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

<b>let </b><font color="Maroon" title="c1">T</font> be  <a href="orders_2.html#V3" title="ORDERS_2:attr.3">reflexive</a> <a href="orders_2.html#V4" title="ORDERS_2:attr.4">transitive</a> <a href="orders_2.html#V5" title="ORDERS_2:attr.5">antisymmetric</a> <a href="lattice3.html#V1" title="LATTICE3:attr.1">with_suprema</a> <a href="abcmiz_0.html#V1" title="ABCMIZ_0:attr.1">Noetherian</a> non <a href="abcmiz_0.html#V4" title="ABCMIZ_0:attr.4">void</a> <a href="abcmiz_0.html#V9" title="ABCMIZ_0:attr.9">adj-structured</a>  <a href="abcmiz_0.html#L2" title="ABCMIZ_0:struct.2">TA-structure</a> ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b>for <font color="Olive" title="b1">t</font> being  <a href="abcmiz_0.html#NM1" title="ABCMIZ_0:NM.1">type</a> of <font color="Maroon" title="c1">T</font><br/> for <font color="Olive" title="b2">v</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="abcmiz_0.html#U1" title="ABCMIZ_0:sel.1">adjectives</a> of <font color="Maroon" title="c1">T</font>  st $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">v</font> &amp; <font color="Olive" title="b2">v</font> <a href="abcmiz_0.html#R3" title="ABCMIZ_0:pred.3">is_applicable_to</a> <font color="Olive" title="b1">t</font> holds <br/>for <font color="Olive" title="b3">A</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of the <a href="abcmiz_0.html#U1" title="ABCMIZ_0:sel.1">adjectives</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Olive" title="b2">v</font> holds <br/><font color="Olive" title="b2">v</font> <a href="abcmiz_0.html#K8" title="ABCMIZ_0:func.8">ast</a> <font color="Olive" title="b1">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">A</font> <a href="abcmiz_0.html#K6" title="ABCMIZ_0:func.6">ast</a> <font color="Olive" title="b1">t</font>;<br/>
<a NAME="E1:92"/><i><font color="Green" title="E56">A1</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<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" title="c2">t</font> be   <a href="abcmiz_0.html#NM1" title="ABCMIZ_0:NM.1">type</a> of <font color="Maroon" title="c1">T</font>;<br/><b>let </b><font color="Maroon" title="c3">v</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="abcmiz_0.html#U1" title="ABCMIZ_0:sel.1">adjectives</a> of <font color="Maroon" title="c1">T</font>;<br/>



<b>assume </b><a NAME="E1:92_1"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">v</font>
 ;<br/>

<a NAME="E2:92_1"/><b>then </b>
( <font color="Maroon" title="c3">v</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a> the <a href="abcmiz_0.html#U1" title="ABCMIZ_0:sel.1">adjectives</a> of <font color="Maroon" title="c1">T</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T25" title="FINSEQ_1:th.25">FINSEQ_1:25</a></i>;<br/>
<a NAME="E3:92_1"/><b>then </b>
( <font color="Maroon" title="c3">v</font> <a href="abcmiz_0.html#K8" title="ABCMIZ_0:func.8">ast</a> <font color="Maroon" title="c2">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">t</font> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> the <a href="abcmiz_0.html#U1" title="ABCMIZ_0:sel.1">adjectives</a> of <font color="Maroon" title="c1">T</font> )
 
<b>by </b><i><a class="ref" href="abcmiz_0.html#T31" target="_self" title="ABCMIZ_0:th.31">Th31</a></i>;<br/>
<b>hence </b><a NAME="E4:92_1"/>
(  not <font color="Maroon" title="c3">v</font> <a href="abcmiz_0.html#R3" title="ABCMIZ_0:pred.3">is_applicable_to</a> <font color="Maroon" title="c2">t</font> or for <font color="Olive" title="b1">A</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of the <a href="abcmiz_0.html#U1" title="ABCMIZ_0:sel.1">adjectives</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">v</font> holds <br/><font color="Maroon" title="c3">v</font> <a href="abcmiz_0.html#K8" title="ABCMIZ_0:func.8">ast</a> <font color="Maroon" title="c2">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">A</font> <a href="abcmiz_0.html#K6" title="ABCMIZ_0:func.6">ast</a> <font color="Maroon" title="c2">t</font> )
 <b>by </b><i><a class="ref" href="abcmiz_0.html#T28" target="_self" title="ABCMIZ_0:th.28">Th28</a></i>;<br/>


</div><b>end;</b></div>
<div><i><font color="Green" title="E57">A2</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" title="c2">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:92_2"/><i><font color="Green" title="E58">A3</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">n</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" title="c3">t</font> be   <a href="abcmiz_0.html#NM1" title="ABCMIZ_0:NM.1">type</a> of <font color="Maroon" title="c1">T</font>;<br/><b>let </b><font color="Maroon" title="c4">v</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="abcmiz_0.html#U1" title="ABCMIZ_0:sel.1">adjectives</a> of <font color="Maroon" title="c1">T</font>;<br/><b>assume </b><a NAME="E1:92_2_1"/><i><font color="Green" title="E59">A4</font></i>: 
( <font color="Maroon" title="c2">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">v</font> &amp; <font color="Maroon" title="c4">v</font> <a href="abcmiz_0.html#R3" title="ABCMIZ_0:pred.3">is_applicable_to</a> <font color="Maroon" title="c3">t</font> )
 ;<br/><b>consider </b><font color="Maroon" title="c5">v1</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="abcmiz_0.html#U1" title="ABCMIZ_0:sel.1">adjectives</a> of <font color="Maroon" title="c1">T</font>, <font color="Maroon" title="c6">a</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="abcmiz_0.html#U1" title="ABCMIZ_0:sel.1">adjectives</a> of <font color="Maroon" title="c1">T</font><b> such that </b><br/><a NAME="E3:92_2_1"/><i><font color="Green" title="E60">A5</font></i>: 
<font color="Maroon" title="c4">v</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c5">v1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c6">a</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="abcmiz_0.html#E1:92_2_1"><i><font color="Green" title="E59">A4</font></i></a>, <a class="ref" href="finseq_2.html#T22" title="FINSEQ_2:th.22">FINSEQ_2:22</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c7">a</font> = <font color="Maroon" title="c6">a</font> as   <a href="abcmiz_0.html#NM2" title="ABCMIZ_0:NM.2">adjective</a> of <font color="Maroon" title="c1">T</font> ;<br/><b>reconsider </b><font color="Maroon" title="c8">B</font> =  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c5">v1</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of the <a href="abcmiz_0.html#U1" title="ABCMIZ_0:sel.1">adjectives</a> of <font color="Maroon" title="c1">T</font> ;<br/><a NAME="E6:92_2_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c7">a</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="ref" href="finseq_1.html#T57" title="FINSEQ_1:th.57">FINSEQ_1:57</a></i>;<br/><a NAME="E7:92_2_1"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">v1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="abcmiz_0.html#E3:92_2_1"><i><font color="Green" title="E60">A5</font></i></a>, <a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></i>;<br/><a NAME="E8:92_2_1"/><b>then </b>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font> &amp; <font color="Maroon" title="c5">v1</font> <a href="abcmiz_0.html#R3" title="ABCMIZ_0:pred.3">is_applicable_to</a> <font color="Maroon" title="c3">t</font> &amp; <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c7">a</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="abcmiz_0.html#R3" title="ABCMIZ_0:pred.3">is_applicable_to</a> <font color="Maroon" title="c5">v1</font> <a href="abcmiz_0.html#K8" title="ABCMIZ_0:func.8">ast</a> <font color="Maroon" title="c3">t</font> )
 <b>by </b><i><a class="txt" href="abcmiz_0.html#E1:92_2_1"><i><font color="Green" title="E59">A4</font></i></a>, <a class="txt" href="abcmiz_0.html#E3:92_2_1"><i><font color="Green" title="E60">A5</font></i></a>, <a class="ref" href="abcmiz_0.html#T41" target="_self" title="ABCMIZ_0:th.41">Th41</a></i>;<br/><a NAME="E9:92_2_1"/><b>then </b><i><font color="Green" title="E61">A6</font></i>: 
( <font color="Maroon" title="c5">v1</font> <a href="abcmiz_0.html#K8" title="ABCMIZ_0:func.8">ast</a> <font color="Maroon" title="c3">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">B</font> <a href="abcmiz_0.html#K6" title="ABCMIZ_0:func.6">ast</a> <font color="Maroon" title="c3">t</font> &amp; <font color="Maroon" title="c7">a</font> <a href="abcmiz_0.html#R1" title="ABCMIZ_0:pred.1">is_applicable_to</a> <font color="Maroon" title="c5">v1</font> <a href="abcmiz_0.html#K8" title="ABCMIZ_0:func.8">ast</a> <font color="Maroon" title="c3">t</font> )
 <b>by </b><i><a class="txt" href="abcmiz_0.html#E1:92_2"><i><font color="Green" title="E58">A3</font></i></a>, <a class="ref" href="abcmiz_0.html#T40" target="_self" title="ABCMIZ_0:th.40">Th40</a></i>;<br/><b>let </b><font color="Maroon" title="c9">A</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of the <a href="abcmiz_0.html#U1" title="ABCMIZ_0:sel.1">adjectives</a> of <font color="Maroon" title="c1">T</font>;<br/><b>assume </b><a NAME="E10:92_2_1"/><i><font color="Green" title="E62">A7</font></i>: 
<font color="Maroon" title="c9">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">v</font>
 ;<br/><b>then </b><i><font color="Green" title="E63">A8</font></i>: <font color="Maroon" title="c9">A</font> = 
<font color="Maroon" title="c8">B</font> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c7">a</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="txt" href="abcmiz_0.html#E3:92_2_1"><i><font color="Green" title="E60">A5</font></i></a>, <a class="ref" href="finseq_1.html#T44" title="FINSEQ_1:th.44">FINSEQ_1:44</a></i>
<br/>.= 
<font color="Maroon" title="c8">B</font> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c7">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
<b>by </b><i><a class="ref" href="finseq_1.html#T55" title="FINSEQ_1:th.55">FINSEQ_1:55</a></i>
;<br/><a NAME="E12:92_2_1"/><i><font color="Green" title="E64">A9</font></i>: 
<font color="Maroon" title="c9">A</font> <a href="abcmiz_0.html#R2" title="ABCMIZ_0:pred.2">is_applicable_to</a> <font color="Maroon" title="c3">t</font>
 <b>by </b><i><a class="txt" href="abcmiz_0.html#E1:92_2_1"><i><font color="Green" title="E59">A4</font></i></a>, <a class="txt" href="abcmiz_0.html#E10:92_2_1"><i><font color="Green" title="E62">A7</font></i></a>, <a class="ref" href="abcmiz_0.html#T46" target="_self" title="ABCMIZ_0:th.46">Th46</a></i>;<br/><b>thus </b><font color="Maroon" title="c4">v</font> <a href="abcmiz_0.html#K8" title="ABCMIZ_0:func.8">ast</a> <font color="Maroon" title="c3">t</font> = 
<span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c7">a</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="abcmiz_0.html#K8" title="ABCMIZ_0:func.8">ast</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="abcmiz_0.html#K8" title="ABCMIZ_0:func.8">ast</a> <font color="Maroon" title="c3">t</font></span>)</span>
<b>by </b><i><a class="txt" href="abcmiz_0.html#E3:92_2_1"><i><font color="Green" title="E60">A5</font></i></a>, <a class="ref" href="abcmiz_0.html#T38" target="_self" title="ABCMIZ_0:th.38">Th38</a></i>
<br/>.= 
<font color="Maroon" title="c7">a</font> <a href="abcmiz_0.html#K5" title="ABCMIZ_0:func.5">ast</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">B</font> <a href="abcmiz_0.html#K6" title="ABCMIZ_0:func.6">ast</a> <font color="Maroon" title="c3">t</font></span>)</span>
<b>by </b><i><a class="txt" href="abcmiz_0.html#E9:92_2_1"><i><font color="Green" title="E61">A6</font></i></a>, <a class="ref" href="abcmiz_0.html#T32" target="_self" title="ABCMIZ_0:th.32">Th32</a></i>
<br/>.= 
<font color="Maroon" title="c9">A</font> <a href="abcmiz_0.html#K6" title="ABCMIZ_0:func.6">ast</a> <font color="Maroon" title="c3">t</font>
<b>by </b><i><a class="txt" href="abcmiz_0.html#E11:92_2_1"><i><font color="Green" title="E63">A8</font></i></a>, <a class="txt" href="abcmiz_0.html#E12:92_2_1"><i><font color="Green" title="E64">A9</font></i></a>, <a class="ref" href="abcmiz_0.html#T56" target="_self" title="ABCMIZ_0:th.56">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" title="c2">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 ;<br/></div><b>end;</b></div>
<a NAME="E3:92"/><i><font color="Green" title="E58">A10</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="abcmiz_0.html#E1:92"><i><font color="Green" title="E56">A1</font></i></a>, <a class="txt" href="abcmiz_0.html#E2:92"><i><font color="Green" title="E57">A2</font></i></a>);<br/></i>
<b>let </b><font color="Maroon" title="c2">t</font> be   <a href="abcmiz_0.html#NM1" title="ABCMIZ_0:NM.1">type</a> of <font color="Maroon" title="c1">T</font>;<br/><b>let </b><font color="Maroon" title="c3">v</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="abcmiz_0.html#U1" title="ABCMIZ_0:sel.1">adjectives</a> of <font color="Maroon" title="c1">T</font>;<br/>



<b>thus </b><a NAME="E4:92"/>
( <font color="Maroon" title="c3">v</font> <a href="abcmiz_0.html#R3" title="ABCMIZ_0:pred.3">is_applicable_to</a> <font color="Maroon" title="c2">t</font> implies for <font color="Olive" title="b1">A</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of the <a href="abcmiz_0.html#U1" title="ABCMIZ_0:sel.1">adjectives</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">v</font> holds <br/><font color="Maroon" title="c3">v</font> <a href="abcmiz_0.html#K8" title="ABCMIZ_0:func.8">ast</a> <font color="Maroon" title="c2">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">A</font> <a href="abcmiz_0.html#K6" title="ABCMIZ_0:func.6">ast</a> <font color="Maroon" title="c2">t</font> )
 <b>by </b><i><a class="txt" href="abcmiz_0.html#E3:92"><i><font color="Green" title="E58">A10</font></i></a></i>;<br/>


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