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<b>let </b><font color="Maroon">f</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a> ;<br/>

<b>assume </b><a NAME="E1:8"/><i><font color="Green">E52</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/>

<a NAME="E2:8"/><b>then </b><i><font color="Green">E53</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <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"><a href="calcul_1.html#K1" target="_self">Ant</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i/>;<br/>
<a NAME="E3:8"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <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"><a href="calcul_1.html#K1" target="_self">Ant</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="calcul_1.html#K2" target="_self">Suc</a> <font color="Maroon">f</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>;<br/>
<a NAME="E4:8"/><b>then </b><i><font color="Green">E54</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="calcul_1.html#K1" target="_self">Ant</a> <font color="Maroon">f</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="calcul_1.html#K2" target="_self">Suc</a> <font color="Maroon">f</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>;<br/>
<div><i><font color="Green">E55</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">j</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:8_1"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span>
 ;<br/><a NAME="E2:8_1"/><i><font color="Green">E61</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="calcul_1.html#K1" target="_self">Ant</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><div><i><font color="Green">E62</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:8_1_1"/>
<font color="Maroon">j</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="calcul_1.html#K1" target="_self">Ant</a> <font color="Maroon">f</font></span>)</span>
 ;<br/><a NAME="E2:8_1_1"/><b>then </b><i><font color="Green">E65</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="calcul_1.html#K1" target="_self">Ant</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E3:8_1_1"/>
 <a href="calcul_1.html#K1" target="_self">Ant</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="relset_1.html#K8">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="calcul_1.html#K1" target="_self">Ant</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E4:8_1_1"/><b>then </b>
 <a href="calcul_1.html#K1" target="_self">Ant</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="relset_1.html#K8">|</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><a href="calcul_1.html#K1" target="_self">Ant</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E5:8_1_1"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="calcul_1.html#K1" target="_self">Ant</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/><b>hence </b><a NAME="E6:8_1_1"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="calcul_1.html#K1" target="_self">Ant</a> <font color="Maroon">f</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="calcul_1.html#K2" target="_self">Suc</a> <font color="Maroon">f</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:8_1_2"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">j</font> <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"><a href="calcul_1.html#K1" target="_self">Ant</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
 ;<br/><a NAME="E2:8_1_2"/><b>then </b>
<font color="Maroon">j</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font>
 <b>by </b><i>, </i>;<br/><a NAME="E3:8_1_2"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a>  <a href="calcul_1.html#K2" target="_self">Suc</a> <font color="Maroon">f</font>
 <b>by </b><i>, </i>;<br/><a NAME="E4:8_1_2"/><b>then </b><i><font color="Green">E67</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="calcul_1.html#K2" target="_self">Suc</a> <font color="Maroon">f</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="funct_1.html#K1">.</a> 1
 <b>by </b><i><a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>;<br/><a NAME="E5:8_1_2"/>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 1
 <b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E6:8_1_2"/><b>then </b>
1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="calcul_1.html#K2" target="_self">Suc</a> <font color="Maroon">f</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T55">FINSEQ_1:55</a></i>;<br/><b>hence </b><a NAME="E7:8_1_2"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="calcul_1.html#K1" target="_self">Ant</a> <font color="Maroon">f</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="calcul_1.html#K2" target="_self">Suc</a> <font color="Maroon">f</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:8_1"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="calcul_1.html#K1" target="_self">Ant</a> <font color="Maroon">f</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="calcul_1.html#K2" target="_self">Suc</a> <font color="Maroon">f</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 <b>by </b><i>, , <a class="ref" href="nat_1.html#T26">NAT_1:26</a></i>;<br/></div><b>end;</b></div>
<a NAME="E6:8"/><b>then </b>
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="calcul_1.html#K1" target="_self">Ant</a> <font color="Maroon">f</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="calcul_1.html#K2" target="_self">Suc</a> <font color="Maroon">f</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
 
<b>by </b><i>, <a class="ref" href="finseq_2.html#T10">FINSEQ_2:10</a></i>;<br/>
<a NAME="E7:8"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><a href="calcul_1.html#K1" target="_self">Ant</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="calcul_1.html#K2" target="_self">Suc</a> <font color="Maroon">f</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T44">FINSEQ_1:44</a></i>;<br/>
<b>hence </b><a NAME="E8:8"/>
( <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="calcul_1.html#K1" target="_self">Ant</a> <font color="Maroon">f</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="calcul_1.html#K2" target="_self">Suc</a> <font color="Maroon">f</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><a href="calcul_1.html#K1" target="_self">Ant</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="calcul_1.html#K2" target="_self">Suc</a> <font color="Maroon">f</font></span>)</span></span><a href="domain_1.html#K6">}</a></span> )
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#T55">FINSEQ_1:55</a>, <a class="ref" href="finseq_2.html#T10">FINSEQ_2:10</a></i>;<br/>


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