<?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> 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="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">A</font>] <b>means </b><span class="p1"><a href="monoid_1.html#K20" title="MONOID_1:func.20">|.</a><span class="default">$1</span><a href="monoid_1.html#K20" title="MONOID_1:func.20">.|</a></span> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="monoid_1.html#K19" title="MONOID_1:func.19">finite-MultiSet_over</a> <font color="Maroon" title="c1">A</font></span>)</span>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</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">p</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">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">p</font>];<br/>
<a NAME="E1:70"/><i><font color="Green" title="E39">A1</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">p</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="E1:70_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/>

<a NAME="E2:70_1"/><b>then </b>
<font color="Maroon" title="c3">p</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>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T32" title="FINSEQ_1:th.32">FINSEQ_1:32</a></i>;<br/>
<b>then </b><span class="p1"><a href="monoid_1.html#K20" title="MONOID_1:func.20">|.</a><span class="default"><font color="Maroon" title="c3">p</font></span><a href="monoid_1.html#K20" title="MONOID_1:func.20">.|</a></span> = 
<font color="Maroon" title="c1">A</font> <a href="monoid_1.html#K17" title="MONOID_1:func.17">--&gt;</a> 0
<b>by </b><i><a class="ref" href="monoid_1.html#T38" target="_self" title="MONOID_1:th.38">Th38</a></i>
<br/>.= 
<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></i>
<br/>.= 
<font color="Maroon">H<sub>2</sub></font>( <a href="monoid_1.html#K19" title="MONOID_1:func.19">finite-MultiSet_over</a> <font color="Maroon" title="c1">A</font>)
<b>by </b><i><a class="ref" href="monoid_0.html#D24" title="MONOID_0:def.24">MONOID_0:def 24</a></i>
;<br/>
<b>hence </b><a NAME="E4:70_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">p</font>]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E2:70"/><i><font color="Green" title="E40">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>2</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">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="E1:70_2"/><i><font color="Green" title="E41">A3</font></i>: 
for <font color="Olive" title="b1">p</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">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">p</font>]
 ;<br/>

<b>let </b><font color="Maroon" title="c4">p</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="E2:70_2"/><i><font color="Green" title="E42">A4</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ;<br/>

<a NAME="E3:70_2"/><b>then </b>
<font color="Maroon" title="c4">p</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="c5">r</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>, <font color="Maroon" title="c6">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:70_2"/><i><font color="Green" title="E43">A5</font></i>: 
<font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">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="c6">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="c7">r</font> = <font color="Maroon" title="c5">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#E5:70_2"><i><font color="Green" title="E43">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="E7:70_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="c6">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="E8:70_2"/><b>then </b><i><font color="Green" title="E44">A6</font></i>: 
<font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c7">r</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="txt" href="monoid_1.html#E2:70_2"><i><font color="Green" title="E42">A4</font></i></a>, <a class="txt" href="monoid_1.html#E5:70_2"><i><font color="Green" title="E43">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="E9:70_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="c6">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="c6">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="E10:70_2"/><b>then </b>
( <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">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="c4">p</font> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">p</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#E5:70_2"><i><font color="Green" title="E43">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="E11:70_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">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="c8">x</font> = <font color="Maroon" title="c6">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>set </b><font color="Maroon" title="c9">M</font> =  <a href="monoid_1.html#K19" title="MONOID_1:func.19">finite-MultiSet_over</a> <font color="Maroon" title="c1">A</font>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">r'</font> = <span class="p1"><a href="monoid_1.html#K20" title="MONOID_1:func.20">|.</a><span class="default"><font color="Maroon" title="c7">r</font></span><a href="monoid_1.html#K20" title="MONOID_1:func.20">.|</a></span>, <font color="Maroon" title="c11">a</font> =  <a href="monoid_1.html#K18" title="MONOID_1:func.18">chi</a> <font color="Maroon" title="c8">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="monoid_1.html#K19" title="MONOID_1:func.19">finite-MultiSet_over</a> <font color="Maroon" title="c1">A</font></span>)</span> <b>by </b><i><a class="txt" href="monoid_1.html#E1:70_2"><i><font color="Green" title="E41">A3</font></i></a>, <a class="txt" href="monoid_1.html#E8:70_2"><i><font color="Green" title="E44">A6</font></i></a>, <a class="ref" href="monoid_1.html#T34" target="_self" title="MONOID_1:th.34">Th34</a></i>;<br/>
<a NAME="E14:70_2"/>
 <a href="monoid_1.html#K19" title="MONOID_1:func.19">finite-MultiSet_over</a> <font color="Maroon" title="c1">A</font> is    <a href="monoid_0.html#M2" title="MONOID_0:mode.2">SubStr</a> of  <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_0.html#T23" title="MONOID_0:th.23">MONOID_0:23</a></i>;<br/>
<b>then </b><font color="Maroon" title="c10">r'</font> <a href="monoid_0.html#K2" title="MONOID_0:func.2">[*]</a> <font color="Maroon" title="c11">a</font> = 
<span class="p1"><a href="monoid_1.html#K20" title="MONOID_1:func.20">|.</a><span class="default"><font color="Maroon" title="c7">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="p1">(<span class="default"><a href="monoid_1.html#K18" title="MONOID_1:func.18">chi</a> <font color="Maroon" title="c8">x</font></span>)</span>
<b>by </b><i><a class="ref" href="monoid_0.html#T27" title="MONOID_0:th.27">MONOID_0:27</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="c4">p</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#E5:70_2"><i><font color="Green" title="E43">A5</font></i></a>, <a class="ref" href="monoid_1.html#T40" target="_self" title="MONOID_1:th.40">Th40</a></i>
;<br/>
<b>hence </b><a NAME="E16:70_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">p</font>]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:70"/><i><font color="Green" title="E41">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>2</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#E1:70"><i><font color="Green" title="E39">A1</font></i></a>, <a class="txt" href="monoid_1.html#E2:70"><i><font color="Green" title="E40">A2</font></i></a>);<br/></i>
<a NAME="E4:70"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">p</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="c2">p</font>
 
;<br/>
<b>hence </b><a NAME="E5:70"/>
<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> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="monoid_1.html#K19" title="MONOID_1:func.19">finite-MultiSet_over</a> <font color="Maroon" title="c1">A</font></span>)</span>
 <b>by </b><i><a class="txt" href="monoid_1.html#E3:70"><i><font color="Green" title="E41">A7</font></i></a></i>;<br/>


</div>
