscomp_1.html#K23" target="_self">W-most</a> <font color="Maroon">X</font>
 <b>by </b><i>, , , <a class="ref" href="euclid.html#T57">EUCLID:57</a></i>;<br/>

<a NAME="E12:130"/>
<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#K23" target="_self">W-most</a> <font color="Maroon">X</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#K23" target="_self">W-most</a> <font color="Maroon">X</font></span>)</span></span>)</span> is <a href="pscomp_1.html#V1" target="_self">with_max</a>
 
<b>by </b><i/>;<br/>
<a NAME="E13:130"/><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#K23" target="_self">W-most</a> <font color="Maroon">X</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#K23" target="_self">W-most</a> <font color="Maroon">X</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#K23" target="_self">W-most</a> <font color="Maroon">X</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#K23" target="_self">W-most</a> <font color="Maroon">X</font></span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">p</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E15:130"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">p</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#K23" target="_self">W-most</a> <font color="Maroon">X</font></span>)</span></span>)</span> &amp; <font color="Maroon">p</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#K23" target="_self">W-most</a> <font color="Maroon">X</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#K23" target="_self">W-most</a> <font color="Maroon">X</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#K23" target="_self">W-most</a> <font color="Maroon">X</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#K23" target="_self">W-most</a> <font color="Maroon">X</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">p</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">p</font> = <font color="Maroon">p</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>, </i>;<br/>
<a NAME="E17:130"/><i><font color="Green">E36</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#K23" target="_self">W-most</a> <font color="Maroon">X</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> 
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E18:130"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">p</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#K19" target="_self">SW-corner</a> <font color="Maroon">X</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K20" target="_self">NW-corner</a> <font color="Maroon">X</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E19:130"/>
( <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#K15" target="_self">W-bound</a> <font color="Maroon">X</font> &amp; <span class="p1">(<span class="default"><a href="pscomp_1.html#K20" target="_self">NW-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#K15" target="_self">W-bound</a> <font color="Maroon">X</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E20:130"/><b>then </b>
<font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K15" target="_self">W-bound</a> <font color="Maroon">X</font>
 
<b>by </b><i>, <a class="ref" href="goboard7.html#T5">GOBOARD7:5</a></i>;<br/>
<b>hence </b><a NAME="E21:130"/>
 <a href="pscomp_1.html#K28" target="_self">W-max</a> <font color="Maroon">X</font> <a href="hidden.html#R2">in</a>  <a href="pscomp_1.html#K23" target="_self">W-most</a> <font color="Maroon">X</font>
 <b>by </b><i>, , , <a class="ref" href="euclid.html#T57">EUCLID:57</a></i>;<br/>


</div>
