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

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:84_1_1_1"/>
<font color="Maroon">v</font> is <a href="glib_000.html#V11" target="_self">endvertex</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="E3:84_1_1_1"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">v</font> <a href="glib_000.html#K29" target="_self">.edgesInOut()</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">e</font></span><a href="tarski.html#K1">}</a></span> &amp; not <font color="Maroon">e</font> <a href="glib_000.html#R1" target="_self">Joins</a> <font color="Maroon">v</font>,<font color="Maroon">v</font>,<font color="Maroon">G</font> )
 <b>by </b><i/>;<br/><a NAME="E4:84_1_1_1"/><i><font color="Green">E29</font></i>: 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">e</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="glib_000.html#K27" target="_self">.edgesIn()</a> </span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="glib_000.html#K28" target="_self">.edgesOut()</a> </span>)</span>
 <b>by </b><i><a class="txt" href="glib_000.html#E17"><i><font color="Green">Lemma38</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="84_1_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="E1:84_1_1_1_1_1"/>
( ( <font color="Maroon">v</font> <a href="glib_000.html#K27" target="_self">.edgesIn()</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">e</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">v</font> <a href="glib_000.html#K28" target="_self">.edgesOut()</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">e</font></span><a href="tarski.html#K1">}</a></span> ) or ( <font color="Maroon">v</font> <a href="glib_000.html#K27" target="_self">.edgesIn()</a>  <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">v</font> <a href="glib_000.html#K28" target="_self">.edgesOut()</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">e</font></span><a href="tarski.html#K1">}</a></span> ) or ( <font color="Maroon">v</font> <a href="glib_000.html#K27" target="_self">.edgesIn()</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">e</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">v</font> <a href="glib_000.html#K28" target="_self">.edgesOut()</a>  <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  ) )
 </span><b>by </b><i>, <a class="ref" href="zfmisc_1.html#T43">ZFMISC_1:43</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="84_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:84_1_1_1_1_1_1"/>
( <font color="Maroon">v</font> <a href="glib_000.html#K27" target="_self">.edgesIn()</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">e</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">v</font> <a href="glib_000.html#K28" target="_self">.edgesOut()</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">e</font></span><a href="tarski.html#K1">}</a></span> )
 ;<br/><div class="add"><a NAME="E2:84_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E30</font></i>: 
( <font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">v</font> <a href="glib_000.html#K27" target="_self">.edgesIn()</a>  &amp; <font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">v</font> <a href="glib_000.html#K28" target="_self">.edgesOut()</a>  )
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E3:84_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E41</font></i>: 
( <font color="Maroon">e</font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K7" target="_self">the_Edges_of</a> <font color="Maroon">G</font> &amp; <span class="p1">(<span class="default"><a href="glib_000.html#K11" target="_self">the_Target_of</a> <font color="Maroon">G</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> )
 <b>by </b><i><a class="txt" href="glib_000.html#E14"><i><font color="Green">Lemma33</font></i></a></i>;<br/><a NAME="E4:84_1_1_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="glib_000.html#K10" target="_self">the_Source_of</a> <font color="Maroon">G</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E5:84_1_1_1_1_1_1"/>
<font color="Maroon">v</font> <a href="glib_000.html#K36" target="_self">.degree()</a>  <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="glib_000.html#E17"><i><font color="Green">Lemma38</font></i></a>, <a class="txt" href="glib_000.html#E3:84_1_1_1_1_1_1"><i><font color="Green">E41</font></i></a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="84_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:84_1_1_1_1_1_2"/>
( <font color="Maroon">v</font> <a href="glib_000.html#K27" target="_self">.edgesIn()</a>  <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">v</font> <a href="glib_000.html#K28" target="_self">.edgesOut()</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">e</font></span><a href="tarski.html#K1">}</a></span> )
 ;<br/><div class="add"><a NAME="E2:84_1_1_1_1_1_2"/><b>then </b>
(  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="glib_000.html#K27" target="_self">.edgesIn()</a> </span>)</span> <a href="hidden.html#R1">=</a> 0 &amp;  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="glib_000.html#K28" target="_self">.edgesOut()</a> </span>)</span> <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i><a class="ref" href="card_1.html#T78">CARD_1:78</a>, <a class="ref" href="card_1.html#T79">CARD_1:79</a></i>;<br/><a NAME="E3:84_1_1_1_1_1_2"/><b>then </b>
( <font color="Maroon">v</font> <a href="glib_000.html#K33" target="_self">.inDegree()</a>  <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">v</font> <a href="glib_000.html#K34" target="_self">.outDegree()</a>  <a href="hidden.html#R1">=</a> 1 )
 ;<br/><a NAME="E4:84_1_1_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="glib_000.html#K33" target="_self">.inDegree()</a> </span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="glib_000.html#K34" target="_self">.outDegree()</a> </span>)</span> <a href="hidden.html#R1">=</a> 0 <a href="nat_1.html#K1">+</a> 1
 ;<br/><b>hence </b><a NAME="E5:84_1_1_1_1_1_2"/>
<font color="Maroon">v</font> <a href="glib_000.html#K36" target="_self">.degree()</a>  <a href="hidden.html#R1">=</a> 1
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="84_1_1_1_1_1_3"><b>suppose </b></a><a NAME="E1:84_1_1_1_1_1_3"/>
( <font color="Maroon">v</font> <a href="glib_000.html#K27" target="_self">.edgesIn()</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">e</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">v</font> <a href="glib_000.html#K28" target="_self">.edgesOut()</a>  <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/><div class="add"><a NAME="E2:84_1_1_1_1_1_3"/><b>then </b>
(  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="glib_000.html#K27" target="_self">.edgesIn()</a> </span>)</span> <a href="hidden.html#R1">=</a> 1 &amp;  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="glib_000.html#K28" target="_self">.edgesOut()</a> </span>)</span> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i><a class="ref" href="card_1.html#T78">CARD_1:78</a>, <a class="ref" href="card_1.html#T79">CARD_1:79</a></i>;<br/><a NAME="E3:84_1_1_1_1_1_3"/><b>then </b>
( <font color="Maroon">v</font> <a href="glib_000.html#K33" target="_self">.inDegree()</a>  <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">v</font> <a href="glib_000.html#K34" target="_self">.outDegree()</a>  <a href="hidden.html#R1">=</a> 0 )
 ;<br/><a NAME="E4:84_1_1_1_1_1_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="glib_000.html#K33" target="_self">.inDegree()</a> </span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="glib_000.html#K34" target="_self">.outDegree()</a> </span>)</span> <a href="hidden.html#R1">=</a> 1 <a href="nat_1.html#K1">+</a> 0
 ;<br/><b>hence </b><a NAME="E5:84_1_1_1_1_1_3"/>
<font color="Maroon">v</font> <a href="glib_000.html#K36" target="_self">.degree()</a>  <a href="hidden.html#R1">=</a> 1
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E6:84_1_1_1"/>
<font color="Maroon">v</font> <a href="glib_000.html#K36" target="_self">.degree()</a>  <a href="hidden.html#R1">=</a> 1
 ;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:84_1_1"/>
<font color="Maroon">v</font> <a href="glib_000.html#K36" target="_self">.degree()</a>  <a href="hidden.html#R1">=</a> 1
 ;<br/>

<a NAME="E3:84_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="glib_000.html#K33" target="_self">.inDegree()</a> </span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="glib_000.html#K34" target="_self">.outDegree()</a> </span>)</span> <a href="hidden.html#R1">=</a> 1
 
;<br/>
<a NAME="E4:84_1_1"/><b>then </b><i><font color="Green">E42</font></i>: 
<span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="glib_000.html#K27" target="_self">.edgesIn()</a> </span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="glib_000.html#K34" target="_self">.outDegree()</a> </span>)</span> <a href="hidden.html#R1">=</a> 1
 
;<br/>
<a NAME="E5:84_1_1"/><b>then </b><i><font color="Green">E43</font></i>: 
<span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="glib_000.html#K27" target="_self">.edgesIn()</a> </span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="glib_000.html#K28" target="_self">.edgesOut()</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1
 
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_1_1_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="84_1_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:84_1_1_2_1"/>
(  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="glib_000.html#K27" target="_self">.edgesIn()</a> </span>)</span> <a href="hidden.html#R1">=</a> 0 or  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="glib_000.html#K27" target="_self">.edgesIn()</a> </span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="84_1_1_2_1_1"><b>suppose </b></a><a NAME="E1:84_1_1_2_1_1"/><i><font color="Green">E64</font></i>: 
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="glib_000.html#K27" target="_self">.edgesIn()</a> </span>)</span> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:84_1_1_2_1_1"/><b>then </b><i><font color="Green">E65</font></i>: 
<font color="Maroon">v</font> <a href="glib_000.html#K27" target="_self">.edgesIn()</a>  <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="card_2.html#T59">CARD_2:59</a></i>;<br/><a NAME="E3:84_1_1_2_1_1"/>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="glib_000.html#K28" target="_self">.edgesOut()</a> </span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="glib_000.html#E17"><i><font color="Green">Lemma38</font></i></a>, </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="E5:84_1_1_2_1_1"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">v</font> <a href="glib_000.html#K28" target="_self">.edgesOut()</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">e</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="card_2.html#T60">CARD_2:60</a></i>;<br/><a NAME="E6:84_1_1_2_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="glib_000.html#K27" target="_self">.edgesIn()</a> </span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="glib_000.html#K28" target="_self">.edgesOut()</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">e</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, <a class="txt" href="glib_000.html#E4:84_1_1"><i><font color="Green">E42</font></i></a></i>;<br/><a NAME="E7:84_1_1_2_1_1"/><b>then </b><i><font color="Green">E67</font></i>: 
<font color="Maroon">v</font> <a href="glib_000.html#K29" target="_self">.edgesInOut()</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">e</font></span><a href="tarski.html#K1">}</a></span>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_1_1_2_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:84_1_1_2_1_1_1"/>
<font color="Maroon">e</font> <a href="glib_000.html#R1" target="_self">Joins</a> <font color="Maroon">v</font>,<font color="Maroon">v</font>,<font color="Maroon">G</font>
 ;<br/><a NAME="E2:84_1_1_2_1_1_1"/><b>then </b>
( <font color="Maroon">e</font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K7" target="_self">the_Edges_of</a> <font color="Maroon">G</font> &amp; <span class="p1">(<span class="default"><a href="glib_000.html#K10" target="_self">the_Source_of</a> <font color="Maroon">G</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> &amp; <span class="p1">(<span class="default"><a href="glib_000.html#K11" target="_self">the_Target_of</a> <font color="Maroon">G</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:84_1_1_2_1_1_1"/>
contradiction
 <b>by </b><i>, <a class="txt" href="glib_000.html#E14"><i><font color="Green">Lemma33</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E9:84_1_1_2_1_1"/>
<font color="Maroon">v</font> is <a href="glib_000.html#V11" target="_self">endvertex</a>
 <b>by </b><i><a class="txt" href="glib_000.html#E5:84_1_1"><i><font color="Green">E43</font></i></a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="84_1_1_2_1_2"><b>suppose </b></a><a NAME="E1:84_1_1_2_1_2"/>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="glib_000.html#K27" target="_self">.edgesIn()</a> </span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:84_1_1_2_1_2"/><b>then </b>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="glib_000.html#K27" target="_self">.edgesIn()</a> </span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T19">NAT_1:19</a></i>;<br/><a NAME="E3:84_1_1_2_1_2"/><b>then </b><i><font color="Green">E103</font></i>: 
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="glib_000.html#K27" target="_self">.edgesIn()</a> </span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E4:84_1_1_2_1_2"/>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="glib_000.html#K27" target="_self">.edgesIn()</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1
 <b>by </b><i><a class="txt" href="glib_000.html#E17"><i><font color="Green">Lemma38</font></i></a>, <a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E5:84_1_1_2_1_2"/><b>then </b><i><font color="Green">E104</font></i>: 
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="glib_000.html#K27" target="_self">.edgesIn()</a> </span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></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="E7:84_1_1_2_1_2"/><i><font color="Green">E105</font></i>: 
<font color="Maroon">v</font> <a href="glib_000.html#K27" target="_self">.edgesIn()</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">e</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="card_2.html#T60">CARD_2:60</a></i>;<br/><a NAME="E8:84_1_1_2_1_2"/>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="glib_000.html#K28" target="_self">.edgesOut()</a> </span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, </i>;<br/><a NAME="E9:84_1_1_2_1_2"/><b>then </b><i><font color="Green">E106</font></i>: 
<font color="Maroon">v</font> <a href="glib_000.html#K28" target="_self">.edgesOut()</a>  <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="card_2.html#T59">CARD_2:59</a></i>;<br/><a NAME="E10:84_1_1_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="glib_000.html#K27" target="_self">.edgesIn()</a> </span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="glib_000.html#K28" target="_self">.edgesOut()</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">e</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="glib_000.html#E4:84_1_1"><i><font color="Green">E42</font></i></a></i>;<br/><a NAME="E11:84_1_1_2_1_2"/><b>then </b><i><font color="Green">E107</font></i>: 
<font color="Maroon">v</font> <a href="glib_000.html#K29" target="_self">.edgesInOut()</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">e</font></span><a href="tarski.html#K1">}</a></span>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_1_1_2_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:84_1_1_2_1_2_1"/>
<font color="Maroon">e</font> <a href="glib_000.html#R1" target="_self">Joins</a> <font color="Maroon">v</font>,<font color="Maroon">v</font>,<font color="Maroon">G</font>
 ;<br/><a NAME="E2:84_1_1_2_1_2_1"/><b>then </b>
( <font color="Maroon">e</font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K7" target="_self">the_Edges_of</a> <font color="Maroon">G</font> &amp; <span class="p1">(<span class="default"><a href="glib_000.html#K10" target="_self">the_Source_of</a> <font color="Maroon">G</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> &amp; <span class="p1">(<span class="default"><a href="glib_000.html#K11" target="_self">the_Target_of</a> <font color="Maroon">G</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:84_1_1_2_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="glib_000.html#E5:84_1_1"><i><font color="Green">E43</font></i></a>, </i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E13:84_1_1_2_1_2"/>
<font color="Maroon">v</font> is <a href="glib_000.html#V11" target="_self">endvertex</a>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E7:84_1_1"/>
<font color="Maroon">v</font> is <a href="glib_000.html#V11" target="_self">endvertex</a>
 ;<br/>


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