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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#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">v1</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>, <font color="Maroon">v2</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="E1:84"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">v1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">v2</font> <a href="abcmiz_0.html#R3" target="_self">is_applicable_to</a> <font color="Maroon">t</font>
 ;<br/>

<a NAME="E2:84"/><b>then </b><i><font color="Green">E35</font></i>: 
<font color="Maroon">v2</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">v1</font> <a href="abcmiz_0.html#R3" target="_self">is_applicable_to</a> <font color="Maroon">t</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:84"/><i><font color="Green">E43</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> <span class="p2">(<span class="default"><font color="Maroon">v1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">v2</font></span>)</span>,<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> <span class="p2">(<span class="default"><font color="Maroon">v1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">v2</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 &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> <span class="p2">(<span class="default"><font color="Maroon">v2</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">v1</font></span>)</span>,<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> <span class="p2">(<span class="default"><font color="Maroon">v2</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">v1</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 )
 
<b>by </b><i/>;<br/>
<a NAME="E4:84"/><i><font color="Green">E46</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">v1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">v2</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">v1</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">v2</font></span>)</span> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">v2</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">v1</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">v1</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">v2</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>;<br/>
<a NAME="E5:84"/><i><font color="Green">E47</font></i>: 
(  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">v1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">v2</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">v1</font></span>)</span> <a href="finsub_1.html#K1">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">v2</font></span>)</span> &amp;  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">v2</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">v1</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">v1</font></span>)</span> <a href="finsub_1.html#K1">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">v2</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T44">FINSEQ_1:44</a></i>;<br/>
<a NAME="E6:84"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">v1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">v2</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E7:84"/><b>then </b>
( <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">v1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">v2</font></span>)</span></span>)</span> <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="abcmiz_0.html#K7" target="_self">apply</a> <span class="p2">(<span class="default"><font color="Maroon">v1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">v2</font></span>)</span>,<font color="Maroon">t</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">v1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">v2</font></span>)</span></span>)</span> <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="abcmiz_0.html#K7" target="_self">apply</a> <span class="p2">(<span class="default"><font color="Maroon">v2</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">v1</font></span>)</span>,<font color="Maroon">t</font></span>)</span> )
 
<b>by </b><i>, , <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E8:84"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">v1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">v2</font></span>)</span> <a href="abcmiz_0.html#K8" target="_self">ast</a> <font color="Maroon">t</font> <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> <span class="p2">(<span class="default"><font color="Maroon">v1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">v2</font></span>)</span>,<font color="Maroon">t</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">v2</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">v1</font></span>)</span> <a href="abcmiz_0.html#K8" target="_self">ast</a> <font color="Maroon">t</font> <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> <span class="p2">(<span class="default"><font color="Maroon">v2</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">v1</font></span>)</span>,<font color="Maroon">t</font></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/>
<a NAME="E9:84"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">v1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">v2</font></span>)</span> <a href="abcmiz_0.html#K8" target="_self">ast</a> <font color="Maroon">t</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">v2</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">v1</font></span>)</span> <a href="abcmiz_0.html#K8" target="_self">ast</a> <font color="Maroon">t</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">v2</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">v1</font></span>)</span> <a href="abcmiz_0.html#K8" target="_self">ast</a> <font color="Maroon">t</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">v1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">v2</font></span>)</span> <a href="abcmiz_0.html#K8" target="_self">ast</a> <font color="Maroon">t</font> )
 
<b>by </b><i>, , <a class="ref" href="abcmiz_0.html#T1" target="_self">Th1</a>, </i>;<br/>
<b>hence </b><a NAME="E10:84"/>
<span class="p1">(<span class="default"><font color="Maroon">v1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">v2</font></span>)</span> <a href="abcmiz_0.html#K8" target="_self">ast</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">v2</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">v1</font></span>)</span> <a href="abcmiz_0.html#K8" target="_self">ast</a> <font color="Maroon">t</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#D3">YELLOW_0:def 3</a></i>;<br/>


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