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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>





<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means for <font color="Olive">b<sub>1</sub></font> being   <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">c<sub>1</sub></font> st  <a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> holds <br/><span class="p1"><a href="monoid_1.html#K20" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="monoid_0.html#K1">^</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span><a href="monoid_1.html#K20" target="_self">.|</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="monoid_1.html#K20" target="_self">|.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="monoid_1.html#K20" target="_self">.|</a></span> <a href="monoid_0.html#K2">[*]</a> <span class="p1"><a href="monoid_1.html#K20" target="_self">|.</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="monoid_1.html#K20" target="_self">.|</a></span>;<br/>
<a NAME="E1:68"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E1:68_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/>

<a NAME="E2:68_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/>
<a NAME="E3:68_1"/><b>then </b>
( <font color="Maroon">c<sub>2</sub></font> <a href="monoid_0.html#K1">^</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K6">&lt;*&gt;</a> <font color="Maroon">c<sub>1</sub></font> &amp; <span class="p1"><a href="monoid_1.html#K20" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K6">&lt;*&gt;</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="monoid_1.html#K20" target="_self">.|</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="monoid_1.html#K17" target="_self">--&gt;</a> 0 &amp; <font color="Maroon">c<sub>1</sub></font> <a href="monoid_1.html#K17" target="_self">--&gt;</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>( <a href="monoid_1.html#K13" target="_self">MultiSet_over</a> <font color="Maroon">c<sub>1</sub></font>) )
 
<b>by </b><i><a class="ref" href="monoid_1.html#T27" target="_self">Th27</a>, <a class="ref" href="monoid_1.html#T38" target="_self">Th38</a>, <a class="ref" href="finseq_1.html#T47">FINSEQ_1:47</a></i>;<br/>
<b>hence </b><a NAME="E4:68_1"/>
<span class="p1"><a href="monoid_1.html#K20" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="monoid_0.html#K1">^</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="monoid_1.html#K20" target="_self">.|</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="monoid_1.html#K20" target="_self">|.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="monoid_1.html#K20" target="_self">.|</a></span> <a href="monoid_0.html#K2">[*]</a> <span class="p1"><a href="monoid_1.html#K20" target="_self">|.</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="monoid_1.html#K20" target="_self">.|</a></span>
 <b>by </b><i><a class="ref" href="monoid_0.html#T18">MONOID_0:18</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:68"/><i><font color="Green">E40</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:68_2"/><i><font color="Green">E41</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">c<sub>1</sub></font> st  <a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> holds <br/><span class="p1"><a href="monoid_1.html#K20" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="monoid_0.html#K1">^</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span><a href="monoid_1.html#K20" target="_self">.|</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="monoid_1.html#K20" target="_self">|.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="monoid_1.html#K20" target="_self">.|</a></span> <a href="monoid_0.html#K2">[*]</a> <span class="p1"><a href="monoid_1.html#K20" target="_self">|.</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="monoid_1.html#K20" target="_self">.|</a></span>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E2:68_2"/><i><font color="Green">E42</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1
 ;<br/>

<a NAME="E3:68_2"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="finseq_1.html#NM1">FinSequence</a>, <font color="Maroon">c<sub>7</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:68_2"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="finseq_1.html#K9">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T63">FINSEQ_1:63</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>6</sub></font> as    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="monoid_1.html#E5:68_2"><i><font color="Green">E43</font></i></a>, <a class="ref" href="finseq_1.html#T50">FINSEQ_1:50</a></i>;<br/>
<a NAME="E7:68_2"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="finseq_1.html#K9">*&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></i>;<br/>
<a NAME="E8:68_2"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <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">c<sub>8</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="txt" href="monoid_1.html#E2:68_2"><i><font color="Green">E42</font></i></a>, <a class="txt" href="monoid_1.html#E5:68_2"><i><font color="Green">E43</font></i></a>, <a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>;<br/>
<a NAME="E9:68_2"/><b>then </b><i><font color="Green">E44</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 
;<br/>
<a NAME="E10:68_2"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>;<br/>
<a NAME="E11:68_2"/><b>then </b>
( <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>5</sub></font> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="txt" href="monoid_1.html#E5:68_2"><i><font color="Green">E43</font></i></a>, <a class="ref" href="finseq_1.html#T43">FINSEQ_1:43</a></i>;<br/>
<a NAME="E12:68_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>7</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<b>thus </b><span class="p1"><a href="monoid_1.html#K20" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="monoid_0.html#K1">^</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="monoid_1.html#K20" target="_self">.|</a></span> = 
<span class="p1"><a href="monoid_1.html#K20" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="monoid_0.html#K1">^</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="monoid_0.html#K1">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>9</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span></span><a href="monoid_1.html#K20" target="_self">.|</a></span>
<b>by </b><i><a class="txt" href="monoid_1.html#E5:68_2"><i><font color="Green">E43</font></i></a>, <a class="ref" href="finseq_1.html#T45">FINSEQ_1:45</a></i>
<br/>.= 
<span class="p1"><a href="monoid_1.html#K20" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="monoid_0.html#K1">^</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span><a href="monoid_1.html#K20" target="_self">.|</a></span> <a href="monoid_0.html#K2">[*]</a> <span class="p1">(<span class="default"><a href="monoid_1.html#K18" target="_self">chi</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="monoid_1.html#T40" target="_self">Th40</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="monoid_1.html#K20" target="_self">|.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="monoid_1.html#K20" target="_self">.|</a></span> <a href="monoid_0.html#K2">[*]</a> <span class="p2"><a href="monoid_1.html#K20" target="_self">|.</a><span class="default"><font color="Maroon">c<sub>8</sub></font></span><a href="monoid_1.html#K20" target="_self">.|</a></span></span>)</span> <a href="monoid_0.html#K2">[*]</a> <span class="p1">(<span class="default"><a href="monoid_1.html#K18" target="_self">chi</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="monoid_1.html#E1:68_2"><i><font color="Green">E41</font></i></a>, <a class="txt" href="monoid_1.html#E9:68_2"><i><font color="Green">E44</font></i></a></i>
<br/>.= 
<span class="p1"><a href="monoid_1.html#K20" target="_self">|.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="monoid_1.html#K20" target="_self">.|</a></span> <a href="monoid_0.html#K2">[*]</a> <span class="p1">(<span class="default"><span class="p2"><a href="monoid_1.html#K20" target="_self">|.</a><span class="default"><font color="Maroon">c<sub>8</sub></font></span><a href="monoid_1.html#K20" target="_self">.|</a></span> <a href="monoid_0.html#K2">[*]</a> <span class="p2">(<span class="default"><a href="monoid_1.html#K18" target="_self">chi</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D4">GROUP_1:def 4</a></i>
<br/>.= 
<span class="p1"><a href="monoid_1.html#K20" target="_self">|.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="monoid_1.html#K20" target="_self">.|</a></span> <a href="monoid_0.html#K2">[*]</a> <span class="p1"><a href="monoid_1.html#K20" target="_self">|.</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="monoid_1.html#K20" target="_self">.|</a></span>
<b>by </b><i><a class="txt" href="monoid_1.html#E5:68_2"><i><font color="Green">E43</font></i></a>, <a class="ref" href="monoid_1.html#T40" target="_self">Th40</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E3:68"/><i><font color="Green">E41</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="monoid_1.html#E1:68"><i><font color="Green">E39</font></i></a>, <a class="txt" href="monoid_1.html#E2:68"><i><font color="Green">E40</font></i></a>);<br/></i>
<a NAME="E4:68"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font>
 
;<br/>
<b>hence </b><a NAME="E5:68"/>
<span class="p1"><a href="monoid_1.html#K20" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="monoid_0.html#K1">^</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="monoid_1.html#K20" target="_self">.|</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="monoid_1.html#K20" target="_self">|.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="monoid_1.html#K20" target="_self">.|</a></span> <a href="monoid_0.html#K2">[*]</a> <span class="p1"><a href="monoid_1.html#K20" target="_self">|.</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="monoid_1.html#K20" target="_self">.|</a></span>
 <b>by </b><i><a class="txt" href="monoid_1.html#E3:68"><i><font color="Green">E41</font></i></a></i>;<br/>


</div>
