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

<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#L2" target="_self">TA-structure</a> ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means for <font color="Olive">b<sub>1</sub></font> being  <a href="abcmiz_0.html#NM1">type</a> of <font color="Maroon">c<sub>1</sub></font><br/>for <font color="Olive">b<sub>2</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> st <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="abcmiz_0.html#R3" target="_self">is_applicable_to</a> <font color="Olive">b<sub>1</sub></font> holds <br/>for <font color="Olive">b<sub>3</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>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K5">rng</a> <font color="Olive">b<sub>2</sub></font> holds <br/><font color="Olive">b<sub>2</sub></font> <a href="abcmiz_0.html#K8" target="_self">ast</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> <a href="abcmiz_0.html#K6" target="_self">ast</a> <font color="Olive">b<sub>1</sub></font>;<br/>
<a NAME="E1:92"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="92_1"><b>proof </b></a><div class="add">

<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="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>;<br/>



<b>assume </b><a NAME="E1:92_1"/>
0 <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<a NAME="E2:92_1"/><b>then </b>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K6">&lt;*&gt;</a> the <a href="abcmiz_0.html#U1" target="_self">adjectives</a> of <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="relat_1.html#K2">rng</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/>
<a NAME="E3:92_1"/><b>then </b>
( <font color="Maroon">c<sub>3</sub></font> <a href="abcmiz_0.html#K8" target="_self">ast</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="subset_1.html#K1">{}</a> the <a href="abcmiz_0.html#U1" target="_self">adjectives</a> of <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="ref" href="abcmiz_0.html#T31" target="_self">Th31</a></i>;<br/>
<b>hence </b><a NAME="E4:92_1"/>
(  not <font color="Maroon">c<sub>3</sub></font> <a href="abcmiz_0.html#R3" target="_self">is_applicable_to</a> <font color="Maroon">c<sub>2</sub></font> or 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="hidden.html#R1">=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>3</sub></font> holds <br/><font color="Maroon">c<sub>3</sub></font> <a href="abcmiz_0.html#K8" 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> )
 <b>by </b><i><a class="ref" href="abcmiz_0.html#T28" target="_self">Th28</a></i>;<br/>


</div><b>end;</b></div>
<div><i><font color="Green">E57</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="92_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:92_2"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>2</sub></font>]
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="92_2_1"><b>now </b></a><div class="add"><b>let </b><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="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>;<br/><b>assume </b><a NAME="E1:92_2_1"/><i><font color="Green">E59</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="abcmiz_0.html#R3" target="_self">is_applicable_to</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/><b>consider </b><font color="Maroon">c<sub>5</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>, <font color="Maroon">c<sub>6</sub></font> being    <a href="subset_1.html#M1">Element</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="E3:92_2_1"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="abcmiz_0.html#E1:92_2_1"><i><font color="Green">E59</font></i></a>, <a class="ref" href="finseq_2.html#T22">FINSEQ_2:22</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font> as   <a href="abcmiz_0.html#NM2">adjective</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/><b>reconsider </b><font color="Maroon">c<sub>8</sub></font> =  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>5</sub></font> as   <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/><a NAME="E6:92_2_1"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>;<br/><a NAME="E7:92_2_1"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="abcmiz_0.html#E3:92_2_1"><i><font color="Green">E60</font></i></a>, <a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>;<br/><a NAME="E8:92_2_1"/><b>then </b>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="abcmiz_0.html#R3" target="_self">is_applicable_to</a> <font color="Maroon">c<sub>3</sub></font> &amp; <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="abcmiz_0.html#R3" target="_self">is_applicable_to</a> <font color="Maroon">c<sub>5</sub></font> <a href="abcmiz_0.html#K8" target="_self">ast</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="abcmiz_0.html#E1:92_2_1"><i><font color="Green">E59</font></i></a>, <a class="txt" href="abcmiz_0.html#E3:92_2_1"><i><font color="Green">E60</font></i></a>, <a class="ref" href="abcmiz_0.html#T41" target="_self">Th41</a></i>;<br/><a NAME="E9:92_2_1"/><b>then </b><i><font color="Green">E61</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="abcmiz_0.html#K8" target="_self">ast</a> <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> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="abcmiz_0.html#R1" target="_self">is_applicable_to</a> <font color="Maroon">c<sub>5</sub></font> <a href="abcmiz_0.html#K8" target="_self">ast</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="abcmiz_0.html#E1:92_2"><i><font color="Green">E58</font></i></a>, <a class="ref" href="abcmiz_0.html#T40" target="_self">Th40</a></i>;<br/><b>let </b><font color="Maroon">c<sub>9</sub></font> be   <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="E10:92_2_1"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><b>then </b><i><font color="Green">E63</font></i>: <font color="Maroon">c<sub>9</sub></font> = 
<font color="Maroon">c<sub>8</sub></font> <a href="finsub_1.html#K1">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="txt" href="abcmiz_0.html#E3:92_2_1"><i><font color="Green">E60</font></i></a>, <a class="ref" href="finseq_1.html#T44">FINSEQ_1:44</a></i>
<br/>.= 
<font color="Maroon">c<sub>8</sub></font> <a href="finsub_1.html#K1">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="domain_1.html#K6">}</a></span>
<b>by </b><i><a class="ref" href="finseq_1.html#T55">FINSEQ_1:55</a></i>
;<br/><a NAME="E12:92_2_1"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="abcmiz_0.html#R2" target="_self">is_applicable_to</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="abcmiz_0.html#E1:92_2_1"><i><font color="Green">E59</font></i></a>, <a class="txt" href="abcmiz_0.html#E10:92_2_1"><i><font color="Green">E62</font></i></a>, <a class="ref" href="abcmiz_0.html#T46" target="_self">Th46</a></i>;<br/><b>thus </b><font color="Maroon">c<sub>4</sub></font> <a href="abcmiz_0.html#K8" target="_self">ast</a> <font color="Maroon">c<sub>3</sub></font> = 
<span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="abcmiz_0.html#K8" target="_self">ast</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="abcmiz_0.html#K8" target="_self">ast</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="abcmiz_0.html#E3:92_2_1"><i><font color="Green">E60</font></i></a>, <a class="ref" href="abcmiz_0.html#T38" target="_self">Th38</a></i>
<br/>.= 
<font color="Maroon">c<sub>7</sub></font> <a href="abcmiz_0.html#K5" target="_self">ast</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="abcmiz_0.html#K6" target="_self">ast</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="abcmiz_0.html#E9:92_2_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="abcmiz_0.html#T32" target="_self">Th32</a></i>
<br/>.= 
<font color="Maroon">c<sub>9</sub></font> <a href="abcmiz_0.html#K6" target="_self">ast</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="abcmiz_0.html#E11:92_2_1"><i><font color="Green">E63</font></i></a>, <a class="txt" href="abcmiz_0.html#E12:92_2_1"><i><font color="Green">E64</font></i></a>, <a class="ref" href="abcmiz_0.html#T56" target="_self">Th56</a></i>
;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:92_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1]
 ;<br/></div><b>end;</b></div>
<a NAME="E3:92"/><i><font color="Green">E58</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="abcmiz_0.html#E1:92"><i><font color="Green">E56</font></i></a>, <a class="txt" href="abcmiz_0.html#E2:92"><i><font color="Green">E57</font></i></a>);<br/></i>
<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="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>;<br/>



<a NAME="E4:92"/>
<font color="Maroon">S<sub>1</sub></font>[ <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font>]
 
<b>by </b><i><a class="txt" href="abcmiz_0.html#E3:92"><i><font color="Green">E58</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:92"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="abcmiz_0.html#R3" target="_self">is_applicable_to</a> <font color="Maroon">c<sub>2</sub></font> implies 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="hidden.html#R1">=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>3</sub></font> holds <br/><font color="Maroon">c<sub>3</sub></font> <a href="abcmiz_0.html#K8" 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> )
 ;<br/>


</div>
