<?xml version="1.0"?>
<div class="add">

<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>let </b><font color="Maroon">t</font> be   <a href="abcmiz_0.html#NM1">type</a> of <font color="Maroon">T</font>;<br/><b>let </b><font color="Maroon">A</font> be  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of the <a href="abcmiz_0.html#U1" target="_self">adjectives</a> of <font color="Maroon">T</font>;<br/>





<b>assume </b><a NAME="E1:122"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">A</font> <a href="abcmiz_0.html#R6" target="_self">is_properly_applicable_to</a> <font color="Maroon">t</font>
 ;<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>consider </b><font color="Maroon">A'</font> being   <a href="subset_1.html#NM2">Subset</a> of the <a href="abcmiz_0.html#U1" target="_self">adjectives</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E3:122"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">A'</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> &amp; <font color="Maroon">A'</font> <a href="abcmiz_0.html#R6" target="_self">is_properly_applicable_to</a> <font color="Maroon">t</font> &amp; <font color="Maroon">A</font> <a href="abcmiz_0.html#K6" target="_self">ast</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A'</font> <a href="abcmiz_0.html#K6" target="_self">ast</a> <font color="Maroon">t</font> )
 <b>and </b><br/><a NAME="E4:122"/><i><font color="Green">E43</font></i>: 
for <font color="Olive">C</font> being  <a href="subset_1.html#NM2">Subset</a> of the <a href="abcmiz_0.html#U1" target="_self">adjectives</a> of <font color="Maroon">T</font>  st <font color="Olive">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A'</font> &amp; <font color="Olive">C</font> <a href="abcmiz_0.html#R6" target="_self">is_properly_applicable_to</a> <font color="Maroon">t</font> &amp; <font color="Maroon">A</font> <a href="abcmiz_0.html#K6" target="_self">ast</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Olive">C</font> <a href="abcmiz_0.html#K6" target="_self">ast</a> <font color="Maroon">t</font> holds <br/><font color="Olive">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A'</font>
 <b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">s</font> being  <a href="funct_1.html#V2">one-to-one</a>  <a href="finseq_1.html#M2">FinSequence</a> of the <a href="abcmiz_0.html#U1" target="_self">adjectives</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E6:122"/><i><font color="Green">E46</font></i>: 
(  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A'</font> &amp; <font color="Maroon">s</font> <a href="abcmiz_0.html#R5" target="_self">is_properly_applicable_to</a> <font color="Maroon">t</font> )
 <b>by </b><i>, </i>;<br/>
<a NAME="E7:122"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">s</font> <a href="abcmiz_0.html#R3" target="_self">is_applicable_to</a> <font color="Maroon">t</font>
 
<b>by </b><i><a class="ref" href="abcmiz_0.html#D11" target="_self">Def11</a>, </i>;<br/>
<b>reconsider </b><font color="Maroon">p</font> =  <a href="finseq_5.html#K4">Rev</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K7" target="_self">apply</a> <font color="Maroon">s</font>,<font color="Maroon">t</font></span>)</span> as    <a href="rewrite1.html#M1">RedSequence</a> of <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#D11" target="_self">Def11</a>, <a class="ref" href="abcmiz_0.html#T8" target="_self">Th8</a></i>;<br/>
<b>take </b>
<font color="Maroon">p</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="rewrite1.html#D3">REWRITE1:def 3</a><br/>

<b>thus </b><font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> 1 = 
<span class="p1">(<span class="default"><a href="abcmiz_0.html#K7" target="_self">apply</a> <font color="Maroon">s</font>,<font color="Maroon">t</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="abcmiz_0.html#K7" target="_self">apply</a> <font color="Maroon">s</font>,<font color="Maroon">t</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_5.html#T65">FINSEQ_5:65</a></i>
<br/>.= 
<font color="Maroon">s</font> <a href="abcmiz_0.html#K8" target="_self">ast</a> <font color="Maroon">t</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">A</font> <a href="abcmiz_0.html#K6" target="_self">ast</a> <font color="Maroon">t</font>
<b>by </b><i>, <a class="ref" href="abcmiz_0.html#D11" target="_self">Def11</a>, <a class="ref" href="abcmiz_0.html#T10" target="_self">Th10</a>, </i>
;<br/>

<b>thus </b><font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span> = 
<font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="abcmiz_0.html#K7" target="_self">apply</a> <font color="Maroon">s</font>,<font color="Maroon">t</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_5.html#D3">FINSEQ_5:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="abcmiz_0.html#K7" target="_self">apply</a> <font color="Maroon">s</font>,<font color="Maroon">t</font></span>)</span> <a href="funct_1.html#K1">.</a> 1
<b>by </b><i><a class="ref" href="finseq_5.html#T65">FINSEQ_5:65</a></i>
<br/>.= 
<font color="Maroon">t</font>
<b>by </b><i/>
;<br/>


</div>
