<?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#L2" target="_self">TA-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">v</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">T</font>;<br/>





<b>assume </b><a NAME="E1:77"/><i><font color="Green">E32</font></i>: 
for <font color="Olive">i</font> being <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <br/> for <font color="Olive">a</font> being  <a href="abcmiz_0.html#NM2">adjective</a> of <font color="Maroon">T</font><br/> for <font color="Olive">s</font> being  <a href="abcmiz_0.html#NM1">type</a> of <font color="Maroon">T</font>  st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">v</font> &amp; <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> &amp; <font color="Olive">s</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K7" target="_self">apply</a> <font color="Maroon">v</font>,<font color="Maroon">t</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> holds <br/><font color="Olive">a</font> <a href="abcmiz_0.html#R1" target="_self">is_applicable_to</a> <font color="Olive">s</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="abcmiz_0.html#D21" target="_self">ABCMIZ_0:def 21</a><br/>

<b>let </b><font color="Maroon">i1</font> be  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> , <font color="Maroon">i2</font> be  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>assume </b><a NAME="E2:77"/><i><font color="Green">E35</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i1</font> &amp; <font color="Maroon">i1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">i2</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">v</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/>

<b>consider </b><font color="Maroon">j</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:77"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i1</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">j</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/>
<a NAME="E5:77"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E6:77"/><i><font color="Green">E47</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K7" target="_self">apply</a> <font color="Maroon">v</font>,<font color="Maroon">t</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">v</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i/>;<br/>
<b>let </b><font color="Maroon">s1</font> be   <a href="abcmiz_0.html#NM1">type</a> of <font color="Maroon">T</font>, <font color="Maroon">s2</font> be   <a href="abcmiz_0.html#NM1">type</a> of <font color="Maroon">T</font>;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>( <font color="Maroon">i1</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K7" target="_self">apply</a> <font color="Maroon">v</font>,<font color="Maroon">t</font></span>)</span> implies for <font color="Olive">s</font> being  <a href="abcmiz_0.html#NM1">type</a> of <font color="Maroon">T</font>  st <font color="Olive">s</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K7" target="_self">apply</a> <font color="Maroon">v</font>,<font color="Maroon">t</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> holds <br/><font color="Olive">s</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">s1</font> );<br/>
<b>assume </b><a NAME="E7:77"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">s1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K7" target="_self">apply</a> <font color="Maroon">v</font>,<font color="Maroon">t</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i1</font> &amp; <font color="Maroon">s2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K7" target="_self">apply</a> <font color="Maroon">v</font>,<font color="Maroon">t</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i2</font> )
 ;<br/>

<a NAME="E8:77"/><b>then </b><i><font color="Green">E49</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
;<br/>
<a NAME="E9:77"/><i><font color="Green">E50</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">i</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="77_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:77_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">i</font>]
 <b>and </b><br/><a NAME="E2:77_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">i1</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K7" target="_self">apply</a> <font color="Maroon">v</font>,<font color="Maroon">t</font></span>)</span>
 ;<br/>

<a NAME="E3:77_1"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">i1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i1</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i1</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">i</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 )
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E4:77_1"/><b>then </b><i><font color="Green">E110</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i1</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i1</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">v</font> &amp; <font color="Maroon">i1</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">i</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  )
 
<b>by </b><i>, <a class="txt" href="abcmiz_0.html#E7:77"><i><font color="Green">E48</font></i></a>, <a class="txt" href="abcmiz_0.html#E3:77_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a>, <a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E5:77_1"/><b>then </b><i><font color="Green">E111</font></i>: 
<font color="Maroon">i1</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">v</font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E6:77_1"/><b>then </b>
( <font color="Maroon">v</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">v</font> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">v</font> <a href="tarski.html#R1">c=</a> the <a href="abcmiz_0.html#U1" target="_self">adjectives</a> of <font color="Maroon">T</font> )
 
<b>by </b><i><a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">a</font> = <font color="Maroon">v</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">i</font></span>)</span> as   <a href="abcmiz_0.html#NM2">adjective</a> of <font color="Maroon">T</font> ;<br/>
<a NAME="E8:77_1"/>
<font color="Maroon">i1</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">v</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="txt" href="abcmiz_0.html#E7:77"><i><font color="Green">E48</font></i></a>, <a class="txt" href="abcmiz_0.html#E3:77_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="abcmiz_0.html#T1" target="_self">Th1</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E9:77_1"/><b>then </b>
<font color="Maroon">i1</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K7" target="_self">apply</a> <font color="Maroon">v</font>,<font color="Maroon">t</font></span>)</span>
 
<b>by </b><i><a class="txt" href="abcmiz_0.html#E7:77"><i><font color="Green">E48</font></i></a>, , <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E10:77_1"/><b>then </b>
( <span class="p1">(<span class="default"><a href="abcmiz_0.html#K7" target="_self">apply</a> <font color="Maroon">v</font>,<font color="Maroon">t</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K7" target="_self">apply</a> <font color="Maroon">v</font>,<font color="Maroon">t</font></span>)</span> &amp;  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K7" target="_self">apply</a> <font color="Maroon">v</font>,<font color="Maroon">t</font></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> )
 
<b>by </b><i><a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">s</font> = <span class="p1">(<span class="default"><a href="abcmiz_0.html#K7" target="_self">apply</a> <font color="Maroon">v</font>,<font color="Maroon">t</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">i</font></span>)</span> as   <a href="abcmiz_0.html#NM1">type</a> of <font color="Maroon">T</font> ;<br/>
<a NAME="E12:77_1"/><i><font color="Green">E112</font></i>: 
<font color="Maroon">a</font> <a href="abcmiz_0.html#R1" target="_self">is_applicable_to</a> <font color="Maroon">s</font>
 
<b>by </b><i><a class="ref" href="abcmiz_0.html#D2" target="_self">Def2</a>, </i>;<br/>
<a NAME="E13:77_1"/><i><font color="Green">E115</font></i>: 
<span class="p1">(<span class="default"><a href="abcmiz_0.html#K7" target="_self">apply</a> <font color="Maroon">v</font>,<font color="Maroon">t</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">i</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="abcmiz_0.html#K5" target="_self">ast</a> <font color="Maroon">s</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E14:77_1"/>
( <font color="Maroon">s</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">s1</font> &amp; <font color="Maroon">a</font> <a href="abcmiz_0.html#K5" target="_self">ast</a> <font color="Maroon">s</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">s</font> )
 
<b>by </b><i><a class="txt" href="abcmiz_0.html#E2:77_1"><i><font color="Green">E52</font></i></a>, <a class="txt" href="abcmiz_0.html#E3:77_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="abcmiz_0.html#T1" target="_self">Th1</a>, , , <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<b>hence </b><a NAME="E15:77_1"/>
for <font color="Olive">s</font> being  <a href="abcmiz_0.html#NM1">type</a> of <font color="Maroon">T</font>  st <font color="Olive">s</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K7" target="_self">apply</a> <font color="Maroon">v</font>,<font color="Maroon">t</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> holds <br/><font color="Olive">s</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">s1</font>
 <b>by </b><i><a class="ref" href="abcmiz_0.html#D4" target="_self">Def4</a>, <a class="ref" href="yellow_0.html#D2">YELLOW_0:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:77"/>
for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">i</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#E9:77"><i><font color="Green">E50</font></i></a>, );<br/></i>
<b>hence </b><a NAME="E11:77"/>
<font color="Maroon">s2</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">s1</font>
 <b>by </b><i><a class="txt" href="abcmiz_0.html#E6:77"><i><font color="Green">E47</font></i></a>, , <a class="txt" href="abcmiz_0.html#E5:77"><i><font color="Green">E46</font></i></a>, <a class="txt" href="abcmiz_0.html#E7:77"><i><font color="Green">E48</font></i></a>, <a class="txt" href="abcmiz_0.html#E8:77"><i><font color="Green">E49</font></i></a></i>;<br/>


</div>
