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<b>let </b><font color="Maroon">T</font> be  non <a href="struct_0.html#V3">empty</a> <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>set </b><font color="Maroon">Q</font> = the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">T</font>;<br/>
<a NAME="E1:119"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">T</font> <a href="abcmiz_0.html#K11" target="_self">@--&gt;</a>  <a href="tarski.html#R1">c=</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">T</font>
 
<b>by </b><i/>;<br/>
<a NAME="E2:119"/><b>then </b><i><font color="Green">E35</font></i>: 
 <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> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K3">field</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">T</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T31">RELAT_1:31</a></i>;<br/>
<a NAME="E3:119"/>
<font color="Maroon">T</font> <a href="abcmiz_0.html#K11" target="_self">@--&gt;</a>  is <a href="rewrite1.html#V1">co-well_founded</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="119_1"><b>proof </b></a><div class="add">

<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="rewrite1.html#D16">REWRITE1:def 16</a><br/>

<b>assume </b><a NAME="E1:119_1"/><i><font color="Green">E43</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">T</font> <a href="abcmiz_0.html#K11" target="_self">@--&gt;</a> </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="E2:119_1"/><b>then </b>
<font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K3">field</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">T</font>
 
<b>by </b><i><a class="ref" href="abcmiz_0.html#T8" target="_self">Th8</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="E4:119_1"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; ( for <font color="Olive">b</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b</font> holds <br/> not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Olive">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">T</font> ) )
 <b>by </b><i><a class="ref" href="abcmiz_0.html#T9" target="_self">Th9</a>, <a class="ref" href="rewrite1.html#D16">REWRITE1:def 16</a></i>;<br/>
<b>take </b>
<font color="Maroon">a</font>
;<br/>

<b>thus </b><a NAME="E5:119_1"/>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &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 <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> or  not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Olive">b<sub>1</sub></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>  ) ) )
 <b>by </b><i><a class="ref" href="abcmiz_0.html#D9" target="_self">Def9</a>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:119"/><b>then </b>
<font color="Maroon">T</font> <a href="abcmiz_0.html#K11" target="_self">@--&gt;</a>  is  <a href="rewrite1.html#V1">co-well_founded</a> <a href="relat_2.html#V2">irreflexive</a> <a href="relat_1.html#NM1">Relation</a>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E5:119"/>
<font color="Maroon">T</font> <a href="abcmiz_0.html#K11" target="_self">@--&gt;</a>  is <a href="rewrite1.html#V3">strongly-normalizing</a>
 ;<br/>


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