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<b>let </b><font color="Maroon">T</font> be  <a href="orders_2.html#V2">reflexive</a> <a href="orders_2.html#V3">transitive</a> <a href="orders_2.html#V4">antisymmetric</a> <a href="lattice3.html#V1">with_suprema</a> <a href="abcmiz_0.html#V1" target="_self">Noetherian</a> non <a href="abcmiz_0.html#V4" target="_self">void</a> <a href="abcmiz_0.html#V9" target="_self">adj-structured</a>  <a href="abcmiz_0.html#L3" target="_self">TAS-structure</a> ;<br/>

<b>set </b><font color="Maroon">R</font> = <font color="Maroon">T</font> <a href="abcmiz_0.html#K11" target="_self">@--&gt;</a> ;<br/>
<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="relat_2.html#D2">RELAT_2:def 2</a>,<a class="ref" href="relat_2.html#D10">RELAT_2:def 10</a><br/>

<b>assume </b><a NAME="E1:118"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="abcmiz_0.html#K11" target="_self">@--&gt;</a> </span>)</span>
 ;<br/>

<b>assume </b><a NAME="E2:118"/><i><font color="Green">E32</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">T</font> <a href="abcmiz_0.html#K11" target="_self">@--&gt;</a> 
 ;<br/>

<b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="abcmiz_0.html#NM1">type</a> of <font color="Maroon">T</font> <b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>consider </b><font color="Maroon">a</font> being   <a href="abcmiz_0.html#NM2">adjective</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E5:118"/><i><font color="Green">E35</font></i>: 
not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="abcmiz_0.html#K2" target="_self">adjs</a> <font color="Maroon">x</font>
 <b>and </b><br/><a NAME="E6:118"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">a</font> <a href="abcmiz_0.html#R4" target="_self">is_properly_applicable_to</a> <font color="Maroon">x</font> &amp; <font color="Maroon">a</font> <a href="abcmiz_0.html#K5" target="_self">ast</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> )
 <b>by </b><i>, </i>;<br/>
<a NAME="E7:118"/>
<font color="Maroon">a</font> <a href="abcmiz_0.html#R1" target="_self">is_applicable_to</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="abcmiz_0.html#T8" target="_self">Th8</a>, </i>;<br/>
<b>hence </b><a NAME="E8:118"/>
contradiction
 <b>by </b><i><a class="ref" href="abcmiz_0.html#D9" target="_self">Def9</a>, <a class="ref" href="abcmiz_0.html#T8" target="_self">Th8</a>, </i>;<br/>


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