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<b>let </b><font color="Maroon">T</font> be  non <a href="struct_0.html#V3">empty</a> <a href="orders_2.html#V2">reflexive</a> <a href="orders_2.html#V3">transitive</a> non <a href="abcmiz_0.html#V4" target="_self">void</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>set </b><font color="Maroon">v</font> = <font color="Maroon">v1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">v2</font>;<br/>
<a NAME="E1:70"/><i><font color="Green">E32</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">v1</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">v1</font></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> <font color="Maroon">v2</font>,<span class="p2">(<span class="default"><font color="Maroon">v1</font> <a href="abcmiz_0.html#K8" target="_self">ast</a> <font color="Maroon">t</font></span>)</span></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">v2</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 )
 
<b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">tt</font> being    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>, <font color="Maroon">q</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E3:70"/><i><font color="Green">E35</font></i>: 
 <a href="abcmiz_0.html#K7" target="_self">apply</a> <font color="Maroon">v1</font>,<font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">tt</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">q</font></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="hilbert2.html#T4">HILBERT2:4</a></i>;<br/>
<i><font color="Green">E43</font></i>:  <a href="abcmiz_0.html#K7" target="_self">apply</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>,<font color="Maroon">t</font> = 
<span class="p1">(<span class="default"><a href="abcmiz_0.html#K7" target="_self">apply</a> <font color="Maroon">v1</font>,<font color="Maroon">t</font></span>)</span> <a href="rewrite1.html#K1">$^</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K7" target="_self">apply</a> <font color="Maroon">v2</font>,<span class="p2">(<span class="default"><font color="Maroon">v1</font> <a href="abcmiz_0.html#K8" target="_self">ast</a> <font color="Maroon">t</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">tt</font> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K7" target="_self">apply</a> <font color="Maroon">v2</font>,<span class="p2">(<span class="default"><font color="Maroon">v1</font> <a href="abcmiz_0.html#K8" target="_self">ast</a> <font color="Maroon">t</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="rewrite1.html#T2">REWRITE1:2</a></i>
;<br/>
<a NAME="E5:70"/>
 <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">q</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#T56">FINSEQ_1:56</a></i>;<br/>
<a NAME="E6:70"/><b>then </b>
<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> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">tt</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i>, , <a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>;<br/>
<a NAME="E7:70"/><b>then </b><i><font color="Green">E46</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">v1</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">tt</font>
 
;<br/>
<b>let </b><font color="Maroon">i</font> be  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>assume </b><a NAME="E8:70"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">v1</font>
 ;<br/>

<a NAME="E9:70"/><b>then </b>
( <font color="Maroon">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">tt</font> &amp; <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="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E10:70"/><b>then </b>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">tt</font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E11:70"/><b>then </b>
( <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="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">tt</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">tt</font> <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">v1</font>,<font color="Maroon">t</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> )
 
<b>by </b><i>, <a class="ref" href="abcmiz_0.html#D1" target="_self">Def1</a>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/>
<b>hence </b><a NAME="E12:70"/>
<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="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">v1</font>,<font color="Maroon">t</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 ;<br/>


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