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

<b>let </b><font color="Maroon" title="c1">A</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">p</font>, <font color="Maroon" title="c3">q</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">A</font>;<br/>





<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b>for <font color="Olive" title="b1">q</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">A</font>  st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 holds <br/><span class="p1"><a href="monoid_1.html#K20" title="MONOID_1:func.20">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="monoid_0.html#K1" title="MONOID_0:func.1">^</a> <font color="Olive" title="b1">q</font></span>)</span></span><a href="monoid_1.html#K20" title="MONOID_1:func.20">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="monoid_1.html#K20" title="MONOID_1:func.20">|.</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="monoid_1.html#K20" title="MONOID_1:func.20">.|</a></span> <a href="monoid_0.html#K2" title="MONOID_0:func.2">[*]</a> <span class="p1"><a href="monoid_1.html#K20" title="MONOID_1:func.20">|.</a><span class="default"><font color="Olive" title="b1">q</font></span><a href="monoid_1.html#K20" title="MONOID_1:func.20">.|</a></span>;<br/>
<a NAME="E1:68"/><i><font color="Green" title="E38">A1</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<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" title="c4">q</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">A</font>;<br/>

<b>assume </b><a NAME="E1:68_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/>

<a NAME="E2:68_1"/><b>then </b>
<font color="Maroon" title="c4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#T25" title="FINSEQ_1:th.25">FINSEQ_1:25</a></i>;<br/>
<a NAME="E3:68_1"/><b>then </b>
( <font color="Maroon" title="c2">p</font> <a href="monoid_0.html#K1" title="MONOID_0:func.1">^</a> <font color="Maroon" title="c4">q</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c2">p</font> &amp; <font color="Maroon" title="c4">q</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a> <font color="Maroon" title="c1">A</font> &amp; <span class="p1"><a href="monoid_1.html#K20" title="MONOID_1:func.20">|.</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a> <font color="Maroon" title="c1">A</font></span>)</span></span><a href="monoid_1.html#K20" title="MONOID_1:func.20">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">A</font> <a href="monoid_1.html#K17" title="MONOID_1:func.17">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c1">A</font> <a href="monoid_1.html#K17" title="MONOID_1:func.17">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>( <a href="monoid_1.html#K13" title="MONOID_1:func.13">MultiSet_over</a> <font color="Maroon" title="c1">A</font>) )
 
<b>by </b><i><a class="ref" href="monoid_1.html#T27" target="_self" title="MONOID_1:th.27">Th27</a>, <a class="ref" href="monoid_1.html#T38" target="_self" title="MONOID_1:th.38">Th38</a>, <a class="ref" href="finseq_1.html#T47" title="FINSEQ_1:th.47">FINSEQ_1:47</a></i>;<br/>
<b>hence </b><a NAME="E4:68_1"/>
<span class="p1"><a href="monoid_1.html#K20" title="MONOID_1:func.20">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="monoid_0.html#K1" title="MONOID_0:func.1">^</a> <font color="Maroon" title="c4">q</font></span>)</span></span><a href="monoid_1.html#K20" title="MONOID_1:func.20">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="monoid_1.html#K20" title="MONOID_1:func.20">|.</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="monoid_1.html#K20" title="MONOID_1:func.20">.|</a></span> <a href="monoid_0.html#K2" title="MONOID_0:func.2">[*]</a> <span class="p1"><a href="monoid_1.html#K20" title="MONOID_1:func.20">|.</a><span class="default"><font color="Maroon" title="c4">q</font></span><a href="monoid_1.html#K20" title="MONOID_1:func.20">.|</a></span>
 <b>by </b><i><a class="ref" href="monoid_0.html#T18" title="MONOID_0:th.18">MONOID_0:18</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:68"/><i><font color="Green" title="E39">A2</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</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" title="c4">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:68_2"/><i><font color="Green" title="E40">A3</font></i>: 
for <font color="Olive" title="b1">q</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">A</font>  st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">n</font> holds <br/><span class="p1"><a href="monoid_1.html#K20" title="MONOID_1:func.20">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="monoid_0.html#K1" title="MONOID_0:func.1">^</a> <font color="Olive" title="b1">q</font></span>)</span></span><a href="monoid_1.html#K20" title="MONOID_1:func.20">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="monoid_1.html#K20" title="MONOID_1:func.20">|.</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="monoid_1.html#K20" title="MONOID_1:func.20">.|</a></span> <a href="monoid_0.html#K2" title="MONOID_0:func.2">[*]</a> <span class="p1"><a href="monoid_1.html#K20" title="MONOID_1:func.20">|.</a><span class="default"><font color="Olive" title="b1">q</font></span><a href="monoid_1.html#K20" title="MONOID_1:func.20">.|</a></span>
 ;<br/>

<b>let </b><font color="Maroon" title="c5">q</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">A</font>;<br/>

<b>assume </b><a NAME="E2:68_2"/><i><font color="Green" title="E41">A4</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ;<br/>

<a NAME="E3:68_2"/><b>then </b>
<font color="Maroon" title="c5">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#T25" title="FINSEQ_1:th.25">FINSEQ_1:25</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c6">r</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>, <font color="Maroon" title="c7">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:68_2"/><i><font color="Green" title="E42">A5</font></i>: 
<font color="Maroon" title="c5">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">r</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c7">x</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T63" title="FINSEQ_1:th.63">FINSEQ_1:63</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">r</font> = <font color="Maroon" title="c6">r</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">A</font> <b>by </b><i><a class="txt" href="monoid_1.html#E5:68_2"><i><font color="Green" title="E42">A5</font></i></a>, <a class="ref" href="finseq_1.html#T50" title="FINSEQ_1:th.50">FINSEQ_1:50</a></i>;<br/>
<a NAME="E7:68_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c7">x</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="ref" href="finseq_1.html#T57" title="FINSEQ_1:th.57">FINSEQ_1:57</a></i>;<br/>
<a NAME="E8:68_2"/><b>then </b><i><font color="Green" title="E43">A6</font></i>: 
<font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">r</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="txt" href="monoid_1.html#E2:68_2"><i><font color="Green" title="E41">A4</font></i></a>, <a class="txt" href="monoid_1.html#E5:68_2"><i><font color="Green" title="E42">A5</font></i></a>, <a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></i>;<br/>
<a NAME="E9:68_2"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c7">x</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c7">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T56" title="FINSEQ_1:th.56">FINSEQ_1:56</a></i>;<br/>
<a NAME="E10:68_2"/><b>then </b>
( <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c7">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c5">q</font> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c5">q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">A</font> )
 
<b>by </b><i><a class="txt" href="monoid_1.html#E5:68_2"><i><font color="Green" title="E42">A5</font></i></a>, <a class="ref" href="finseq_1.html#T43" title="FINSEQ_1:th.43">FINSEQ_1:43</a></i>;<br/>
<a NAME="E11:68_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c7">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">A</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c9">x</font> = <font color="Maroon" title="c7">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> <b>by </b><i><a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></i>;<br/>
<b>thus </b><span class="p1"><a href="monoid_1.html#K20" title="MONOID_1:func.20">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="monoid_0.html#K1" title="MONOID_0:func.1">^</a> <font color="Maroon" title="c5">q</font></span>)</span></span><a href="monoid_1.html#K20" title="MONOID_1:func.20">.|</a></span> = 
<span class="p1"><a href="monoid_1.html#K20" title="MONOID_1:func.20">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="monoid_0.html#K1" title="MONOID_0:func.1">^</a> <font color="Maroon" title="c8">r</font></span>)</span> <a href="monoid_0.html#K1" title="MONOID_0:func.1">^</a> <span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c9">x</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span></span><a href="monoid_1.html#K20" title="MONOID_1:func.20">.|</a></span>
<b>by </b><i><a class="txt" href="monoid_1.html#E5:68_2"><i><font color="Green" title="E42">A5</font></i></a>, <a class="ref" href="finseq_1.html#T45" title="FINSEQ_1:th.45">FINSEQ_1:45</a></i>
<br/>.= 
<span class="p1"><a href="monoid_1.html#K20" title="MONOID_1:func.20">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="monoid_0.html#K1" title="MONOID_0:func.1">^</a> <font color="Maroon" title="c8">r</font></span>)</span></span><a href="monoid_1.html#K20" title="MONOID_1:func.20">.|</a></span> <a href="monoid_0.html#K2" title="MONOID_0:func.2">[*]</a> <span class="p1">(<span class="default"><a href="monoid_1.html#K18" title="MONOID_1:func.18">chi</a> <font color="Maroon" title="c9">x</font></span>)</span>
<b>by </b><i><a class="ref" href="monoid_1.html#T40" target="_self" title="MONOID_1:th.40">Th40</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="monoid_1.html#K20" title="MONOID_1:func.20">|.</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="monoid_1.html#K20" title="MONOID_1:func.20">.|</a></span> <a href="monoid_0.html#K2" title="MONOID_0:func.2">[*]</a> <span class="p2"><a href="monoid_1.html#K20" title="MONOID_1:func.20">|.</a><span class="default"><font color="Maroon" title="c8">r</font></span><a href="monoid_1.html#K20" title="MONOID_1:func.20">.|</a></span></span>)</span> <a href="monoid_0.html#K2" title="MONOID_0:func.2">[*]</a> <span class="p1">(<span class="default"><a href="monoid_1.html#K18" title="MONOID_1:func.18">chi</a> <font color="Maroon" title="c9">x</font></span>)</span>
<b>by </b><i><a class="txt" href="monoid_1.html#E1:68_2"><i><font color="Green" title="E40">A3</font></i></a>, <a class="txt" href="monoid_1.html#E8:68_2"><i><font color="Green" title="E43">A6</font></i></a></i>
<br/>.= 
<span class="p1"><a href="monoid_1.html#K20" title="MONOID_1:func.20">|.</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="monoid_1.html#K20" title="MONOID_1:func.20">.|</a></span> <a href="monoid_0.html#K2" title="MONOID_0:func.2">[*]</a> <span class="p1">(<span class="default"><span class="p2"><a href="monoid_1.html#K20" title="MONOID_1:func.20">|.</a><span class="default"><font color="Maroon" title="c8">r</font></span><a href="monoid_1.html#K20" title="MONOID_1:func.20">.|</a></span> <a href="monoid_0.html#K2" title="MONOID_0:func.2">[*]</a> <span class="p2">(<span class="default"><a href="monoid_1.html#K18" title="MONOID_1:func.18">chi</a> <font color="Maroon" title="c9">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">GROUP_1:def 4</a></i>
<br/>.= 
<span class="p1"><a href="monoid_1.html#K20" title="MONOID_1:func.20">|.</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="monoid_1.html#K20" title="MONOID_1:func.20">.|</a></span> <a href="monoid_0.html#K2" title="MONOID_0:func.2">[*]</a> <span class="p1"><a href="monoid_1.html#K20" title="MONOID_1:func.20">|.</a><span class="default"><font color="Maroon" title="c5">q</font></span><a href="monoid_1.html#K20" title="MONOID_1:func.20">.|</a></span>
<b>by </b><i><a class="txt" href="monoid_1.html#E5:68_2"><i><font color="Green" title="E42">A5</font></i></a>, <a class="ref" href="monoid_1.html#T40" target="_self" title="MONOID_1:th.40">Th40</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E3:68"/><i><font color="Green" title="E40">A7</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="monoid_1.html#E1:68"><i><font color="Green" title="E38">A1</font></i></a>, <a class="txt" href="monoid_1.html#E2:68"><i><font color="Green" title="E39">A2</font></i></a>);<br/></i>
<a NAME="E4:68"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">q</font>
 
;<br/>
<b>hence </b><a NAME="E5:68"/>
<span class="p1"><a href="monoid_1.html#K20" title="MONOID_1:func.20">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="monoid_0.html#K1" title="MONOID_0:func.1">^</a> <font color="Maroon" title="c3">q</font></span>)</span></span><a href="monoid_1.html#K20" title="MONOID_1:func.20">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="monoid_1.html#K20" title="MONOID_1:func.20">|.</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="monoid_1.html#K20" title="MONOID_1:func.20">.|</a></span> <a href="monoid_0.html#K2" title="MONOID_0:func.2">[*]</a> <span class="p1"><a href="monoid_1.html#K20" title="MONOID_1:func.20">|.</a><span class="default"><font color="Maroon" title="c3">q</font></span><a href="monoid_1.html#K20" title="MONOID_1:func.20">.|</a></span>
 <b>by </b><i><a class="txt" href="monoid_1.html#E3:68"><i><font color="Green" title="E40">A7</font></i></a></i>;<br/>


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