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<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#L3" target="_self">TAS-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">A</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">T</font>;<br/>





<b>assume </b><a NAME="E1:121"/><i><font color="Green">E32</font></i>: 
for <font color="Olive">C</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">T</font>  st <font color="Olive">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> &amp; <font color="Olive">C</font> <a href="abcmiz_0.html#R6" target="_self">is_properly_applicable_to</a> <font color="Maroon">t</font> &amp; <font color="Maroon">A</font> <a href="abcmiz_0.html#K6" target="_self">ast</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Olive">C</font> <a href="abcmiz_0.html#K6" target="_self">ast</a> <font color="Maroon">t</font> holds <br/><font color="Olive">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font>
 ;<br/>

<b>let </b><font color="Maroon">s</font> be  <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">T</font>;<br/>

<b>assume </b><a NAME="E2:121"/><i><font color="Green">E35</font></i>: 
(  <a href="relset_1.html#K5">rng</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> &amp; <font color="Maroon">s</font> <a href="abcmiz_0.html#R5" target="_self">is_properly_applicable_to</a> <font color="Maroon">t</font> )
 ;<br/>

<a NAME="E3:121"/><i><font color="Green">E43</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p2">(<span class="default"><a href="abcmiz_0.html#K7" target="_self">apply</a> <font color="Maroon">s</font>,<font color="Maroon">t</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</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">s</font>,<font color="Maroon">t</font></span>)</span> &amp;  <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">s</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">s</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 &amp;  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p2">(<span class="default"><a href="abcmiz_0.html#K7" target="_self">apply</a> <font color="Maroon">s</font>,<font color="Maroon">t</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</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">s</font>,<font color="Maroon">t</font></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="finseq_5.html#T60">FINSEQ_5:60</a>, <a class="ref" href="finseq_5.html#D3">FINSEQ_5:def 3</a></i>;<br/>
<b>hence </b><a NAME="E4:121"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p2">(<span class="default"><a href="abcmiz_0.html#K7" target="_self">apply</a> <font color="Maroon">s</font>,<font color="Maroon">t</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="rewrite1.html#D2">REWRITE1:def 2</a><br/>

<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><a NAME="E5:121"/>
( <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="finseq_5.html#K4">Rev</a> <span class="p2">(<span class="default"><a href="abcmiz_0.html#K7" target="_self">apply</a> <font color="Maroon">s</font>,<font color="Maroon">t</font></span>)</span></span>)</span> &amp; <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p2">(<span class="default"><a href="abcmiz_0.html#K7" target="_self">apply</a> <font color="Maroon">s</font>,<font color="Maroon">t</font></span>)</span></span>)</span> )
 ;<br/>

<a NAME="E6:121"/><b>then </b><i><font color="Green">E46</font></i>: 
( <font color="Maroon">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <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">s</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 &amp; <span class="p1">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p2">(<span class="default"><a href="abcmiz_0.html#K7" target="_self">apply</a> <font color="Maroon">s</font>,<font color="Maroon">t</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</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">s</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"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">s</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">i</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> &amp; <span class="p1">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p2">(<span class="default"><a href="abcmiz_0.html#K7" target="_self">apply</a> <font color="Maroon">s</font>,<font color="Maroon">t</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</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"><a href="abcmiz_0.html#K7" target="_self">apply</a> <font color="Maroon">s</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"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">s</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> )
 
<b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a>, <a class="ref" href="finseq_5.html#D3">FINSEQ_5:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">j</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E8:121"/><i><font color="Green">E47</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">s</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <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="E9:121"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">s</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</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> <font color="Maroon">j</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">s</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> 1 &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">j</font> )
 
<b>by </b><i><a class="ref" href="abcmiz_0.html#D11" target="_self">Def11</a></i>;<br/>
<a NAME="E10:121"/><b>then </b>
( <font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">s</font> )
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a>, <a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<b>hence </b><a NAME="E11:121"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p4">(<span class="default"><a href="abcmiz_0.html#K7" target="_self">apply</a> <font color="Maroon">s</font>,<font color="Maroon">t</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p4">(<span class="default"><a href="abcmiz_0.html#K7" target="_self">apply</a> <font color="Maroon">s</font>,<font color="Maroon">t</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">T</font> <a href="abcmiz_0.html#K11" target="_self">@--&gt;</a> 
 <b>by </b><i><a class="ref" href="abcmiz_0.html#T9" target="_self">Th9</a>, , , <a class="ref" href="abcmiz_0.html#D11" target="_self">Def11</a>, <a class="ref" href="abcmiz_0.html#D9" target="_self">Def9</a></i>;<br/>


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