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<b>let </b><font color="Maroon">G</font> be   <a href="glib_000.html#NM2">_Graph</a>;<br/><b>let </b><font color="Maroon">a</font> be   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font>, <font color="Maroon">b</font> be   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:95"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font>
 <b>and </b><br/><a NAME="E2:95"/><i><font color="Green">E33</font></i>: 
not <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="chord.html#R1" target="_self">are_adjacent</a> 
 ;<br/>

<b>let </b><font color="Maroon">S</font> be    <a href="chord.html#M2" target="_self">VertexSeparator</a> of <font color="Maroon">a</font>,<font color="Maroon">b</font>;<br/>

<b>assume </b><a NAME="E3:95"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">S</font> is <a href="chord.html#V4" target="_self">minimal</a>
 ;<br/>

<b>let </b><font color="Maroon">x</font> be   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font>;<br/>

<b>assume </b><a NAME="E4:95"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 ;<br/>

<b>assume </b><a NAME="E5:95"/><i><font color="Green">E59</font></i>: 
for <font color="Olive">W</font> being   <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">G</font>  holds <br/>(  not <font color="Olive">W</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> or  not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">W</font> <a href="glib_001.html#K13">.vertices()</a>  )
 ;<br/>

<b>set </b><font color="Maroon">T</font> = <font color="Maroon">S</font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span>;<br/>
<a NAME="E6:95"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E7:95"/><b>then </b><i><font color="Green">E66</font></i>: 
<font color="Maroon">S</font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">S</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E8:95"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">S</font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T36">XBOOLE_1:36</a></i>;<br/>
<a NAME="E9:95"/><i><font color="Green">E68</font></i>: 
( not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span> &amp; not <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span> )
 
<b>by </b><i>, , , <a class="txt" href="chord.html#E7:95"><i><font color="Green">E66</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">W</font> be    <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">G</font>;<br/><b>assume </b><a NAME="E1:95_1"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">W</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>
 ;<br/><b>consider </b><font color="Maroon">y</font> being   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E3:95_1"/><i><font color="Green">E70</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> <a href="glib_001.html#K13">.vertices()</a>  )
 <b>by </b><i>, , , <a class="txt" href="chord.html#E9:95"><i><font color="Green">E68</font></i></a></i>;<br/><b>take </b><font color="Maroon">y</font> = <font color="Maroon">y</font>;<br/><a NAME="E4:95_1"/>
<font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font>
 <b>by </b><i>, , <a class="ref" href="chord.html#T18" target="_self">Th18</a></i>;<br/><a NAME="E5:95_1"/><b>then </b>
not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E6:95_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="chord.html#T18" target="_self">Th18</a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><b>thus </b><a NAME="E7:95_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> <a href="glib_001.html#K13">.vertices()</a> 
 <b>by </b><i><a class="ref" href="chord.html#T18" target="_self">Th18</a></i>;<br/></div><b>end;</b></div>
<a NAME="E11:95"/><b>then </b>
<font color="Maroon">S</font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span> is    <a href="chord.html#M2" target="_self">VertexSeparator</a> of <font color="Maroon">a</font>,<font color="Maroon">b</font>
 
<b>by </b><i>, , , <a class="txt" href="chord.html#E9:95"><i><font color="Green">E68</font></i></a></i>;<br/>
<b>hence </b><a NAME="E12:95"/>
contradiction
 <b>by </b><i>, , , </i>;<br/>


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