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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="sgraph1.html#M1" target="_self">SimpleGraph</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a> ;<br/>







<b>assume </b><a NAME="E1:41"/><i><font color="Green">E23</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> &amp; 1 <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="sgraph1.html#K4" target="_self">degree</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>4</sub></font> )
 ;<br/>

<a NAME="E2:41"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="sgraph1.html#T27" target="_self">Th27</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>1</sub></font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="txt" href="sgraph1.html#E1:41"><i><font color="Green">E23</font></i></a></i>;<br/>
<b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>4</sub></font>;<br/>

<b>assume </b><a NAME="E4:41"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/>

<b>take </b>
<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K2">}</a></span>
;<br/>

<div><b>hereby </b>
<div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:41_1_1"/><i><font color="Green">E25</font></i>: 
not <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></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">c<sub>2</sub></font>
 ;<br/><b>consider </b><font color="Maroon">c<sub>7</sub></font> being  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:41_1_1"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>5</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> &amp; <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></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">c<sub>2</sub></font> ) </span>}</span>  &amp;  <a href="sgraph1.html#K4" target="_self">degree</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="txt" href="sgraph1.html#E1:41"><i><font color="Green">E23</font></i></a>, <a class="ref" href="sgraph1.html#T39" target="_self">Th39</a></i>;<br/><div><i><font color="Green">E27</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>8</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:41_1_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>9</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>5</sub></font><b> such that </b><br/><a NAME="E3:41_1_1_1"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> &amp; <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>9</sub></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">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="sgraph1.html#E3:41_1_1"><i><font color="Green">E26</font></i></a></i>;<br/><b>thus </b><a NAME="E4:41_1_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="sgraph1.html#E3:41_1_1_1"><i><font color="Green">E28</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E5:41_1_1"/><i><font color="Green">E28</font></i>: 
( not <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> &amp; not <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1_2"><b>proof </b></a><div class="add">

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:41_1_1_2_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>8</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>5</sub></font><b> such that </b><br/><a NAME="E3:41_1_1_2_1"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> &amp; <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>8</sub></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">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="sgraph1.html#E3:41_1_1"><i><font color="Green">E26</font></i></a></i>;<br/><a NAME="E4:41_1_1_2_1"/>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R2">in</a> the <a href="sgraph1.html#U1" target="_self">SEdges</a> of <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="sgraph1.html#E3:41_1_1_2_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="enumset1.html#T69">ENUMSET1:69</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>9</sub></font> being  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>4</sub></font><b> such that </b><br/><a NAME="E6:41_1_1_2_1"/><i><font color="Green">E30</font></i>: 
( <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> &amp;  <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> 2 )
 <b>by </b><i><a class="txt" href="sgraph1.html#E1:41"><i><font color="Green">E23</font></i></a>, <a class="ref" href="sgraph1.html#T9" target="_self">Th9</a></i>;<br/><b>thus </b><a NAME="E7:41_1_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="sgraph1.html#E6:41_1_1_2_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="card_1.html#T79">CARD_1:79</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:41_1_1_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>8</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>5</sub></font><b> such that </b><br/><a NAME="E4:41_1_1_2"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> &amp; <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>8</sub></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">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="sgraph1.html#E3:41_1_1"><i><font color="Green">E26</font></i></a></i>;<br/>
<b>thus </b><a NAME="E5:41_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="sgraph1.html#E1:41_1_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="sgraph1.html#E4:41_1_1_2"><i><font color="Green">E29</font></i></a></i>;<br/>


</div><b>end;</b></div><b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="tarski.html#K1">}</a></span> as  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  ;<br/><a NAME="E7:41_1_1"/><i><font color="Green">E29</font></i>: 
 <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> 1 <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="sgraph1.html#E5:41_1_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="card_2.html#T54">CARD_2:54</a></i>;<br/><a NAME="E8:41_1_1"/>
( <font color="Maroon">c<sub>8</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1_3"><b>proof </b></a><div class="add">

<a NAME="E1:41_1_1_3"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="sgraph1.html#E4:41_1_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E2:41_1_1_3"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="tarski.html#K1">}</a></span> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="sgraph1.html#E1:41"><i><font color="Green">E23</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E3:41_1_1_3"/><b>then </b>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>hence </b><a NAME="E4:41_1_1_3"/>
<font color="Maroon">c<sub>8</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="sgraph1.html#E1:41_1_1_3"><i><font color="Green">E30</font></i></a>, <a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/>

<a NAME="E5:41_1_1_3"/><i><font color="Green">E31</font></i>: 
not <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="txt" href="sgraph1.html#E4:41"><i><font color="Green">E24</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>assume </b><a NAME="E6:41_1_1_3"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<b>hence </b><a NAME="E7:41_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="sgraph1.html#E5:41_1_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="sgraph1.html#E5:41_1_1_3"><i><font color="Green">E31</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


</div><b>end;</b></div><a NAME="E9:41_1_1"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="xboole_0.html#R2">c&lt;</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D8">XBOOLE_0:def 8</a></i>;<br/><b>hence </b><a NAME="E10:41_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="sgraph1.html#E1:41"><i><font color="Green">E23</font></i></a>, <a class="txt" href="sgraph1.html#E3:41_1_1"><i><font color="Green">E26</font></i></a>, <a class="txt" href="sgraph1.html#E7:41_1_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="card_2.html#T67">CARD_2:67</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E2:41_1"/>
<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></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">c<sub>2</sub></font>
 ;<br/></div>
<b>end;</b></div>

<b>thus </b><a NAME="E6:41"/>
<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K2">}</a></span>
 ;<br/>


</div>
