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<b>let </b><font color="Maroon">c<sub>1</sub></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">c<sub>2</sub></font> be   <a href="abcmiz_0.html#NM1">type</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></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">c<sub>1</sub></font>;<br/>





<b>assume </b><a NAME="E1:122"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="abcmiz_0.html#R6" target="_self">is_properly_applicable_to</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>set </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>1</sub></font> <a href="abcmiz_0.html#K11" target="_self">@--&gt;</a> ;<br/>
<b>consider </b><font color="Maroon">c<sub>5</sub></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">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:122"/><i><font color="Green">E77</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="abcmiz_0.html#R6" target="_self">is_properly_applicable_to</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="abcmiz_0.html#K6" target="_self">ast</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="abcmiz_0.html#K6" target="_self">ast</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>and </b><br/><a NAME="E4:122"/><i><font color="Green">E78</font></i>: 
for <font color="Olive">b<sub>1</sub></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">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="abcmiz_0.html#R6" target="_self">is_properly_applicable_to</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="abcmiz_0.html#K6" target="_self">ast</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="abcmiz_0.html#K6" target="_self">ast</a> <font color="Maroon">c<sub>2</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="abcmiz_0.html#E1:122"><i><font color="Green">E76</font></i></a>, <a class="ref" href="abcmiz_0.html#T65" target="_self">Th65</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>6</sub></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">c<sub>1</sub></font><b> such that </b><br/><a NAME="E6:122"/><i><font color="Green">E79</font></i>: 
(  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="abcmiz_0.html#R5" target="_self">is_properly_applicable_to</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="abcmiz_0.html#E3:122"><i><font color="Green">E77</font></i></a>, <a class="ref" href="abcmiz_0.html#T66" target="_self">Th66</a></i>;<br/>
<a NAME="E7:122"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="abcmiz_0.html#R3" target="_self">is_applicable_to</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="abcmiz_0.html#E6:122"><i><font color="Green">E79</font></i></a>, <a class="ref" href="abcmiz_0.html#T58" target="_self">Th58</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>7</sub></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">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> as    <a href="rewrite1.html#M1">RedSequence</a> of <font color="Maroon">c<sub>1</sub></font> <a href="abcmiz_0.html#K11" target="_self">@--&gt;</a>  <b>by </b><i><a class="txt" href="abcmiz_0.html#E3:122"><i><font color="Green">E77</font></i></a>, <a class="txt" href="abcmiz_0.html#E4:122"><i><font color="Green">E78</font></i></a>, <a class="txt" href="abcmiz_0.html#E6:122"><i><font color="Green">E79</font></i></a>, <a class="ref" href="abcmiz_0.html#T72" target="_self">Th72</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>7</sub></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">c<sub>7</sub></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">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></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">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_5.html#T65">FINSEQ_5:65</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="abcmiz_0.html#K7" target="_self">apply</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="abcmiz_0.html#D19" target="_self">Def19</a></i>
<br/>.= 
<font color="Maroon">c<sub>6</sub></font> <a href="abcmiz_0.html#K8" target="_self">ast</a> <font color="Maroon">c<sub>2</sub></font>

<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="abcmiz_0.html#K6" target="_self">ast</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="abcmiz_0.html#E3:122"><i><font color="Green">E77</font></i></a>, <a class="txt" href="abcmiz_0.html#E6:122"><i><font color="Green">E79</font></i></a>, <a class="txt" href="abcmiz_0.html#E7:122"><i><font color="Green">E80</font></i></a>, <a class="ref" href="abcmiz_0.html#T57" target="_self">Th57</a></i>
;<br/>

<b>thus </b><font color="Maroon">c<sub>7</sub></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">c<sub>7</sub></font></span>)</span> = 
<font color="Maroon">c<sub>7</sub></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">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></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">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></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">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="abcmiz_0.html#D19" target="_self">Def19</a></i>
;<br/>


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