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<b>let </b><font color="Maroon" title="c1">G</font> be   <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a>;<br/><b>let </b><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font> be   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:95"/><i><font color="Green" title="E78">A1</font></i>: 
<font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">b</font>
 <b>and </b><br/><a NAME="E2:95"/><i><font color="Green" title="E79">A2</font></i>: 
not <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a> 
 ;<br/>

<b>let </b><font color="Maroon" title="c4">S</font> be    <a href="chord.html#M2" title="CHORD:mode.2">VertexSeparator</a> of <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>;<br/>

<b>assume </b><a NAME="E3:95"/><i><font color="Green" title="E80">A3</font></i>: 
<font color="Maroon" title="c4">S</font> is <a href="chord.html#V4" title="CHORD:attr.4">minimal</a>
 ;<br/>

<b>let </b><font color="Maroon" title="c5">x</font> be   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font>;<br/>

<b>assume </b><a NAME="E4:95"/><i><font color="Green" title="E81">A4</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">S</font>
 ;<br/>

<b>assume </b><a NAME="E5:95"/><i><font color="Green" title="E82">A5</font></i>: 
for <font color="Olive" title="b1">W</font> being   <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c1">G</font>  holds <br/>(  not <font color="Olive" title="b1">W</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> or  not <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">W</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  )
 ;<br/>

<b>set </b><font color="Maroon" title="c6">T</font> = <font color="Maroon" title="c4">S</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>;<br/>
<a NAME="E6:95"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E7:95"/><b>then </b><i><font color="Green" title="E83">A6</font></i>: 
<font color="Maroon" title="c4">S</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">S</font>
 
<b>by </b><i><a class="txt" href="chord.html#E4:95"><i><font color="Green" title="E81">A4</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E8:95"/><i><font color="Green" title="E84">A7</font></i>: 
<font color="Maroon" title="c4">S</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">S</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T36" title="XBOOLE_1:th.36">XBOOLE_1:36</a></i>;<br/>
<a NAME="E9:95"/><b>then </b><i><font color="Green" title="E85">A8</font></i>: 
( not <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">S</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; not <font color="Maroon" title="c3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">S</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 
<b>by </b><i><a class="txt" href="chord.html#E1:95"><i><font color="Green" title="E78">A1</font></i></a>, <a class="txt" href="chord.html#E2:95"><i><font color="Green" title="E79">A2</font></i></a>, <a class="ref" href="chord.html#D8" target="_self" title="CHORD:def.8">Def8</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" title="c7">W</font> be    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c1">G</font>;<br/><b>assume </b><a NAME="E1:95_1"/><i><font color="Green" title="E86">A9</font></i>: 
<font color="Maroon" title="c7">W</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>
 ;<br/><b>consider </b><font color="Maroon" title="c8">y</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E3:95_1"/><i><font color="Green" title="E87">A10</font></i>: 
( <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">S</font> &amp; <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">W</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  )
 <b>by </b><i><a class="txt" href="chord.html#E1:95"><i><font color="Green" title="E78">A1</font></i></a>, <a class="txt" href="chord.html#E2:95"><i><font color="Green" title="E79">A2</font></i></a>, <a class="txt" href="chord.html#E1:95_1"><i><font color="Green" title="E86">A9</font></i></a>, <a class="ref" href="chord.html#T71" target="_self" title="CHORD:th.71">Th71</a></i>;<br/><b>take </b><font color="Maroon" title="c9">y</font> = <font color="Maroon" title="c8">y</font>;<br/><a NAME="E4:95_1"/>
<font color="Maroon" title="c9">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">x</font>
 <b>by </b><i><a class="txt" href="chord.html#E5:95"><i><font color="Green" title="E82">A5</font></i></a>, <a class="txt" href="chord.html#E1:95_1"><i><font color="Green" title="E86">A9</font></i></a>, <a class="txt" href="chord.html#E3:95_1"><i><font color="Green" title="E87">A10</font></i></a></i>;<br/><a NAME="E5:95_1"/><b>then </b>
not <font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E6:95_1"/>
<font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">S</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="txt" href="chord.html#E3:95_1"><i><font color="Green" title="E87">A10</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><b>thus </b><a NAME="E7:95_1"/>
<font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">W</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 <b>by </b><i><a class="txt" href="chord.html#E3:95_1"><i><font color="Green" title="E87">A10</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E11:95"/><b>then </b>
<font color="Maroon" title="c4">S</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> is    <a href="chord.html#M2" title="CHORD:mode.2">VertexSeparator</a> of <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>
 
<b>by </b><i><a class="txt" href="chord.html#E1:95"><i><font color="Green" title="E78">A1</font></i></a>, <a class="txt" href="chord.html#E2:95"><i><font color="Green" title="E79">A2</font></i></a>, <a class="txt" href="chord.html#E9:95"><i><font color="Green" title="E85">A8</font></i></a>, <a class="ref" href="chord.html#T71" target="_self" title="CHORD:th.71">Th71</a></i>;<br/>
<b>hence </b><a NAME="E12:95"/>
contradiction
 <b>by </b><i><a class="txt" href="chord.html#E3:95"><i><font color="Green" title="E80">A3</font></i></a>, <a class="txt" href="chord.html#E7:95"><i><font color="Green" title="E83">A6</font></i></a>, <a class="txt" href="chord.html#E8:95"><i><font color="Green" title="E84">A7</font></i></a>, <a class="ref" href="chord.html#D9" target="_self" title="CHORD:def.9">Def9</a></i>;<br/>


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