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<b>let </b><font color="Maroon">G</font> be  <a href="chord.html#V2" target="_self">complete</a> <a href="glib_000.html#NM2">_Graph</a>;<br/><b>let </b><font color="Maroon">S</font> be   <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/>





<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="74_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:74_1"/>
( <font color="Maroon">S</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">S</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="74_1_1"><b>suppose </b></a><a NAME="E1:74_1_1"/>
<font color="Maroon">S</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:74_1_1"/><b>then </b>
<font color="Maroon">H</font> <a href="glib_000.html#R5">==</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>hence </b><a NAME="E3:74_1_1"/>
<font color="Maroon">H</font> is <a href="chord.html#V2" target="_self">complete</a>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="74_1_2"><b>suppose </b></a><a NAME="E1:74_1_2"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">S</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:74_1_2"/><i><font color="Green">E33</font></i>: 
 <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <font color="Maroon">S</font>
 <b>by </b><i>, <a class="ref" href="glib_000.html#D39">GLIB_000:def 39</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">u</font> be   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">H</font>, <font color="Maroon">v</font> be   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">H</font>;<br/><b>assume </b><a NAME="E1:74_1_2_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">v</font>
 ;<br/><b>reconsider </b><font color="Maroon">u2</font> = <font color="Maroon">u</font> as   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font> <b>by </b><i><a class="ref" href="chord.html#T15" target="_self">Th15</a>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon">v2</font> = <font color="Maroon">v</font> as   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font> <b>by </b><i><a class="ref" href="chord.html#T15" target="_self">Th15</a>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><a NAME="E4:74_1_2_1"/>
<font color="Maroon">u2</font>,<font color="Maroon">v2</font> <a href="chord.html#R1" target="_self">are_adjacent</a> 
 <b>by </b><i>, </i>;<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="E6:74_1_2_1"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">e</font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">u2</font>,<font color="Maroon">v2</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="E7:74_1_2_1"/>
<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">H</font>
 <b>by </b><i><a class="ref" href="chord.html#T15" target="_self">Th15</a>, , </i>;<br/><b>hence </b><a NAME="E8:74_1_2_1"/>
<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/></div><b>end;</b></div><b>hence </b><a NAME="E4:74_1_2"/>
<font color="Maroon">H</font> is <a href="chord.html#V2" target="_self">complete</a>
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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