<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">G</font> be   <a href="glib_000.html#NM2">_Graph</a>;<br/><b>let </b><font color="Maroon">W</font> be    <a href="glib_001.html#M3" target="_self">Walk</a> of <font color="Maroon">G</font>;<br/><b>let </b><font color="Maroon">e</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>







<b>set </b><font color="Maroon">W2</font> = <font color="Maroon">G</font> <a href="glib_001.html#K2" target="_self">.walkOf</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K4" target="_self">.last()</a> </span>)</span>,<font color="Maroon">e</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K4" target="_self">.last()</a> </span>)</span> <a href="glib_000.html#K30">.adj</a> <font color="Maroon">e</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">W3</font> = <font color="Maroon">W</font> <a href="glib_001.html#K10" target="_self">.addEdge</a> <font color="Maroon">e</font>;<br/>
<b>set </b><font color="Maroon">WV</font> = <font color="Maroon">W</font> <a href="glib_001.html#K13" target="_self">.vertices()</a> ;<br/>
<b>assume </b><a NAME="E1:195"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">e</font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">W</font> <a href="glib_001.html#K4" target="_self">.last()</a> ,<font color="Maroon">x</font>,<font color="Maroon">G</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font> as   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font> <b>by </b><i><a class="ref" href="glib_001.html#D27" target="_self">Def27</a>, <a class="ref" href="glib_000.html#T16">GLIB_000:16</a></i>;<br/>
<a NAME="E3:195"/><i><font color="Green">E40</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K4" target="_self">.last()</a> </span>)</span> <a href="glib_000.html#K30">.adj</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x'</font>
 
<b>by </b><i><a class="ref" href="glib_001.html#D27" target="_self">Def27</a>, <a class="ref" href="glib_000.html#T69">GLIB_000:69</a></i>;<br/>
<a NAME="E4:195"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_001.html#K2" target="_self">.walkOf</a> <span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K4" target="_self">.last()</a> </span>)</span>,<font color="Maroon">e</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K4" target="_self">.last()</a> </span>)</span> <a href="glib_000.html#K30">.adj</a> <font color="Maroon">e</font></span>)</span></span>)</span> <a href="glib_001.html#K3" target="_self">.first()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> <a href="glib_001.html#K4" target="_self">.last()</a> 
 
<b>by </b><i><a class="ref" href="glib_001.html#D27" target="_self">Def27</a>, </i>;<br/>
<a NAME="E5:195"/><b>then </b><i><font color="Green">E42</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K10" target="_self">.addEdge</a> <font color="Maroon">e</font></span>)</span> <a href="glib_001.html#K13" target="_self">.vertices()</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K13" target="_self">.vertices()</a> </span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="glib_001.html#K2" target="_self">.walkOf</a> <span class="p3">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K4" target="_self">.last()</a> </span>)</span>,<font color="Maroon">e</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K4" target="_self">.last()</a> </span>)</span> <a href="glib_000.html#K30">.adj</a> <font color="Maroon">e</font></span>)</span></span>)</span> <a href="glib_001.html#K13" target="_self">.vertices()</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="glib_001.html#D25" target="_self">Def25</a></i>;<br/>
<a NAME="E6:195"/><i><font color="Green">E43</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_001.html#K2" target="_self">.walkOf</a> <span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K4" target="_self">.last()</a> </span>)</span>,<font color="Maroon">e</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K4" target="_self">.last()</a> </span>)</span> <a href="glib_000.html#K30">.adj</a> <font color="Maroon">e</font></span>)</span></span>)</span> <a href="glib_001.html#K13" target="_self">.vertices()</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K4" target="_self">.last()</a> </span>)</span>,<font color="Maroon">x</font></span><a href="tarski.html#K2">}</a></span>
 
<b>by </b><i><a class="ref" href="glib_001.html#D27" target="_self">Def27</a>, <a class="ref" href="glib_001.html#D28" target="_self">Def28</a>, <a class="ref" href="glib_001.html#D24" target="_self">Def24</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="195_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:195_1_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K13" target="_self">.vertices()</a> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K4" target="_self">.last()</a> </span>)</span>,<font color="Maroon">x</font></span><a href="tarski.html#K2">}</a></span>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="195_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="195_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:195_1_1_1_1"/>
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> <a href="glib_001.html#K13" target="_self">.vertices()</a>  or <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K4" target="_self">.last()</a> </span>)</span>,<font color="Maroon">x</font></span><a href="tarski.html#K2">}</a></span> )
 </span><b>by </b><i><a class="ref" href="glib_001.html#D31" target="_self">Def31</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="195_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:195_1_1_1_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> <a href="glib_001.html#K13" target="_self">.vertices()</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:195_1_1_1_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K13" target="_self">.vertices()</a> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="195_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:195_1_1_1_1_2"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K4" target="_self">.last()</a> </span>)</span>,<font color="Maroon">x</font></span><a href="tarski.html#K2">}</a></span>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="195_1_1_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="195_1_1_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:195_1_1_1_1_2_1_1"/>
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> <a href="glib_001.html#K4" target="_self">.last()</a>  or <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> )
 </span><b>by </b><i><a class="txt" href="glib_001.html#E40"><i><font color="Green">Lemma142</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="195_1_1_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:195_1_1_1_1_2_1_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> <a href="glib_001.html#K4" target="_self">.last()</a> 
 ;<br/><div class="add"><a NAME="E2:195_1_1_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> <a href="glib_001.html#K13" target="_self">.vertices()</a> 
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:195_1_1_1_1_2_1_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K13" target="_self">.vertices()</a> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="195_1_1_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:195_1_1_1_1_2_1_1_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 ;<br/><div class="add"><a NAME="E2:195_1_1_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:195_1_1_1_1_2_1_1_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K13" target="_self">.vertices()</a> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:195_1_1_1_1_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K13" target="_self">.vertices()</a> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:195_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K13" target="_self">.vertices()</a> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 ;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:195_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K13" target="_self">.vertices()</a> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="195_1_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="195_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:195_1_2_1"/>
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> <a href="glib_001.html#K13" target="_self">.vertices()</a>  or <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> )
 </span><b>by </b><i><a class="ref" href="glib_001.html#D31" target="_self">Def31</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="195_1_2_1_1"><b>suppose </b></a><a NAME="E1:195_1_2_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> <a href="glib_001.html#K13" target="_self">.vertices()</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:195_1_2_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K13" target="_self">.vertices()</a> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K4" target="_self">.last()</a> </span>)</span>,<font color="Maroon">x</font></span><a href="tarski.html#K2">}</a></span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="195_1_2_1_2"><b>suppose </b></a><a NAME="E1:195_1_2_1_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 ;<br/><div class="add"><a NAME="E2:195_1_2_1_2"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E3:195_1_2_1_2"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K4" target="_self">.last()</a> </span>)</span>,<font color="Maroon">x</font></span><a href="tarski.html#K2">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><b>hence </b><a NAME="E4:195_1_2_1_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K13" target="_self">.vertices()</a> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K4" target="_self">.last()</a> </span>)</span>,<font color="Maroon">x</font></span><a href="tarski.html#K2">}</a></span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:195_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K13" target="_self">.vertices()</a> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K4" target="_self">.last()</a> </span>)</span>,<font color="Maroon">x</font></span><a href="tarski.html#K2">}</a></span>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:195"/>
<span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K10" target="_self">.addEdge</a> <font color="Maroon">e</font></span>)</span> <a href="glib_001.html#K13" target="_self">.vertices()</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K13" target="_self">.vertices()</a> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="glib_001.html#D29" target="_self">Def29</a>, <a class="ref" href="glib_001.html#D30" target="_self">Def30</a>, <a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>


</div>
