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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="E1:84"/><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="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/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:84_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">S</font> is    <a href="chord.html#M2" target="_self">VertexSeparator</a> of <font color="Maroon">a</font>,<font color="Maroon">b</font>
 ;<br/><b>thus </b><a NAME="E2:84_1"/>
( not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> &amp; not <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> )
 <b>by </b><i>, , </i>;<br/><a NAME="E3:84_1"/><b>then </b><i><font color="Green">E53</font></i>: 
not <span class="p1">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">S</font> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><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="E4:84_1"/><i><font color="Green">E56</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/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:84_1_1"/><i><font color="Green">E59</font></i>: 
for <font color="Olive">x</font> being  <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font>  holds <br/>(  not <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> or  not <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> <a href="glib_001.html#K13">.vertices()</a>  )
 ;<br/><b>let </b><font color="Maroon">G2</font> be   <a href="glib_000.html#NM7">removeVertices</a> of <font color="Maroon">G</font>,<font color="Maroon">S</font>;<br/><a NAME="E2:84_1_1"/><i><font color="Green">E66</font></i>: 
 <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">G2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">S</font>
 <b>by </b><i><a class="txt" href="chord.html#E3:84"><i><font color="Green">E71</font></i></a>, <a class="ref" href="glib_000.html#D39">GLIB_000:def 39</a></i>;<br/><a NAME="E3:84_1_1"/><i><font color="Green">E67</font></i>: 
 <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">G2</font> <a href="hidden.html#R1">=</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#K25">the_Vertices_of</a> <font color="Maroon">G2</font></span>)</span>
 <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="84_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:84_1_1_1"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> <a href="glib_001.html#K13">.vertices()</a> 
 ;<br/><a NAME="E2:84_1_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> )
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E3:84_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">G2</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/></div><b>end;</b></div><a NAME="E5:84_1_1"/><b>then </b><i><font color="Green">E69</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><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><a NAME="E6:84_1_1"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">W</font> <a href="glib_001.html#K14">.edges()</a>  <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"><font color="Maroon">W</font> <a href="glib_001.html#K13">.vertices()</a> </span>)</span>
 <b>by </b><i><a class="ref" href="glib_001.html#T110">GLIB_001:110</a></i>;<br/><a NAME="E7:84_1_1"/>
<font color="Maroon">G</font> <a href="glib_000.html#K22">.edgesBetween</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K13">.vertices()</a> </span>)</span> <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#K25">the_Vertices_of</a> <font color="Maroon">G2</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="glib_000.html#T39">GLIB_000:39</a></i>;<br/><a NAME="E8:84_1_1"/><b>then </b>
<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><a class="ref" href="chord.html#T17" target="_self">Th17</a>, , <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>then 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="glib_001.html#T171">GLIB_001:171</a></i>;<br/><a NAME="E10:84_1_1"/>
( <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="E11:84_1_1"/><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="E12:84_1_1"/><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="E13:84_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="chord.html#E5:84_1_1"><i><font color="Green">E69</font></i></a>, <a class="txt" href="chord.html#E6:84_1_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="chord.html#E2:84_1_1"><i><font color="Green">E66</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E6:84_1"/>
 ex <font color="Olive">x</font> being  <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font> st <br/>( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> <a href="glib_001.html#K13">.vertices()</a>  )
 ;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E3:84"/><i><font color="Green">E71</font></i>: 
( not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> &amp; not <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> &amp; ( for <font color="Olive">W</font> being   <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">G</font>  st <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> holds <br/> ex <font color="Olive">x</font> being  <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font> st <br/>( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">W</font> <a href="glib_001.html#K13">.vertices()</a>  ) ) )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">G2</font> be   <a href="glib_000.html#NM7">removeVertices</a> of <font color="Maroon">G</font>,<font color="Maroon">S</font>;<br/><a NAME="E1:84_2"/>
not <span class="p1">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">S</font> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E2:84_2"/><b>then </b><i><font color="Green">E72</font></i>: 
 <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">G2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K6">\</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/><b>given </b><font color="Maroon">W</font> being    <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">G2</font><b> such that </b><a NAME="E3:84_2"/><i><font color="Green">E73</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>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">G</font> <b>by </b><i><a class="ref" href="glib_001.html#T168">GLIB_001:168</a></i>;<br/><a NAME="E5:84_2"/>
( <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="txt" href="chord.html#E2:84_2"><i><font color="Green">E72</font></i></a>, <a class="ref" href="glib_001.html#D23">GLIB_001:def 23</a></i>;<br/><a NAME="E6:84_2"/><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="E7:84_2"/><b>then </b><i><font color="Green">E74</font></i>: 
<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/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:84_2_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W2</font> <a href="glib_001.html#K13">.vertices()</a> 
 ;<br/><a NAME="E2:84_2_1_1"/><b>then </b>
 ex <font color="Olive">n</font> being <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font> &amp; <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="glib_001.html#T88">GLIB_001:88</a></i>;<br/><b>hence </b><a NAME="E3:84_2_1_1"/>
<font color="Maroon">x</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="glib_001.html#T88">GLIB_001:88</a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:84_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> <a href="glib_001.html#K13">.vertices()</a> 
 ;<br/><a NAME="E3:84_2_1"/><b>then </b>
 ex <font color="Olive">n</font> being <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">W2</font> &amp; <font color="Maroon">W2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="glib_001.html#T88">GLIB_001:88</a></i>;<br/><b>hence </b><a NAME="E4:84_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W2</font> <a href="glib_001.html#K13">.vertices()</a> 
 <b>by </b><i><a class="ref" href="glib_001.html#T88">GLIB_001:88</a></i>;<br/></div><b>end;</b></div><a NAME="E9:84_2"/><b>then </b><i><font color="Green">E75</font></i>: 
<font color="Maroon">W2</font> <a href="glib_001.html#K13">.vertices()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> <a href="glib_001.html#K13">.vertices()</a> 
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/><a NAME="E10:84_2"/>
for <font color="Olive">x</font> being  <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font>  holds <br/>(  not <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> or  not <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W2</font> <a href="glib_001.html#K13">.vertices()</a>  )
 <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a>, <a class="txt" href="chord.html#E7:84_2"><i><font color="Green">E74</font></i></a>, <a class="txt" href="chord.html#E3:84"><i><font color="Green">E71</font></i></a></i>;<br/><b>hence </b><a NAME="E11:84_2"/>
contradiction
 <b>by </b><i>, <a class="txt" href="chord.html#E3:84_2"><i><font color="Green">E73</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:84"/>
<font color="Maroon">S</font> 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>, , </i>;<br/>


</div>
