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





<b>assume </b><b>that </b><br/><a NAME="E1:5"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">W</font>
 <b>and </b><br/><a NAME="E2:5"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">W</font>
 <b>and </b><br/><a NAME="E3:5"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">W</font> is <a href="wellord1.html#V2">well-ordering</a>
 ;<br/>

<a NAME="E4:5"/>
<font color="Maroon">W</font> is <a href="relat_2.html#V4">antisymmetric</a>
 
<b>by </b><i>, <a class="ref" href="wellord1.html#D4">WELLORD1:def 4</a></i>;<br/>
<a NAME="E5:5"/><b>then </b><i><font color="Green">E40</font></i>: 
<font color="Maroon">W</font> <a href="relat_2.html#R4">is_antisymmetric_in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">W</font>
 
<b>by </b><i><a class="ref" href="relat_2.html#D12">RELAT_2:def 12</a></i>;<br/>
<b>assume </b><a NAME="E6:5"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">y</font>
 ;<br/>

<a NAME="E7:5"/><b>then </b><i><font color="Green">E41</font></i>: 
( <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#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> <font color="Maroon">W</font> )
 
<b>by </b><i><a class="ref" href="wellord1.html#D1">WELLORD1:def 1</a></i>;<br/>
<b>assume </b><a NAME="E8:5"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font>
 ;<br/>

<b>hence </b><a NAME="E9:5"/>
contradiction
 <b>by </b><i>, , , , <a class="ref" href="relat_2.html#D4">RELAT_2:def 4</a></i>;<br/>


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