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<b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">R</font> be   <a href="relat_1.html#NM1">Relation</a>;<br/>



<b>assume </b><a NAME="E1:31"/><i><font color="Green">E19</font></i>: 
for <font color="Olive">Y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">Y</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font> &amp; <font color="Olive">Y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  holds <br/> ex <font color="Olive">a</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> &amp; <font color="Maroon">R</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Olive">a</font> <a href="xboole_0.html#R1">misses</a> <font color="Olive">Y</font> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="wellord1.html#D2" target="_self">WELLORD1:def 2</a><br/>

<b>let </b><font color="Maroon">Y</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="wellord1.html#D2" target="_self">WELLORD1:def 2</a><br/>

<b>assume </b><a NAME="E2:31"/><i><font color="Green">E20</font></i>: 
( <font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K2" target="_self">|_2</a> <font color="Maroon">X</font></span>)</span> &amp; <font color="Maroon">Y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/>

<a NAME="E3:31"/>
 <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K2" target="_self">|_2</a> <font color="Maroon">X</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font>
 
<b>by </b><i><a class="txt" href="wellord1.html#E1:31"><i><font color="Green">E19</font></i></a></i>;<br/>
<a NAME="E4:31"/><b>then </b>
<font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font>
 
<b>by </b><i><a class="ref" href="wellord1.html#D2" target="_self">Def2</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>then consider </b><font color="Maroon">a</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:31"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">R</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">a</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">Y</font> )
 <b>by </b><i><a class="ref" href="wellord1.html#T2" target="_self">Th2</a>, <a class="ref" href="wellord1.html#D2" target="_self">Def2</a></i>;<br/>
<b>take </b>
<font color="Maroon">a</font>
;<br/>

<b>thus </b><a NAME="E7:31"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="wellord1.html#D3" target="_self">Def3</a></i>;<br/>

<b>assume </b><a NAME="E8:31"/>
not <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K2" target="_self">|_2</a> <font color="Maroon">X</font></span>)</span> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">a</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">Y</font>
 ;<br/>

<b>then consider </b><font color="Maroon">b</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E10:31"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K2" target="_self">|_2</a> <font color="Maroon">X</font></span>)</span> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">a</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E11:31"/>
<span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K2" target="_self">|_2</a> <font color="Maroon">X</font></span>)</span> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">a</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">R</font> <a href="wellord1.html#K1" target="_self">-Seg</a> <font color="Maroon">a</font>
 
<b>by </b><i><a class="txt" href="wellord1.html#E2:31"><i><font color="Green">E20</font></i></a></i>;<br/>
<b>hence </b><a NAME="E12:31"/>
contradiction
 <b>by </b><i><a class="ref" href="wellord1.html#D3" target="_self">Def3</a>, <a class="ref" href="wellord1.html#T5" target="_self">Th5</a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>


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