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

<b>let </b><font color="Maroon">Fr</font> be   <a href="afinsq_1.html#NM2">XFinSequence</a> of  <a href="numbers.html#K1">REAL</a> ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b><font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">Fr</font> <a href="catalan2.html#K5" target="_self">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>;<br/>
<a NAME="E1:71"/><i><font color="Green">E51</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">Fr</font> holds <br/> ex <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">i</font>,<font color="Olive">x</font>]
 
;<br/>
<b>consider </b><font color="Maroon">absFr</font> being   <a href="afinsq_1.html#NM2">XFinSequence</a> of  <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E3:71"/><i><font color="Green">E53</font></i>: 
 <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">absFr</font> <a href="hidden.html#R1">=</a>  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">Fr</font>
 <b>and </b><br/><a NAME="E4:71"/><i><font color="Green">E54</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">Fr</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">i</font>,<font color="Maroon">absFr</font> <a href="catalan2.html#K5" target="_self">.</a> <font color="Olive">i</font>]
 <b>from </b><i><a class="ref" href="stirl2_1.html#S6">STIRL2_1:sch 6</a>();<br/></i>
<b>take </b>
<font color="Maroon">absFr</font>
;<br/>

<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>( <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">Fr</font> implies  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="catalan2.html#K4" target="_self">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">Fr</font> <a href="card_fin.html#K4">|</a> <font color="Maroon">a<sub>1</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="catalan2.html#K4" target="_self">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">absFr</font> <a href="card_fin.html#K4">|</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> );<br/>
<a NAME="E5:71"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1"><b>proof </b></a><div class="add">

<a NAME="E1:71_1"/>
( <font color="Maroon">Fr</font> <a href="card_fin.html#K4">|</a> 0 <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">absFr</font> <a href="card_fin.html#K4">|</a> 0 <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i><a class="ref" href="relat_1.html#T110">RELAT_1:110</a></i>;<br/>
<a NAME="E2:71_1"/><b>then </b>
(  <a href="catalan2.html#K4" target="_self">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">Fr</font> <a href="card_fin.html#K4">|</a> 0</span>)</span> <a href="hidden.html#R1">=</a> 0 &amp;  <a href="catalan2.html#K4" target="_self">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">absFr</font> <a href="card_fin.html#K4">|</a> 0</span>)</span> <a href="hidden.html#R1">=</a> 0 )
 
<b>by </b><i>, <a class="ref" href="relat_1.html#T60">RELAT_1:60</a></i>;<br/>
<b>hence </b><a NAME="E3:71_1"/>
<font color="Maroon">S<sub>2</sub></font>[0]
 <b>by </b><i><a class="ref" href="complex1.html#T130">COMPLEX1:130</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:71"/><i><font color="Green">E58</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">i</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">i</font> <a href="binop_2.html#K23">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:71_2"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">i</font>]
 ;<br/>

<b>set </b><font color="Maroon">i1</font> = <font color="Maroon">i</font> <a href="binop_2.html#K23">+</a> 1;<br/>
<b>assume </b><a NAME="E2:71_2"/>
<font color="Maroon">i</font> <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">Fr</font>
 ;<br/>

<a NAME="E3:71_2"/><b>then </b>
( <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">Fr</font> &amp;  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">Fr</font> <a href="hidden.html#R1">=</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">Fr</font> )
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a>, <a class="ref" href="afinsq_1.html#D1">AFINSQ_1:def 1</a></i>;<br/>
<a NAME="E4:71_2"/><b>then </b>
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">Fr</font> &amp; <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">Fr</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">Fr</font> )
 
<b>by </b><i><a class="ref" href="euler_1.html#T1">EULER_1:1</a></i>;<br/>
<a NAME="E5:71_2"/><b>then </b>
(  <a href="catalan2.html#K4" target="_self">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">Fr</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">Fr</font> <a href="catalan2.html#K5" target="_self">.</a> <font color="Maroon">i</font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><a href="catalan2.html#K4" target="_self">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">Fr</font> <a href="card_fin.html#K4">|</a> <font color="Maroon">i</font></span>)</span></span>)</span> &amp;  <a href="catalan2.html#K4" target="_self">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">absFr</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">absFr</font> <a href="catalan2.html#K5" target="_self">.</a> <font color="Maroon">i</font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><a href="catalan2.html#K4" target="_self">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">absFr</font> <a href="card_fin.html#K4">|</a> <font color="Maroon">i</font></span>)</span></span>)</span> &amp;  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="catalan2.html#K4" target="_self">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">Fr</font> <a href="card_fin.html#K4">|</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="catalan2.html#K4" target="_self">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">absFr</font> <a href="card_fin.html#K4">|</a> <font color="Maroon">i</font></span>)</span> &amp; <font color="Maroon">absFr</font> <a href="catalan2.html#K5" target="_self">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">Fr</font> <a href="catalan2.html#K5" target="_self">.</a> <font color="Maroon">i</font></span>)</span> )
 
<b>by </b><i>, , , <a class="txt" href="catalan2.html#E1:71"><i><font color="Green">E51</font></i></a></i>;<br/>
<a NAME="E6:71_2"/><b>then </b>
(  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="catalan2.html#K4" target="_self">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">Fr</font> <a href="card_fin.html#K4">|</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">absFr</font> <a href="catalan2.html#K5" target="_self">.</a> <font color="Maroon">i</font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><a href="catalan2.html#K4" target="_self">Sum</a> <span class="p3">(<span class="default"><font color="Maroon">Fr</font> <a href="card_fin.html#K4">|</a> <font color="Maroon">i</font></span>)</span></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">absFr</font> <a href="catalan2.html#K5" target="_self">.</a> <font color="Maroon">i</font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><a href="catalan2.html#K4" target="_self">Sum</a> <span class="p3">(<span class="default"><font color="Maroon">Fr</font> <a href="card_fin.html#K4">|</a> <font color="Maroon">i</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="catalan2.html#K4" target="_self">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">absFr</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="complex1.html#T142">COMPLEX1:142</a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<b>hence </b><a NAME="E7:71_2"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="catalan2.html#K4" target="_self">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">Fr</font> <a href="card_fin.html#K4">|</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="catalan2.html#K4" target="_self">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">absFr</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:71"/>
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">i</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, <a class="ref" href="catalan2.html#T4" target="_self">Th4</a>);<br/></i>
<a NAME="E8:71"/><b>then </b>
(  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="catalan2.html#K4" target="_self">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">Fr</font> <a href="card_fin.html#K4">|</a> <span class="p3">(<span class="default"><a href="afinsq_1.html#K1">len</a> <font color="Maroon">Fr</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="catalan2.html#K4" target="_self">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">absFr</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><a href="afinsq_1.html#K1">len</a> <font color="Maroon">Fr</font></span>)</span></span>)</span> &amp;  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">Fr</font> <a href="hidden.html#R1">=</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">Fr</font> &amp; <font color="Maroon">Fr</font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K2">dom</a> <font color="Maroon">Fr</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">Fr</font> &amp; <font color="Maroon">absFr</font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K2">dom</a> <font color="Maroon">absFr</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">absFr</font> )
 
<b>by </b><i><a class="ref" href="relat_1.html#T98">RELAT_1:98</a>, <a class="ref" href="afinsq_1.html#D1">AFINSQ_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E9:71"/>
(  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">absFr</font> <a href="hidden.html#R1">=</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">Fr</font> &amp;  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="catalan2.html#K4" target="_self">Sum</a> <font color="Maroon">Fr</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="catalan2.html#K4" target="_self">Sum</a> <font color="Maroon">absFr</font> &amp; ( for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">absFr</font> holds <br/><font color="Maroon">absFr</font> <a href="catalan2.html#K5" target="_self">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">Fr</font> <a href="catalan2.html#K5" target="_self">.</a> <font color="Olive">i</font></span>)</span> ) )
 <b>by </b><i>, </i>;<br/>


</div>
