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

<b>let </b><font color="Maroon">G</font> be   <a href="goboard1.html#NM1">Go-board</a>;<br/><b>let </b><font color="Maroon">f</font> be  non <a href="funct_1.html#V3">constant</a> <a href="goboard5.html#V2">standard</a> <a href="goboard5.html#NM1">special_circular_sequence</a>;<br/>



<b>assume </b><a NAME="E1:29"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">f</font> <a href="goboard1.html#R1">is_sequence_on</a> <font color="Maroon">G</font>
 ;<br/>

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E2:29"/><i><font color="Green">E42</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> )
 ;<br/>

<b>set </b><font color="Maroon">rc</font> = <span class="p1">(<span class="default"><a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span>;<br/>
<a NAME="E3:29"/><i><font color="Green">E43</font></i>: 
<span class="p1">(<span class="default"><a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="subset_1.html#T32">SUBSET_1:32</a></i>;<br/>
<a NAME="E4:29"/><i><font color="Green">E44</font></i>: 
<span class="p1">(<span class="default"><a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<font color="Maroon">G</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T36">XBOOLE_1:36</a></i>;<br/>
<a NAME="E5:29"/>
 <a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<font color="Maroon">G</font> <a href="tarski.html#R1">c=</a>  <a href="goboard5.html#K4">right_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>
 
<b>by </b><i>, , <a class="ref" href="gobrd13.html#T34">GOBRD13:34</a></i>;<br/>
<a NAME="E6:29"/><b>then </b>
<span class="p1">(<span class="default"><a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="goboard5.html#K4">right_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E7:29"/><b>then </b><i><font color="Green">E45</font></i>: 
 <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="goboard5.html#K4">right_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font></span>)</span>
 
<b>by </b><i><a class="ref" href="tops_1.html#T48">TOPS_1:48</a></i>;<br/>
<a NAME="E8:29"/>
 <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="goboard5.html#K4">right_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="goboard9.html#K3">RightComp</a> <font color="Maroon">f</font>
 
<b>by </b><i>, <a class="ref" href="goboard9.html#T28">GOBOARD9:28</a></i>;<br/>
<a NAME="E9:29"/><b>then </b><i><font color="Green">E46</font></i>: 
 <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="goboard9.html#K3">RightComp</a> <font color="Maroon">f</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E10:29"/><i><font color="Green">E47</font></i>: 
 <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<font color="Maroon">G</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E11:29"/><i><font color="Green">E48</font></i>: 
 <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<font color="Maroon">G</font></span>)</span> <a href="xboole_0.html#R1">misses</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E12:29"/><i><font color="Green">E59</font></i>: 
 <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<font color="Maroon">G</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<font color="Maroon">G</font>
 
<b>by </b><i><a class="ref" href="tops_1.html#T44">TOPS_1:44</a></i>;<br/>
<a NAME="E13:29"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T39">XBOOLE_1:39</a></i>;<br/>
<a NAME="E14:29"/><b>then </b>
 <a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<font color="Maroon">G</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E15:29"/><b>then </b>
 <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<font color="Maroon">G</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan9.html#E1"><i><font color="Green">Lemma35</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E16:29"/><b>then </b><i><font color="Green">E60</font></i>: 
 <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<font color="Maroon">G</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T73">XBOOLE_1:73</a></i>;<br/>
<a NAME="E17:29"/><b>then </b>
 <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p2">(<span class="default"><a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<font color="Maroon">G</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="tops_1.html#T48">TOPS_1:48</a></i>;<br/>
<a NAME="E18:29"/><b>then </b>
 <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<font color="Maroon">G</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="tops_1.html#T45">TOPS_1:45</a></i>;<br/>
<a NAME="E19:29"/><b>then </b><i><font color="Green">E61</font></i>: 
<span class="p1">(<span class="default"><a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="jordan9.html#T3" target="_self">Th3</a>, <a class="ref" href="xboole_1.html#T68">XBOOLE_1:68</a></i>;<br/>
<a NAME="E20:29"/><i><font color="Green">E62</font></i>: 
<span class="p1">(<span class="default"><a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span> is <a href="connsp_1.html#V2">connected</a>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E21:29"/><i><font color="Green">E64</font></i>: 
 <a href="goboard9.html#K3">RightComp</a> <font color="Maroon">f</font> <a href="connsp_1.html#R4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K3">`</a> 
 
<b>by </b><i><a class="ref" href="goboard9.html#D2">GOBOARD9:def 2</a></i>;<br/>
<a NAME="E22:29"/>
<span class="p1">(<span class="default"><a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K3">`</a> 
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<b>hence </b><a NAME="E23:29"/>
<span class="p1">(<span class="default"><a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="goboard9.html#K3">RightComp</a> <font color="Maroon">f</font>
 <b>by </b><i>, , , , <a class="ref" href="goboard9.html#T6">GOBOARD9:6</a></i>;<br/>

<b>set </b><font color="Maroon">lc</font> = <span class="p1">(<span class="default"><a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span>;<br/>
<a NAME="E24:29"/><i><font color="Green">E65</font></i>: 
<span class="p1">(<span class="default"><a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="subset_1.html#T32">SUBSET_1:32</a></i>;<br/>
<a NAME="E25:29"/><i><font color="Green">E66</font></i>: 
<span class="p1">(<span class="default"><a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<font color="Maroon">G</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T36">XBOOLE_1:36</a></i>;<br/>
<a NAME="E26:29"/>
 <a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<font color="Maroon">G</font> <a href="tarski.html#R1">c=</a>  <a href="goboard5.html#K5">left_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>
 
<b>by </b><i>, , <a class="ref" href="gobrd13.html#T34">GOBRD13:34</a></i>;<br/>
<a NAME="E27:29"/><b>then </b>
<span class="p1">(<span class="default"><a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="goboard5.html#K5">left_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>
 
<b>by </b><i><a class="txt" href="jordan9.html#E1:29"><i><font color="Green">E41</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E28:29"/><b>then </b><i><font color="Green">E68</font></i>: 
 <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="goboard5.html#K5">left_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font></span>)</span>
 
<b>by </b><i><a class="ref" href="tops_1.html#T48">TOPS_1:48</a></i>;<br/>
<a NAME="E29:29"/>
 <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="goboard5.html#K5">left_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="goboard9.html#K2">LeftComp</a> <font color="Maroon">f</font>
 
<b>by </b><i>, <a class="ref" href="goboard9.html#T24">GOBOARD9:24</a></i>;<br/>
<a NAME="E30:29"/><b>then </b><i><font color="Green">E69</font></i>: 
 <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="goboard9.html#K2">LeftComp</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="txt" href="jordan9.html#E2:29"><i><font color="Green">E42</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E31:29"/><i><font color="Green">E70</font></i>: 
 <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<font color="Maroon">G</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E32:29"/><i><font color="Green">E71</font></i>: 
 <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<font color="Maroon">G</font></span>)</span> <a href="xboole_0.html#R1">misses</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E33:29"/><i><font color="Green">E72</font></i>: 
 <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<font color="Maroon">G</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<font color="Maroon">G</font>
 
<b>by </b><i><a class="ref" href="tops_1.html#T44">TOPS_1:44</a></i>;<br/>
<a NAME="E34:29"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T39">XBOOLE_1:39</a></i>;<br/>
<a NAME="E35:29"/><b>then </b>
 <a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<font color="Maroon">G</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E36:29"/><b>then </b>
 <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<font color="Maroon">G</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan9.html#E9:29"><i><font color="Green">E46</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E37:29"/><b>then </b><i><font color="Green">E73</font></i>: 
 <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<font color="Maroon">G</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan9.html#E7:29"><i><font color="Green">E45</font></i></a>, <a class="ref" href="xboole_1.html#T73">XBOOLE_1:73</a></i>;<br/>
<a NAME="E38:29"/><b>then </b>
 <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p2">(<span class="default"><a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<font color="Maroon">G</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="tops_1.html#T48">TOPS_1:48</a></i>;<br/>
<a NAME="E39:29"/><b>then </b>
 <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<font color="Maroon">G</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="tops_1.html#T45">TOPS_1:45</a></i>;<br/>
<a NAME="E40:29"/><b>then </b><i><font color="Green">E74</font></i>: 
<span class="p1">(<span class="default"><a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan9.html#E4:29"><i><font color="Green">E44</font></i></a>, <a class="txt" href="jordan9.html#E10:29"><i><font color="Green">E47</font></i></a>, <a class="ref" href="xboole_1.html#T68">XBOOLE_1:68</a></i>;<br/>
<a NAME="E41:29"/><i><font color="Green">E75</font></i>: 
<span class="p1">(<span class="default"><a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span> is <a href="connsp_1.html#V2">connected</a>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E42:29"/><i><font color="Green">E76</font></i>: 
 <a href="goboard9.html#K2">LeftComp</a> <font color="Maroon">f</font> <a href="connsp_1.html#R4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K3">`</a> 
 
<b>by </b><i><a class="ref" href="goboard9.html#D1">GOBOARD9:def 1</a></i>;<br/>
<a NAME="E43:29"/>
<span class="p1">(<span class="default"><a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K3">`</a> 
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<b>hence </b><a NAME="E44:29"/>
<span class="p1">(<span class="default"><a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="goboard9.html#K2">LeftComp</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="txt" href="jordan9.html#E3:29"><i><font color="Green">E43</font></i></a>, <a class="txt" href="jordan9.html#E11:29"><i><font color="Green">E48</font></i></a>, <a class="ref" href="jordan9.html#T4" target="_self">Th4</a>, <a class="ref" href="jordan9.html#T5" target="_self">Th5</a>, <a class="ref" href="goboard9.html#T6">GOBOARD9:6</a></i>;<br/>


</div>
