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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">G0</font> be    <a href="glib_000.html#M2">Subgraph</a> of <font color="Maroon">G</font>;<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">x</font> be   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font>;<br/><b>let </b><font color="Maroon">G1</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">G2</font> be   <a href="glib_000.html#NM5">inducedSubgraph</a> of <font color="Maroon">G</font>,<span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="subset_1.html#K4">\/</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span>;<br/>











<b>assume </b><b>that </b><br/><a NAME="E1:64"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">G1</font> is <a href="glib_002.html#V1">connected</a>
 <b>and </b><br/><a NAME="E2:64"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="chord.html#K2" target="_self">.AdjacentSet</a> <span class="p1">(<span class="default"><a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">G1</font></span>)</span>
 ;<br/>

<b>let </b><font color="Maroon">u</font> be   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G2</font>, <font color="Maroon">v</font> be   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G2</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="glib_002.html#D1">GLIB_002:def 1</a><br/>

<b>reconsider </b><font color="Maroon">Sx</font> = <font color="Maroon">S</font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span> as   <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/>
<a NAME="E4:64"/>
 <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">G2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Sx</font>
 
<b>by </b><i><a class="ref" href="glib_000.html#D39">GLIB_000:def 39</a></i>;<br/>
<a NAME="E5:64"/><b>then </b><i><font color="Green">E53</font></i>: 
( ( <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> or <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span> ) &amp; ( <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> or <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span> ) )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E6:64"/>
<font color="Maroon">S</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Sx</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E7:64"/><b>then </b><i><font color="Green">E56</font></i>: 
<font color="Maroon">G1</font> is   <a href="glib_000.html#NM5">inducedSubgraph</a> of <font color="Maroon">G2</font>,<font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="chord.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E8:64"/><i><font color="Green">E59</font></i>: 
 <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">G1</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/>
<b>then consider </b><font color="Maroon">xs</font> being   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E10:64"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">xs</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 <b>and </b><br/><a NAME="E11:64"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">x</font>,<font color="Maroon">xs</font> <a href="chord.html#R1" target="_self">are_adjacent</a> 
 <b>by </b><i>, <a class="ref" href="chord.html#T11" target="_self">Th11</a></i>;<br/>
<b>consider </b><font color="Maroon">e</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E13:64"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">e</font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">x</font>,<font color="Maroon">xs</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="E14:64"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">e</font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">xs</font>,<font color="Maroon">x</font>,<font color="Maroon">G</font>
 
<b>by </b><i>, <a class="ref" href="glib_000.html#T17">GLIB_000:17</a></i>;<br/>
<a NAME="E15:64"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E16:64"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Sx</font> &amp; <font color="Maroon">xs</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Sx</font> )
 
<b>by </b><i><a class="txt" href="chord.html#E7:64"><i><font color="Green">E56</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E17:64"/><b>then </b><i><font color="Green">E70</font></i>: 
<font color="Maroon">e</font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">xs</font>,<font color="Maroon">x</font>,<font color="Maroon">G2</font>
 
<b>by </b><i><a class="txt" href="chord.html#E8:64"><i><font color="Green">E59</font></i></a>, </i>;<br/>
<a NAME="E18:64"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">e</font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">x</font>,<font color="Maroon">xs</font>,<font color="Maroon">G2</font>
 
<b>by </b><i><a class="ref" href="chord.html#T15" target="_self">Th15</a>, <a class="ref" href="glib_000.html#T17">GLIB_000:17</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:64_1"/>
( ( <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> &amp; <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> ) or ( <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> &amp; <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> ) or ( <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> ) or ( <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> ) )
 </span><b>by </b><i><a class="txt" href="chord.html#E5:64"><i><font color="Green">E53</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_1"><b>suppose </b></a><a NAME="E1:64_1_1"/><i><font color="Green">E72</font></i>: 
( <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> &amp; <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> )
 ;<br/><div class="add"><a NAME="E2:64_1_1"/>
 <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">G1</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/><b>then consider </b><font color="Maroon">W</font> being    <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">G1</font><b> such that </b><br/><a NAME="E4:64_1_1"/><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">u</font>,<font color="Maroon">v</font>
 <b>by </b><i>, , <a class="ref" href="glib_002.html#D1">GLIB_002:def 1</a></i>;<br/><b>reconsider </b><font color="Maroon">W</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#T168">GLIB_001:168</a></i>;<br/><b>take </b>
<font color="Maroon">W</font>
;<br/><b>thus </b><a NAME="E6:64_1_1"/>
<font color="Maroon">W</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">u</font>,<font color="Maroon">v</font>
 <b>by </b><i>, <a class="ref" href="glib_001.html#T20">GLIB_001:20</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_2"><b>suppose </b></a><a NAME="E1:64_1_2"/><i><font color="Green">E74</font></i>: 
( <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> &amp; <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> )
 ;<br/><div class="add"><b>consider </b><font color="Maroon">W</font> being    <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">G1</font><b> such that </b><br/><a NAME="E3:64_1_2"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">W</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">u</font>,<font color="Maroon">xs</font>
 <b>by </b><i><a class="txt" href="chord.html#E7:64"><i><font color="Green">E56</font></i></a>, , , , <a class="ref" href="glib_002.html#D1">GLIB_002:def 1</a></i>;<br/><b>reconsider </b><font color="Maroon">W</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#T168">GLIB_001:168</a></i>;<br/><b>take </b><font color="Maroon">WW</font> = <font color="Maroon">W</font> <a href="glib_001.html#K7">.append</a> <span class="p1">(<span class="default"><font color="Maroon">G2</font> <a href="glib_001.html#K2">.walkOf</a> <font color="Maroon">xs</font>,<font color="Maroon">e</font>,<font color="Maroon">x</font></span>)</span>;<br/><a NAME="E5:64_1_2"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">W</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">u</font>,<font color="Maroon">xs</font>
 <b>by </b><i>, <a class="ref" href="glib_001.html#T20">GLIB_001:20</a></i>;<br/><a NAME="E6:64_1_2"/>
<font color="Maroon">G2</font> <a href="glib_001.html#K2">.walkOf</a> <font color="Maroon">xs</font>,<font color="Maroon">e</font>,<font color="Maroon">x</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">xs</font>,<font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="chord.html#T15" target="_self">Th15</a>, <a class="ref" href="glib_001.html#T16">GLIB_001:16</a></i>;<br/><b>hence </b><a NAME="E7:64_1_2"/>
<font color="Maroon">WW</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">u</font>,<font color="Maroon">v</font>
 <b>by </b><i>, , <a class="ref" href="glib_001.html#T32">GLIB_001:32</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_3"><b>suppose </b></a><a NAME="E1:64_1_3"/><i><font color="Green">E80</font></i>: 
( <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> )
 ;<br/><div class="add"><b>consider </b><font color="Maroon">W</font> being    <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">G1</font><b> such that </b><br/><a NAME="E3:64_1_3"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">W</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">xs</font>,<font color="Maroon">v</font>
 <b>by </b><i><a class="txt" href="chord.html#E7:64"><i><font color="Green">E56</font></i></a>, , , , <a class="ref" href="glib_002.html#D1">GLIB_002:def 1</a></i>;<br/><b>reconsider </b><font color="Maroon">W</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#T168">GLIB_001:168</a></i>;<br/><b>take </b><font color="Maroon">WW</font> = <span class="p1">(<span class="default"><font color="Maroon">G2</font> <a href="glib_001.html#K2">.walkOf</a> <font color="Maroon">x</font>,<font color="Maroon">e</font>,<font color="Maroon">xs</font></span>)</span> <a href="glib_001.html#K7">.append</a> <font color="Maroon">W</font>;<br/><a NAME="E5:64_1_3"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">W</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">xs</font>,<font color="Maroon">v</font>
 <b>by </b><i>, <a class="ref" href="glib_001.html#T20">GLIB_001:20</a></i>;<br/><a NAME="E6:64_1_3"/>
<font color="Maroon">G2</font> <a href="glib_001.html#K2">.walkOf</a> <font color="Maroon">x</font>,<font color="Maroon">e</font>,<font color="Maroon">xs</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">x</font>,<font color="Maroon">xs</font>
 <b>by </b><i>, <a class="ref" href="glib_001.html#T16">GLIB_001:16</a></i>;<br/><b>hence </b><a NAME="E7:64_1_3"/>
<font color="Maroon">WW</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">u</font>,<font color="Maroon">v</font>
 <b>by </b><i>, , <a class="ref" href="glib_001.html#T32">GLIB_001:32</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_4"><b>suppose </b></a><a NAME="E1:64_1_4"/>
( <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> )
 ;<br/><div class="add"><a NAME="E2:64_1_4"/><b>then </b>
<font color="Maroon">G2</font> <a href="glib_001.html#K1">.walkOf</a> <font color="Maroon">u</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">u</font>,<font color="Maroon">v</font>
 <b>by </b><i><a class="ref" href="glib_001.html#T14">GLIB_001:14</a></i>;<br/><b>hence </b><a NAME="E3:64_1_4"/>
 ex <font color="Olive">b<sub>1</sub></font> being   <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">G2</font> st <font color="Olive">b<sub>1</sub></font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">u</font>,<font color="Maroon">v</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

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