<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">G</font> be   <a href="group_1.html#NM1">Group</a>;<br/><b>let </b><font color="Maroon">I</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K4">INT</a> ;<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>for <font color="Olive">I</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K4">INT</a>   st  <a href="finseq_1.html#K3">len</a> <font color="Olive">I</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="finsop_1.html#K1">|-&gt;</a> <span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">G</font></span>)</span></span>)</span> <a href="group_4.html#K4" target="_self">|^</a> <font color="Olive">I</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> <a href="finsop_1.html#K1">|-&gt;</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">G</font></span>)</span>;<br/>
<a NAME="E1:34"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">I</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K4">INT</a> ;<br/>

<b>assume </b><a NAME="E1:34_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">I</font> <a href="hidden.html#R1">=</a> 0
 ;<br/>

<a NAME="E2:34_1"/><b>then </b>
( <font color="Maroon">I</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K6">&lt;*&gt;</a> <a href="numbers.html#K4">INT</a>  &amp; 0 <a href="finsop_1.html#K1">|-&gt;</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">G</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K6">&lt;*&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T32">FINSEQ_1:32</a>, <a class="ref" href="finseq_2.html#T72">FINSEQ_2:72</a></i>;<br/>
<b>hence </b><a NAME="E3:34_1"/>
<span class="p1">(<span class="default">0 <a href="finsop_1.html#K1">|-&gt;</a> <span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">G</font></span>)</span></span>)</span> <a href="group_4.html#K4" target="_self">|^</a> <font color="Maroon">I</font> <a href="hidden.html#R1">=</a> 0 <a href="finsop_1.html#K1">|-&gt;</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">G</font></span>)</span>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E2:34"/><i><font color="Green">E33</font></i>: 
for <font color="Olive">k</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">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:34_2"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>]
 ;<br/>

<b>let </b><font color="Maroon">I</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K4">INT</a> ;<br/>

<b>assume </b><a NAME="E2:34_2"/><i><font color="Green">E37</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">I</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 ;<br/>

<b>reconsider </b><font color="Maroon">J</font> = <font color="Maroon">I</font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font></span>)</span> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K4">INT</a>  <b>by </b><i><a class="ref" href="finseq_1.html#T23">FINSEQ_1:23</a></i>;<br/>
<a NAME="E4:34_2"/><i><font color="Green">E38</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">J</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 
<b>by </b><i>, <a class="ref" href="finseq_3.html#T59">FINSEQ_3:59</a></i>;<br/>
<a NAME="E5:34_2"/><b>then </b><i><font color="Green">E39</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="finsop_1.html#K1">|-&gt;</a> <span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">G</font></span>)</span></span>)</span> <a href="group_4.html#K4" target="_self">|^</a> <font color="Maroon">J</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="finsop_1.html#K1">|-&gt;</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">G</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E6:34_2"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/>
<a NAME="E7:34_2"/><b>then </b>
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">I</font>
 
<b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E8:34_2"/><b>then </b>
( <font color="Maroon">I</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">I</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">I</font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K4">INT</a>  )
 
<b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">i</font> = <font color="Maroon">I</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> as   <a href="int_1.html#NM1">Integer</a> <b>by </b><i><a class="ref" href="int_1.html#D2">INT_1:def 2</a></i>;<br/>
<a NAME="E10:34_2"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">I</font> <a href="hidden.html#R1">=</a> <font color="Maroon">J</font> <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="group_4.html#K2" target="_self">@</a> <font color="Maroon">i</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
 
<b>by </b><i>, <a class="ref" href="finseq_3.html#T61">FINSEQ_3:61</a></i>;<br/>
<a NAME="E11:34_2"/><i><font color="Green">E41</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="finsop_1.html#K1">|-&gt;</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">G</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="finsop_1.html#K1">|-&gt;</a> <span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">G</font></span>)</span></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="group_1.html#K2">1.</a> <font color="Maroon">G</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
 
<b>by </b><i><a class="ref" href="finseq_2.html#T74">FINSEQ_2:74</a></i>;<br/>
<a NAME="E12:34_2"/>
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="finsop_1.html#K1">|-&gt;</a> <span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">G</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">G</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R1">=</a> 1 &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="group_4.html#K2" target="_self">@</a> <font color="Maroon">i</font></span>)</span></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#T57">FINSEQ_1:57</a>, <a class="ref" href="finseq_2.html#T69">FINSEQ_2:69</a></i>;<br/>
<b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="finsop_1.html#K1">|-&gt;</a> <span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">G</font></span>)</span></span>)</span> <a href="group_4.html#K4" target="_self">|^</a> <font color="Maroon">I</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="finsop_1.html#K1">|-&gt;</a> <span class="p3">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">G</font></span>)</span></span>)</span> <a href="group_4.html#K4" target="_self">|^</a> <font color="Maroon">J</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">G</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="group_4.html#K4" target="_self">|^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="group_4.html#K2" target="_self">@</a> <font color="Maroon">i</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span>
<b>by </b><i>, , , </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="finsop_1.html#K1">|-&gt;</a> <span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">G</font></span>)</span></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"><span class="p3">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">G</font></span>)</span> <a href="group_1.html#K6">|^</a> <font color="Maroon">i</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="finsop_1.html#K1">|-&gt;</a> <span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">G</font></span>)</span></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="group_1.html#K2">1.</a> <font color="Maroon">G</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
<b>by </b><i><a class="ref" href="group_1.html#T61">GROUP_1:61</a></i>
;<br/>
<b>hence </b><a NAME="E14:34_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="finsop_1.html#K1">|-&gt;</a> <span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">G</font></span>)</span></span>)</span> <a href="group_4.html#K4" target="_self">|^</a> <font color="Maroon">I</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="finsop_1.html#K1">|-&gt;</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">G</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_2.html#T74">FINSEQ_2:74</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:34"/>
for <font color="Olive">k</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">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );<br/></i>
<b>hence </b><a NAME="E4:34"/>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">I</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> implies <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="finsop_1.html#K1">|-&gt;</a> <span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">G</font></span>)</span></span>)</span> <a href="group_4.html#K4" target="_self">|^</a> <font color="Maroon">I</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="finsop_1.html#K1">|-&gt;</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">G</font></span>)</span> )
 ;<br/>


</div>
