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

<b>let </b><font color="Maroon">c<sub>1</sub></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#V14" target="_self">commutative</a>  <a href="abcmiz_0.html#L3" target="_self">TAS-structure</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</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>4</sub></font> be   <a href="abcmiz_0.html#NM2">adjective</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>





<b>set </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>1</sub></font> <a href="abcmiz_0.html#K11" target="_self">@--&gt;</a> ;<br/>
<a NAME="E1:132"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="abcmiz_0.html#K11" target="_self">@--&gt;</a>  is  <a href="rewrite1.html#V3">strongly-normalizing</a> <a href="rewrite1.html#V4">with_UN_property</a> <a href="rewrite1.html#V8">with_Church-Rosser_property</a> <a href="relat_1.html#NM1">Relation</a>
 
<b>by </b><i><a class="ref" href="abcmiz_0.html#T70" target="_self">Th70</a>, <a class="ref" href="abcmiz_0.html#T78" target="_self">Th78</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="abcmiz_0.html#NM1">type</a> of <font color="Maroon">c<sub>1</sub></font> : ex <font color="Olive">b<sub>1</sub></font> being <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> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="abcmiz_0.html#R6" target="_self">is_properly_applicable_to</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="abcmiz_0.html#K6" target="_self">ast</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> ) </span>}</span> ;<br/>
<b>assume </b><a NAME="E2:132"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="abcmiz_0.html#R4" target="_self">is_properly_applicable_to</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="abcmiz_0.html#K5" target="_self">ast</a> <font color="Maroon">c<sub>2</sub></font> <a href="orders_2.html#R3">&lt;=</a>  <a href="abcmiz_0.html#K12" target="_self">radix</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>7</sub></font> being  <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><b> such that </b><br/><a NAME="E4:132"/><i><font color="Green">E86</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="abcmiz_0.html#R6" target="_self">is_properly_applicable_to</a> <font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K12" target="_self">radix</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="abcmiz_0.html#K6" target="_self">ast</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K13">"\/"</a> <span class="p2">(<span class="default"><a href="abcmiz_0.html#K12" target="_self">radix</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="abcmiz_0.html#K12" target="_self">radix</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="ref" href="abcmiz_0.html#D30" target="_self">Def30</a></i>;<br/>
<a NAME="E5:132"/>
 <a href="rewrite1.html#K2">nf</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="abcmiz_0.html#K11" target="_self">@--&gt;</a> </span>)</span> <a href="rewrite1.html#R4">is_a_normal_form_of</a> <font color="Maroon">c<sub>3</sub></font>,<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#E1:132"><i><font color="Green">E85</font></i></a>, <a class="ref" href="rewrite1.html#T55">REWRITE1:55</a></i>;<br/>
<a NAME="E6:132"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="abcmiz_0.html#K11" target="_self">@--&gt;</a>  <a href="rewrite1.html#R1">reduces</a> <font color="Maroon">c<sub>3</sub></font>, <a href="rewrite1.html#K2">nf</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="abcmiz_0.html#K11" target="_self">@--&gt;</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="rewrite1.html#D6">REWRITE1:def 6</a></i>;<br/>
<a NAME="E7:132"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="abcmiz_0.html#K11" target="_self">@--&gt;</a>  <a href="rewrite1.html#R1">reduces</a> <font color="Maroon">c<sub>3</sub></font>, <a href="abcmiz_0.html#K12" target="_self">radix</a> <font color="Maroon">c<sub>3</sub></font>
 
;<br/>
<b>then consider </b><font color="Maroon">c<sub>8</sub></font> being  <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><b> such that </b><br/><a NAME="E9:132"/><i><font color="Green">E87</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="abcmiz_0.html#R6" target="_self">is_properly_applicable_to</a>  <a href="abcmiz_0.html#K12" target="_self">radix</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="abcmiz_0.html#K6" target="_self">ast</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K12" target="_self">radix</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> )
 <b>by </b><i><a class="ref" href="abcmiz_0.html#T77" target="_self">Th77</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>9</sub></font> being    <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="E11:132"/><i><font color="Green">E88</font></i>: 
(  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="abcmiz_0.html#R5" target="_self">is_properly_applicable_to</a> <font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K12" target="_self">radix</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="abcmiz_0.html#E4:132"><i><font color="Green">E86</font></i></a>, <a class="ref" href="abcmiz_0.html#D29" target="_self">Def29</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>10</sub></font> being    <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="E13:132"/><i><font color="Green">E89</font></i>: 
(  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="abcmiz_0.html#R5" target="_self">is_properly_applicable_to</a>  <a href="abcmiz_0.html#K12" target="_self">radix</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="abcmiz_0.html#E9:132"><i><font color="Green">E87</font></i></a>, <a class="ref" href="abcmiz_0.html#D29" target="_self">Def29</a></i>;<br/>
<a NAME="E14:132"/><i><font color="Green">E90</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="finsub_1.html#K1">\/</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="abcmiz_0.html#E11:132"><i><font color="Green">E88</font></i></a>, <a class="txt" href="abcmiz_0.html#E13:132"><i><font color="Green">E89</font></i></a>, <a class="ref" href="finseq_1.html#T44">FINSEQ_1:44</a></i>;<br/>
<a NAME="E15:132"/><i><font color="Green">E91</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="abcmiz_0.html#R3" target="_self">is_applicable_to</a> <font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K12" target="_self">radix</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="abcmiz_0.html#R3" target="_self">is_applicable_to</a>  <a href="abcmiz_0.html#K12" target="_self">radix</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="txt" href="abcmiz_0.html#E11:132"><i><font color="Green">E88</font></i></a>, <a class="txt" href="abcmiz_0.html#E13:132"><i><font color="Green">E89</font></i></a>, <a class="ref" href="abcmiz_0.html#T58" target="_self">Th58</a></i>;<br/>
<a NAME="E16:132"/><b>then </b><i><font color="Green">E92</font></i>: 
 <a href="abcmiz_0.html#K12" target="_self">radix</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="abcmiz_0.html#K8" target="_self">ast</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K13">"\/"</a> <span class="p2">(<span class="default"><a href="abcmiz_0.html#K12" target="_self">radix</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="abcmiz_0.html#E4:132"><i><font color="Green">E86</font></i></a>, <a class="txt" href="abcmiz_0.html#E11:132"><i><font color="Green">E88</font></i></a>, <a class="ref" href="abcmiz_0.html#T57" target="_self">Th57</a></i>;<br/>
<a NAME="E17:132"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>10</sub></font> <a href="abcmiz_0.html#R5" target="_self">is_properly_applicable_to</a> <font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K12" target="_self">radix</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="abcmiz_0.html#E11:132"><i><font color="Green">E88</font></i></a>, <a class="txt" href="abcmiz_0.html#E13:132"><i><font color="Green">E89</font></i></a>, <a class="ref" href="abcmiz_0.html#T62" target="_self">Th62</a></i>;<br/>
<a NAME="E18:132"/><b>then </b><i><font color="Green">E93</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="finsub_1.html#K1">\/</a> <font color="Maroon">c<sub>8</sub></font> <a href="abcmiz_0.html#R6" target="_self">is_properly_applicable_to</a> <font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K12" target="_self">radix</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>10</sub></font> <a href="abcmiz_0.html#R3" target="_self">is_applicable_to</a> <font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K12" target="_self">radix</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="abcmiz_0.html#E14:132"><i><font color="Green">E90</font></i></a>, <a class="ref" href="abcmiz_0.html#D29" target="_self">Def29</a>, <a class="ref" href="abcmiz_0.html#T58" target="_self">Th58</a></i>;<br/>
<b>then </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finsub_1.html#K1">\/</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="abcmiz_0.html#K6" target="_self">ast</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K13">"\/"</a> <span class="p2">(<span class="default"><a href="abcmiz_0.html#K12" target="_self">radix</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="abcmiz_0.html#K8" target="_self">ast</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K13">"\/"</a> <span class="p2">(<span class="default"><a href="abcmiz_0.html#K12" target="_self">radix</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="abcmiz_0.html#E14:132"><i><font color="Green">E90</font></i></a>, <a class="ref" href="abcmiz_0.html#T57" target="_self">Th57</a></i>
<br/>.= 
<font color="Maroon">c<sub>10</sub></font> <a href="abcmiz_0.html#K8" target="_self">ast</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K12" target="_self">radix</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="abcmiz_0.html#E16:132"><i><font color="Green">E92</font></i></a>, <a class="ref" href="abcmiz_0.html#T38" target="_self">Th38</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="abcmiz_0.html#E9:132"><i><font color="Green">E87</font></i></a>, <a class="txt" href="abcmiz_0.html#E13:132"><i><font color="Green">E89</font></i></a>, <a class="txt" href="abcmiz_0.html#E15:132"><i><font color="Green">E91</font></i></a>, <a class="ref" href="abcmiz_0.html#T57" target="_self">Th57</a></i>
;<br/>
<a NAME="E20:132"/><b>then </b>
( <font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K12" target="_self">radix</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="abcmiz_0.html#NM1">type</a> of <font color="Maroon">c<sub>1</sub></font> : ex <font color="Olive">b<sub>1</sub></font> being <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> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="abcmiz_0.html#R6" target="_self">is_properly_applicable_to</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="abcmiz_0.html#K6" target="_self">ast</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> ) </span>}</span>  &amp;  <a href="yellow_0.html#R1">ex_sup_of</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="abcmiz_0.html#NM1">type</a> of <font color="Maroon">c<sub>1</sub></font> : ex <font color="Olive">b<sub>1</sub></font> being <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> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="abcmiz_0.html#R6" target="_self">is_properly_applicable_to</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="abcmiz_0.html#K6" target="_self">ast</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> ) </span>}</span> ,<font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="txt" href="abcmiz_0.html#E18:132"><i><font color="Green">E93</font></i></a>, <a class="ref" href="abcmiz_0.html#T81" target="_self">Th81</a></i>;<br/>
<a NAME="E21:132"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K12" target="_self">radix</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#K1">"\/"</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="abcmiz_0.html#NM1">type</a> of <font color="Maroon">c<sub>1</sub></font> : ex <font color="Olive">b<sub>1</sub></font> being <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> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="abcmiz_0.html#R6" target="_self">is_properly_applicable_to</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="abcmiz_0.html#K6" target="_self">ast</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> ) </span>}</span> ,<font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="yellow_4.html#T1">YELLOW_4:1</a></i>;<br/>
<a NAME="E22:132"/><b>then </b>
( <font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K12" target="_self">radix</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="orders_2.html#R3">&lt;=</a>  <a href="abcmiz_0.html#K12" target="_self">radix</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K12" target="_self">radix</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="ref" href="abcmiz_0.html#T81" target="_self">Th81</a>, <a class="ref" href="yellow_0.html#T22">YELLOW_0:22</a></i>;<br/>
<b>hence </b><a NAME="E23:132"/>
<font color="Maroon">c<sub>2</sub></font> <a href="orders_2.html#R3">&lt;=</a>  <a href="abcmiz_0.html#K12" target="_self">radix</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="yellow_0.html#D2">YELLOW_0:def 2</a></i>;<br/>


</div>
