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

<b>let </b><font color="Maroon" title="c1">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>let </b><font color="Maroon" title="c2">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="c3">a</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">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><span class="p1"><a href="monoid_1.html#K20" title="MONOID_1:func.20">|.</a><span class="default"><span class="p2">(<span class="default">$1 <a href="monoid_1.html#K4" title="MONOID_1:func.4">.--&gt;</a> <font color="Maroon" title="c3">a</font></span>)</span></span><a href="monoid_1.html#K20" title="MONOID_1:func.20">.|</a></span> <a href="monoid_1.html#K15" title="MONOID_1:func.15">.</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1;<br/>
<a NAME="E1:69"/><i><font color="Green" title="E39">A1</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">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> </span>)</span> <a href="monoid_1.html#K15" title="MONOID_1:func.15">.</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="monoid_1.html#K4" title="MONOID_1:func.4">.--&gt;</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp;  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a> <font color="Maroon" title="c2">A</font> )
 
<b>by </b><i><a class="ref" href="finseq_2.html#T72" title="FINSEQ_2:th.72">FINSEQ_2:72</a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/>
<a NAME="E2:69"/><b>then </b><i><font color="Green" title="E40">A2</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<b>by </b><i><a class="ref" href="monoid_1.html#T38" target="_self" title="MONOID_1:th.38">Th38</a></i>;<br/>
<a NAME="E3:69"/><i><font color="Green" title="E41">A3</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="69_1"><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="E1:69_1"/><i><font color="Green" title="E42">A4</font></i>: 
<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="c4">n</font> <a href="monoid_1.html#K4" title="MONOID_1:func.4">.--&gt;</a> <font color="Maroon" title="c3">a</font></span>)</span></span><a href="monoid_1.html#K20" title="MONOID_1:func.20">.|</a></span> <a href="monoid_1.html#K15" title="MONOID_1:func.15">.</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">n</font>
 ;<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"><span class="p3">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="monoid_1.html#K4" title="MONOID_1:func.4">.--&gt;</a> <font color="Maroon" title="c3">a</font></span>)</span></span><a href="monoid_1.html#K20" title="MONOID_1:func.20">.|</a></span> <a href="monoid_1.html#K15" title="MONOID_1:func.15">.</a> <font color="Maroon" title="c3">a</font> = 
<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="c4">n</font> <a href="monoid_1.html#K4" title="MONOID_1:func.4">.--&gt;</a> <font color="Maroon" title="c3">a</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="c3">a</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> <a href="monoid_1.html#K15" title="MONOID_1:func.15">.</a> <font color="Maroon" title="c3">a</font>
<b>by </b><i><a class="ref" href="finseq_2.html#T74" title="FINSEQ_2:th.74">FINSEQ_2:74</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"><span class="p3">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="monoid_1.html#K4" title="MONOID_1:func.4">.--&gt;</a> <font color="Maroon" title="c3">a</font></span>)</span></span><a href="monoid_1.html#K20" title="MONOID_1:func.20">.|</a></span> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <span class="p2">(<span class="default"><a href="monoid_1.html#K18" title="MONOID_1:func.18">chi</a> <font color="Maroon" title="c3">a</font></span>)</span></span>)</span> <a href="monoid_1.html#K15" title="MONOID_1:func.15">.</a> <font color="Maroon" title="c3">a</font>
<b>by </b><i><a class="ref" href="monoid_1.html#T40" target="_self" title="MONOID_1:th.40">Th40</a></i>
<br/>.= 
<font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="monoid_1.html#K18" title="MONOID_1:func.18">chi</a> <font color="Maroon" title="c3">a</font></span>)</span> <a href="monoid_1.html#K15" title="MONOID_1:func.15">.</a> <font color="Maroon" title="c3">a</font></span>)</span>
<b>by </b><i><a class="txt" href="monoid_1.html#E1:69_1"><i><font color="Green" title="E42">A4</font></i></a>, <a class="ref" href="monoid_1.html#T30" target="_self" title="MONOID_1:th.30">Th30</a></i>
<br/>.= 
<font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
<b>by </b><i><a class="ref" href="monoid_1.html#T32" target="_self" title="MONOID_1:th.32">Th32</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E4:69"/>
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#E2:69"><i><font color="Green" title="E40">A2</font></i></a>, <a class="txt" href="monoid_1.html#E3:69"><i><font color="Green" title="E41">A3</font></i></a>);<br/></i>
<b>hence </b><a NAME="E5:69"/>
<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="c1">n</font> <a href="monoid_1.html#K4" title="MONOID_1:func.4">.--&gt;</a> <font color="Maroon" title="c3">a</font></span>)</span></span><a href="monoid_1.html#K20" title="MONOID_1:func.20">.|</a></span> <a href="monoid_1.html#K15" title="MONOID_1:func.15">.</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c4">b</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">A</font>;<br/>

<b>assume </b><a NAME="E6:69"/><i><font color="Green" title="E42">A5</font></i>: 
<font color="Maroon" title="c4">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">a</font>
 ;<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><span class="p1"><a href="monoid_1.html#K20" title="MONOID_1:func.20">|.</a><span class="default"><span class="p2">(<span class="default">$1 <a href="monoid_1.html#K4" title="MONOID_1:func.4">.--&gt;</a> <font color="Maroon" title="c3">a</font></span>)</span></span><a href="monoid_1.html#K20" title="MONOID_1:func.20">.|</a></span> <a href="monoid_1.html#K15" title="MONOID_1:func.15">.</a> <font color="Maroon" title="c4">b</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="E7:69"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">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> </span>)</span> <a href="monoid_1.html#K15" title="MONOID_1:func.15">.</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/>
<a NAME="E8:69"/><b>then </b><i><font color="Green" title="E43">A6</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<b>by </b><i><a class="txt" href="monoid_1.html#E1:69"><i><font color="Green" title="E39">A1</font></i></a>, <a class="ref" href="monoid_1.html#T38" target="_self" title="MONOID_1:th.38">Th38</a></i>;<br/>
<a NAME="E9:69"/><i><font color="Green" title="E44">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>   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#K2" title="NAT_1:func.2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">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="E1:69_2"/><i><font color="Green" title="E45">A8</font></i>: 
<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="c5">n</font> <a href="monoid_1.html#K4" title="MONOID_1:func.4">.--&gt;</a> <font color="Maroon" title="c3">a</font></span>)</span></span><a href="monoid_1.html#K20" title="MONOID_1:func.20">.|</a></span> <a href="monoid_1.html#K15" title="MONOID_1:func.15">.</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<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"><span class="p3">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="monoid_1.html#K4" title="MONOID_1:func.4">.--&gt;</a> <font color="Maroon" title="c3">a</font></span>)</span></span><a href="monoid_1.html#K20" title="MONOID_1:func.20">.|</a></span> <a href="monoid_1.html#K15" title="MONOID_1:func.15">.</a> <font color="Maroon" title="c4">b</font> = 
<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="c5">n</font> <a href="monoid_1.html#K4" title="MONOID_1:func.4">.--&gt;</a> <font color="Maroon" title="c3">a</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="c3">a</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> <a href="monoid_1.html#K15" title="MONOID_1:func.15">.</a> <font color="Maroon" title="c4">b</font>
<b>by </b><i><a class="ref" href="finseq_2.html#T74" title="FINSEQ_2:th.74">FINSEQ_2:74</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"><span class="p3">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="monoid_1.html#K4" title="MONOID_1:func.4">.--&gt;</a> <font color="Maroon" title="c3">a</font></span>)</span></span><a href="monoid_1.html#K20" title="MONOID_1:func.20">.|</a></span> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <span class="p2">(<span class="default"><a href="monoid_1.html#K18" title="MONOID_1:func.18">chi</a> <font color="Maroon" title="c3">a</font></span>)</span></span>)</span> <a href="monoid_1.html#K15" title="MONOID_1:func.15">.</a> <font color="Maroon" title="c4">b</font>
<b>by </b><i><a class="ref" href="monoid_1.html#T40" target="_self" title="MONOID_1:th.40">Th40</a></i>
<br/>.= 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="monoid_1.html#K18" title="MONOID_1:func.18">chi</a> <font color="Maroon" title="c3">a</font></span>)</span> <a href="monoid_1.html#K15" title="MONOID_1:func.15">.</a> <font color="Maroon" title="c4">b</font></span>)</span>
<b>by </b><i><a class="txt" href="monoid_1.html#E1:69_2"><i><font color="Green" title="E45">A8</font></i></a>, <a class="ref" href="monoid_1.html#T30" target="_self" title="MONOID_1:th.30">Th30</a></i>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="txt" href="monoid_1.html#E6:69"><i><font color="Green" title="E42">A5</font></i></a>, <a class="ref" href="monoid_1.html#T32" target="_self" title="MONOID_1:th.32">Th32</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E10:69"/>
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#E8:69"><i><font color="Green" title="E43">A6</font></i></a>, <a class="txt" href="monoid_1.html#E9:69"><i><font color="Green" title="E44">A7</font></i></a>);<br/></i>
<b>hence </b><a NAME="E11:69"/>
<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="c1">n</font> <a href="monoid_1.html#K4" title="MONOID_1:func.4">.--&gt;</a> <font color="Maroon" title="c3">a</font></span>)</span></span><a href="monoid_1.html#K20" title="MONOID_1:func.20">.|</a></span> <a href="monoid_1.html#K15" title="MONOID_1:func.15">.</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/>


</div>
