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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><a NAME="E1:86"/><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> &amp; 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="E2:86"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">S</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>consider </b><font color="Maroon">G2</font> being   <a href="glib_000.html#NM7">removeVertices</a> of <font color="Maroon">G</font>,<font color="Maroon">S</font>;<br/>
<a NAME="E4:86"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">G2</font> is    <a href="glib_000.html#M3">inducedSubgraph</a> of <font color="Maroon">G</font>, <a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font>, <a href="glib_000.html#K7">the_Edges_of</a> <font color="Maroon">G</font>
 
<b>by </b><i>, <a class="ref" href="glib_000.html#T37">GLIB_000:37</a></i>;<br/>
<a NAME="E5:86"/>
 <a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font> <a href="tarski.html#R1">c=</a>  <a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font>
 
;<br/>
<a NAME="E6:86"/><b>then </b>
(  <a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font> is  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span> &amp;  <a href="glib_000.html#K7">the_Edges_of</a> <font color="Maroon">G</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font> <a href="glib_000.html#K22">.edgesBetween</a> <span class="p1">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="glib_000.html#T37">GLIB_000:37</a></i>;<br/>
<a NAME="E7:86"/><b>then </b><i><font color="Green">E56</font></i>: 
(  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">G2</font> <a href="hidden.html#R1">=</a>  <a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font> &amp;  <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">G2</font> <a href="hidden.html#R1">=</a>  <a href="glib_000.html#K7">the_Edges_of</a> <font color="Maroon">G</font> )
 
<b>by </b><i>, <a class="ref" href="glib_000.html#D39">GLIB_000:def 39</a></i>;<br/>
<b>given </b><font color="Maroon">W</font> being    <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">G</font><b> such that </b><a NAME="E8:86"/><i><font color="Green">E59</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/>

<a NAME="E9:86"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">W</font> <a href="glib_001.html#K13">.vertices()</a>  <a href="tarski.html#R1">c=</a>  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">G2</font>
 
<b>by </b><i/>;<br/>
<a NAME="E10:86"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">W</font> <a href="glib_001.html#K14">.edges()</a>  <a href="tarski.html#R1">c=</a>  <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">G2</font>
 
<b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">W2</font> = <font color="Maroon">W</font> as    <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">G2</font> <b>by </b><i>, <a class="ref" href="chord.html#T17" target="_self">Th17</a>, <a class="ref" href="glib_001.html#T171">GLIB_001:171</a></i>;<br/>
<a NAME="E12:86"/>
( <font color="Maroon">W</font> <a href="glib_001.html#K3">.first()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">W</font> <a href="glib_001.html#K4">.last()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> )
 
<b>by </b><i>, <a class="ref" href="glib_001.html#D23">GLIB_001:def 23</a></i>;<br/>
<a NAME="E13:86"/><b>then </b>
( <font color="Maroon">W2</font> <a href="glib_001.html#K3">.first()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">W2</font> <a href="glib_001.html#K4">.last()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> )
 
;<br/>
<a NAME="E14:86"/><b>then </b>
<font color="Maroon">W2</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>
 
<b>by </b><i><a class="ref" href="glib_001.html#D23">GLIB_001:def 23</a></i>;<br/>
<b>hence </b><a NAME="E15:86"/>
contradiction
 <b>by </b><i>, <a class="txt" href="chord.html#E9:86"><i><font color="Green">E66</font></i></a></i>;<br/>


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