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

<b>let </b><font color="Maroon">X</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">g</font> be    <a href="sgraph1.html#M1" target="_self">SimpleGraph</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">v</font> be    <a href="hidden.html#M1">set</a> ;<br/>





<b>assume </b><a NAME="E1:39"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">g</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">VV</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">g</font> as  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="txt" href="sgraph1.html#E1"><i><font color="Green">Lemma15</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon">VV</font>
;<br/>

<b>consider </b><font color="Maroon">ww</font> being  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:39"/><i><font color="Green">E22</font></i>: 
( <font color="Maroon">ww</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">w</font> where <font color="Olive">w</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font> : ( <font color="Olive">w</font> <a href="hidden.html#R2">in</a> <font color="Maroon">VV</font> &amp; <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">v</font>,<font color="Olive">w</font></span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R2">in</a> the <a href="sgraph1.html#U1" target="_self">SEdges</a> of <font color="Maroon">g</font> ) </span>}</span>  &amp;  <a href="sgraph1.html#K4" target="_self">degree</a> <font color="Maroon">g</font>,<font color="Maroon">v</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">ww</font> )
 <b>by </b><i><a class="txt" href="sgraph1.html#E4:39"><i><font color="Green">E22</font></i></a></i>;<br/>
<a NAME="E5:39"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">ww</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">VV</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">e</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:39_1"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">ww</font>
 ;<br/>

<b>then consider </b><font color="Maroon">w</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E3:39_1"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w</font> &amp; <font color="Maroon">w</font> <a href="hidden.html#R2">in</a> <font color="Maroon">VV</font> &amp; <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">v</font>,<font color="Maroon">w</font></span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R2">in</a> the <a href="sgraph1.html#U1" target="_self">SEdges</a> of <font color="Maroon">g</font> )
 <b>by </b><i/>;<br/>
<b>thus </b><a NAME="E4:39_1"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">VV</font>
 <b>by </b><i><a class="ref" href="sgraph1.html#T5" target="_self">Th5</a></i>;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:39_2"/>
<font color="Maroon">ww</font> <a href="hidden.html#R1">=</a> <font color="Maroon">VV</font>
 ;<br/><b>then consider </b><font color="Maroon">w</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E3:39_2"/><i><font color="Green">E73</font></i>: 
( <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w</font> &amp; <font color="Maroon">w</font> <a href="hidden.html#R2">in</a> <font color="Maroon">VV</font> &amp; <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">v</font>,<font color="Maroon">w</font></span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R2">in</a> the <a href="sgraph1.html#U1" target="_self">SEdges</a> of <font color="Maroon">g</font> )
 <b>by </b><i>, </i>;<br/><a NAME="E4:39_2"/>
<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">v</font>,<font color="Maroon">v</font></span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">v</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="enumset1.html#T69">ENUMSET1:69</a></i>;<br/><b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:39_2"/><i><font color="Green">E74</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">VV</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">VV</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font> &amp; <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">v</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K2">}</a></span> )
 <b>by </b><i><a class="ref" href="sgraph1.html#T5" target="_self">Th5</a>, </i>;<br/><a NAME="E7:39_2"/>
( <font color="Maroon">v</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">y</font> )
 <b>by </b><i><a class="ref" href="sgraph1.html#T6" target="_self">Th6</a>, <a class="ref" href="zfmisc_1.html#T8">ZFMISC_1:8</a></i>;<br/><b>hence </b><a NAME="E8:39_2"/>
<font color="Maroon">ww</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">VV</font>
 <b>by </b><i><a class="ref" href="sgraph1.html#T6" target="_self">Th6</a></i>;<br/></div><b>end;</b></div>
<a NAME="E7:39"/><b>then </b>
<font color="Maroon">ww</font> <a href="xboole_0.html#R2">c&lt;</a> <font color="Maroon">VV</font>
 
<b>by </b><i><a class="txt" href="sgraph1.html#E5:39"><i><font color="Green">E25</font></i></a>, <a class="ref" href="xboole_0.html#D8">XBOOLE_0:def 8</a></i>;<br/>
<b>hence </b><a NAME="E8:39"/>
( <font color="Maroon">VV</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">g</font> &amp;  <a href="sgraph1.html#K4" target="_self">degree</a> <font color="Maroon">g</font>,<font color="Maroon">v</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">VV</font> )
 <b>by </b><i>, <a class="ref" href="card_2.html#T67">CARD_2:67</a></i>;<br/>


</div>
