<?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>consider </b><font color="Maroon">Y</font> being  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E2:38"/><i><font color="Green">E19</font></i>: 
( ( for <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> iff ( <font color="Olive">z</font> <a href="hidden.html#R2">in</a> the <a href="sgraph1.html#U1" target="_self">SEdges</a> of <font color="Maroon">G</font> &amp; <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Olive">z</font> ) ) ) &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">Y</font> )
 <b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">ww</font> = <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> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</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> ;<br/>
<a NAME="E3:38"/><i><font color="Green">E22</font></i>: 
for <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">z</font> <a href="hidden.html#R2">in</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> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</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>  iff ( <font color="Olive">z</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font> &amp; <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">v</font>,<font color="Olive">z</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> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:38_1_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</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> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</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> 
 ;<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="E3:38_1_1"/><i><font color="Green">E25</font></i>: 
( <font color="Maroon">z</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> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</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> )
 ;<br/><b>thus </b><a NAME="E4:38_1_1"/>
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font> &amp; <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">v</font>,<font color="Maroon">z</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><a class="txt" href="sgraph1.html#E3:38_1_1"><i><font color="Green">E25</font></i></a></i>;<br/></div>
<b>end;</b></div>

<b>thus </b><a NAME="E2:38_1"/>
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font> &amp; <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">v</font>,<font color="Maroon">z</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> implies <font color="Maroon">z</font> <a href="hidden.html#R2">in</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> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</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>  )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:38_1_2"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font> &amp; <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">v</font>,<font color="Maroon">z</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> )
 ;<br/>

<a NAME="E2:38_1_2"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font> is  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>
 
<b>by </b><i><a class="txt" href="sgraph1.html#E1"><i><font color="Green">Lemma15</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:38_1_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</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> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</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> 
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>


</div><b>end;</b></div>
<a NAME="E4:38"/><i><font color="Green">E73</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font> is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i><a class="txt" href="sgraph1.html#E1"><i><font color="Green">Lemma15</font></i></a></i>;<br/>
<a NAME="E5:38"/><i><font color="Green">E74</font></i>: 
<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> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</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>  <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">z</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="E1:38_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</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> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</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> 
 ;<br/>

<b>hence </b><a NAME="E2:38_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">ww</font> = <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> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</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>  as  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i>, <a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/>
<b>take </b>
<font color="Maroon">ww</font>
;<br/>

<a NAME="E7:38"/>
<font color="Maroon">ww</font>,<font color="Maroon">Y</font> <a href="tarski.html#R2">are_equipotent</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_3"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">wwY</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">w</font>,<span class="p3"><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></span><a href="tarski.html#K4">]</a></span> 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">ww</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> <font color="Maroon">Y</font> ) </span>}</span> ;<br/>
<a NAME="E1:38_3"/><i><font color="Green">E75</font></i>: 
for <font color="Olive">w</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">w</font>,<span class="p2"><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></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">w</font>,<span class="p3"><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></span><a href="tarski.html#K4">]</a></span> 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">ww</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> <font color="Maroon">Y</font> ) </span>}</span>  iff ( <font color="Olive">w</font> <a href="hidden.html#R2">in</a> <font color="Maroon">ww</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> <font color="Maroon">Y</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">w</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:38_3_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">w</font>,<span class="p2"><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></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">w</font>,<span class="p3"><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></span><a href="tarski.html#K4">]</a></span> 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">ww</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> <font color="Maroon">Y</font> ) </span>}</span> 
 ;<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="E3:38_3_1_1"/><i><font color="Green">E76</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">w</font>,<span class="p2"><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></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">w'</font>,<span class="p2"><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></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">w'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">ww</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> <font color="Maroon">Y</font> )
 ;<br/><font color="Maroon">w</font> = 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">w'</font>,<span class="p2"><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></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K1">`1</a> 
<b>by </b><i>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
<br/>.= 
<font color="Maroon">w'</font>
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
;<br/><b>hence </b><a NAME="E5:38_3_1_1"/>
( <font color="Maroon">w</font> <a href="hidden.html#R2">in</a> <font color="Maroon">ww</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> <font color="Maroon">Y</font> )
 <b>by </b><i/>;<br/></div>
<b>end;</b></div>

<b>thus </b><a NAME="E2:38_3_1"/>
( <font color="Maroon">w</font> <a href="hidden.html#R2">in</a> <font color="Maroon">ww</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> <font color="Maroon">Y</font> implies <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">w</font>,<span class="p2"><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></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">w</font>,<span class="p3"><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></span><a href="tarski.html#K4">]</a></span> 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">ww</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> <font color="Maroon">Y</font> ) </span>}</span>  )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_3_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:38_3_1_2"/><i><font color="Green">E77</font></i>: 
( <font color="Maroon">w</font> <a href="hidden.html#R2">in</a> <font color="Maroon">ww</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> <font color="Maroon">Y</font> )
 ;<br/>

<a NAME="E2:38_3_1_2"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font> is  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>
 
<b>by </b><i><a class="txt" href="sgraph1.html#E1"><i><font color="Green">Lemma15</font></i></a></i>;<br/>
<a NAME="E3:38_3_1_2"/><b>then </b>
( <font color="Maroon">w</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font> &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> )
 
<b>by </b><i><a class="ref" href="sgraph1.html#T5" target="_self">Th5</a>, </i>;<br/>
<b>then reconsider </b><font color="Maroon">w</font> = <font color="Maroon">w</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font> ;<br/>
<a NAME="E5:38_3_1_2"/>
 ex <font color="Olive">z</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font> st <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">w</font>,<span class="p2"><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></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">z</font>,<span class="p2"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">v</font>,<font color="Olive">z</font></span><a href="tarski.html#K2">}</a></span></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">ww</font> &amp; <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">v</font>,<font color="Olive">z</font></span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E6:38_3_1_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">w</font>,<span class="p2"><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></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">w</font>,<span class="p3"><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></span><a href="tarski.html#K4">]</a></span> 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">ww</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> <font color="Maroon">Y</font> ) </span>}</span> 
 ;<br/>


</div><b>end;</b></div>


</div><b>end;</b></div>
<a NAME="E2:38_3"/><i><font color="Green">E79</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">ww</font> holds <br/> ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">w</font>,<span class="p3"><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></span><a href="tarski.html#K4">]</a></span> 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">ww</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> <font color="Maroon">Y</font> ) </span>}</span>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_3_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:38_3_2"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">ww</font>
 ;<br/>

<a NAME="E2:38_3_2"/><b>then </b><i><font color="Green">E81</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font> &amp; <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">v</font>,<font color="Maroon">x</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/>
<a NAME="E3:38_3_2"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">v</font>,<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>take </b>
<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">v</font>,<font color="Maroon">x</font></span><a href="tarski.html#K2">}</a></span>
;<br/>

<b>thus </b><a NAME="E4:38_3_2"/>
<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">v</font>,<font color="Maroon">x</font></span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>by </b><i>, <a class="txt" href="sgraph1.html#E10"><i><font color="Green">Lemma30</font></i></a>, </i>;<br/>

<b>hence </b><a NAME="E5:38_3_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<span class="p2"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">v</font>,<font color="Maroon">x</font></span><a href="tarski.html#K2">}</a></span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">w</font>,<span class="p3"><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></span><a href="tarski.html#K4">]</a></span> 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">ww</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> <font color="Maroon">Y</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="sgraph1.html#T6" target="_self">Th6</a>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:38_3"/><i><font color="Green">E84</font></i>: 
for <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> holds <br/> ex <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">ww</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">w</font>,<span class="p3"><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></span><a href="tarski.html#K4">]</a></span> 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">ww</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> <font color="Maroon">Y</font> ) </span>}</span>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_3_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:38_3_3"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 ;<br/>

<a NAME="E2:38_3_3"/><b>then </b><i><font color="Green">E90</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="sgraph1.html#U1" target="_self">SEdges</a> of <font color="Maroon">G</font> &amp; <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E3:38_3_3"/>
 ex <font color="Olive">w</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">w</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <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> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_3_3_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">v1</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">v2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E2:38_3_3_1"/><i><font color="Green">E93</font></i>: 
( <font color="Maroon">v1</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font> &amp; <font color="Maroon">v2</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font> &amp; <font color="Maroon">v1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">v2</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">v1</font>,<font color="Maroon">v2</font></span><a href="tarski.html#K2">}</a></span> )
 <b>by </b><i>, </i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_3_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:38_3_3_1_1"/>
( <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font> or <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v2</font> )
 </span><b>by </b><i>, , <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="38_3_3_1_1_1"><b>suppose </b></a><a NAME="E1:38_3_3_1_1_1"/><i><font color="Green">E95</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font>
 ;<br/><div class="add"><b>take </b>
<font color="Maroon">v2</font>
;<br/><b>thus </b><a NAME="E2:38_3_3_1_1_1"/>
( <font color="Maroon">v2</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">v</font>,<font color="Maroon">v2</font></span><a href="tarski.html#K2">}</a></span> )
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_3_3_1_1_2"><b>suppose </b></a><a NAME="E1:38_3_3_1_1_2"/><i><font color="Green">E96</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v2</font>
 ;<br/><div class="add"><b>take </b>
<font color="Maroon">v1</font>
;<br/><b>thus </b><a NAME="E2:38_3_3_1_1_2"/>
<font color="Maroon">v1</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font>
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E3:38_3_3_1_1_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">v</font>,<font color="Maroon">v1</font></span><a href="tarski.html#K2">}</a></span>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>then consider </b><font color="Maroon">w</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:38_3_3"/><i><font color="Green">E97</font></i>: 
( <font color="Maroon">w</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <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> )
 ;<br/>
<b>take </b>
<font color="Maroon">w</font>
;<br/>

<b>thus </b><a NAME="E6:38_3_3"/>
<font color="Maroon">w</font> <a href="hidden.html#R2">in</a> <font color="Maroon">ww</font>
 <b>by </b><i>, , </i>;<br/>

<b>hence </b><a NAME="E7:38_3_3"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">w</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">w</font>,<span class="p3"><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></span><a href="tarski.html#K4">]</a></span> 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">ww</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> <font color="Maroon">Y</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="sgraph1.html#T6" target="_self">Th6</a>, <a class="txt" href="sgraph1.html#E10"><i><font color="Green">Lemma30</font></i></a>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:38_3"/><i><font color="Green">E99</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font>, <font color="Olive">z</font>, <font color="Olive">u</font> being   <a href="hidden.html#M1">set</a>   st <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">w</font>,<span class="p3"><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></span><a href="tarski.html#K4">]</a></span> 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">ww</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> <font color="Maroon">Y</font> ) </span>}</span>  &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">z</font>,<font color="Olive">u</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">w</font>,<span class="p3"><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></span><a href="tarski.html#K4">]</a></span> 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">ww</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> <font color="Maroon">Y</font> ) </span>}</span>  holds <br/>( <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">z</font> iff <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">u</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_3_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">u</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:38_3_4"/><i><font color="Green">E127</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">w</font>,<span class="p3"><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></span><a href="tarski.html#K4">]</a></span> 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">ww</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> <font color="Maroon">Y</font> ) </span>}</span>  &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">z</font>,<font color="Maroon">u</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">w</font>,<span class="p3"><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></span><a href="tarski.html#K4">]</a></span> 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">ww</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> <font color="Maroon">Y</font> ) </span>}</span>  )
 ;<br/>

<b>then consider </b><font color="Maroon">w1</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E3:38_3_4"/><i><font color="Green">E129</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">w1</font>,<span class="p2"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">v</font>,<font color="Maroon">w1</font></span><a href="tarski.html#K2">}</a></span></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">w1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">ww</font> &amp; <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">v</font>,<font color="Maroon">w1</font></span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> )
 ;<br/>
<b>consider </b><font color="Maroon">w2</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E5:38_3_4"/><i><font color="Green">E131</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">z</font>,<font color="Maroon">u</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">w2</font>,<span class="p2"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">v</font>,<font color="Maroon">w2</font></span><a href="tarski.html#K2">}</a></span></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">w2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">ww</font> &amp; <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">v</font>,<font color="Maroon">w2</font></span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> )
 <b>by </b><i/>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:38_3_4_1"/><i><font color="Green">E132</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font>
 ;<br/><font color="Maroon">w1</font> = 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K1">`1</a> 
<b>by </b><i>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
<br/>.= 
<font color="Maroon">z</font>
<b>by </b><i><a class="ref" href="sgraph1.html#T9" target="_self">Th9</a>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">w2</font>,<span class="p2"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">v</font>,<font color="Maroon">w2</font></span><a href="tarski.html#K2">}</a></span></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K1">`1</a> 
<b>by </b><i>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
<br/>.= 
<font color="Maroon">w2</font>
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
;<br/><b>hence </b><font color="Maroon">y</font> = 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">w2</font>,<span class="p2"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">v</font>,<font color="Maroon">w2</font></span><a href="tarski.html#K2">}</a></span></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K2">`2</a> 
<b>by </b><i>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
<br/>.= 
<font color="Maroon">u</font>
<b>by </b><i>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
;<br/><br/></div>
<b>end;</b></div>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:38_3_4_2"/><i><font color="Green">E133</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font>
 ;<br/><i><font color="Green">E134</font></i>: <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">v</font>,<font color="Maroon">w1</font></span><a href="tarski.html#K2">}</a></span> = 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K2">`2</a> 
<b>by </b><i>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
<br/>.= 
<font color="Maroon">u</font>
<b>by </b><i><a class="ref" href="sgraph1.html#T9" target="_self">Th9</a>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">w2</font>,<span class="p2"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">v</font>,<font color="Maroon">w2</font></span><a href="tarski.html#K2">}</a></span></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K2">`2</a> 
<b>by </b><i>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">v</font>,<font color="Maroon">w2</font></span><a href="tarski.html#K2">}</a></span>
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
;<br/><a NAME="E3:38_3_4_2"/>
<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">v</font>,<font color="Maroon">w1</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="E4:38_3_4_2"/><b>then </b>
<font color="Maroon">v</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">w1</font>
 <b>by </b><i/>;<br/><a NAME="E5:38_3_4_2"/><b>then </b>
<font color="Maroon">w1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w2</font>
 <b>by </b><i><a class="ref" href="sgraph1.html#T10" target="_self">Th10</a>, <a class="ref" href="zfmisc_1.html#T10">ZFMISC_1:10</a></i>;<br/><b>hence </b><font color="Maroon">x</font> = 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">z</font>,<font color="Maroon">u</font></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K1">`1</a> 
<b>by </b><i>, , <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
<br/>.= 
<font color="Maroon">z</font>
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
;<br/><br/></div>
<b>end;</b></div>


</div><b>end;</b></div>
<b>take </b>
<span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">w</font>,<span class="p3"><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></span><a href="tarski.html#K4">]</a></span> 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">ww</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> <font color="Maroon">Y</font> ) </span>}</span> 
; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D6">TARSKI:def 6</a><br/>

<b>thus </b><a NAME="E5:38_3"/>
( ( for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>   holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">ww</font> or  ex <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">w</font>,<span class="p3"><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></span><a href="tarski.html#K4">]</a></span> 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">ww</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> <font color="Maroon">Y</font> ) </span>}</span>  ) ) ) &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>   holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> or  ex <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">ww</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">w</font>,<span class="p3"><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></span><a href="tarski.html#K4">]</a></span> 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">ww</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> <font color="Maroon">Y</font> ) </span>}</span>  ) ) ) &amp; ( for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font> being   <a href="hidden.html#M1">set</a>   holds <br/>(  not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">w</font>,<span class="p3"><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></span><a href="tarski.html#K4">]</a></span> 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">ww</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> <font color="Maroon">Y</font> ) </span>}</span>  or  not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>4</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">w</font>,<span class="p3"><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></span><a href="tarski.html#K4">]</a></span> 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">ww</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> <font color="Maroon">Y</font> ) </span>}</span>  or ( (  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> or <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>4</sub></font> ) &amp; (  not <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>4</sub></font> or <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> ) ) ) ) )
 <b>by </b><i>, , <a class="txt" href="sgraph1.html#E10"><i><font color="Green">Lemma30</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:38"/>
( <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> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</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="ref" href="card_1.html#T81">CARD_1:81</a></i>;<br/>


</div>
