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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">S</font> be  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>;<br/><b>let </b><font color="Maroon">H</font> be   <a href="glib_000.html#NM5">inducedSubgraph</a> of <font color="Maroon">G</font>,<font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">u</font> be   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font>, <font color="Maroon">v</font> be   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font>;<br/><b>let </b><font color="Maroon">t</font> be   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">H</font>, <font color="Maroon">w</font> be   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">H</font>;<br/>









<b>assume </b><a NAME="E1:51"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> &amp; <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w</font> )
 ;<br/>

<a NAME="E2:51"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">S</font> <a href="hidden.html#R1">=</a>  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">H</font>
 
<b>by </b><i><a class="ref" href="glib_000.html#D39">GLIB_000:def 39</a></i>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:51_1"/>
<font color="Maroon">u</font>,<font color="Maroon">v</font> <a href="chord.html#R1" target="_self">are_adjacent</a> 
 ;<br/><b>then consider </b><font color="Maroon">e</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:51_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">e</font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">u</font>,<font color="Maroon">v</font>,<font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="chord.html#T4" target="_self">Th4</a></i>;<br/><a NAME="E4:51_1"/>
<font color="Maroon">e</font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">t</font>,<font color="Maroon">w</font>,<font color="Maroon">H</font>
 <b>by </b><i><a class="ref" href="chord.html#T7" target="_self">Th7</a>, <a class="ref" href="chord.html#T10" target="_self">Th10</a>, <a class="ref" href="chord.html#T8" target="_self">Th8</a>, </i>;<br/><b>hence </b><a NAME="E5:51_1"/>
<font color="Maroon">t</font>,<font color="Maroon">w</font> <a href="chord.html#R1" target="_self">are_adjacent</a> 
 <b>by </b><i><a class="ref" href="chord.html#T4" target="_self">Th4</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E4:51"/>
<font color="Maroon">t</font>,<font color="Maroon">w</font> <a href="chord.html#R1" target="_self">are_adjacent</a> 
 ;<br/>

<b>then consider </b><font color="Maroon">e</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:51"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">e</font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">t</font>,<font color="Maroon">w</font>,<font color="Maroon">H</font>
 <b>by </b><i><a class="ref" href="chord.html#T4" target="_self">Th4</a></i>;<br/>
<a NAME="E7:51"/>
<font color="Maroon">e</font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">u</font>,<font color="Maroon">v</font>,<font color="Maroon">G</font>
 
<b>by </b><i><a class="ref" href="chord.html#T10" target="_self">Th10</a>, <a class="ref" href="chord.html#T7" target="_self">Th7</a>, <a class="ref" href="glib_000.html#T75">GLIB_000:75</a></i>;<br/>
<b>hence </b><a NAME="E8:51"/>
<font color="Maroon">u</font>,<font color="Maroon">v</font> <a href="chord.html#R1" target="_self">are_adjacent</a> 
 <b>by </b><i><a class="ref" href="chord.html#T4" target="_self">Th4</a></i>;<br/>


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