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

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>for <font color="Olive">Fn</font> being  <a href="afinsq_1.html#NM2">XFinSequence</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">Fi</font> being  <a href="afinsq_1.html#NM2">XFinSequence</a> of  <a href="numbers.html#K4">INT</a>   st <font color="Olive">Fi</font> <a href="hidden.html#R1">=</a> <font color="Olive">Fn</font> &amp;  <a href="afinsq_1.html#K2">dom</a> <font color="Olive">Fn</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> holds <br/> <a href="card_fin.html#K6" target="_self">Sum</a> <font color="Olive">Fi</font> <a href="hidden.html#R1">=</a>  <a href="stirl2_1.html#K7">Sum</a> <font color="Olive">Fn</font>;<br/>
<a NAME="E1:78"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="78_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">Fn</font> be   <a href="afinsq_1.html#NM2">XFinSequence</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">Fi</font> be   <a href="afinsq_1.html#NM2">XFinSequence</a> of  <a href="numbers.html#K4">INT</a> ;<br/>



<b>assume </b><a NAME="E1:78_1"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">Fi</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Fn</font> &amp;  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">Fn</font> <a href="hidden.html#R1">=</a> 0 )
 ;<br/>

<a NAME="E2:78_1"/>
 <a href="afinsq_1.html#K1">len</a> <font color="Maroon">Fn</font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i>, <a class="ref" href="afinsq_1.html#D1">AFINSQ_1:def 1</a></i>;<br/>
<a NAME="E3:78_1"/><b>then </b>
( <a href="binop_2.html#K47">addnat</a>  <a href="stirl2_1.html#K4">"**"</a> <font color="Maroon">Fn</font> <a href="hidden.html#R1">=</a> 0 &amp; <a href="binop_2.html#K44">addint</a>  <a href="stirl2_1.html#K4">"**"</a> <font color="Maroon">Fi</font> <a href="hidden.html#R1">=</a> 0 &amp; <a href="binop_2.html#K47">addnat</a>  <a href="stirl2_1.html#K4">"**"</a> <font color="Maroon">Fn</font> <a href="hidden.html#R1">=</a>  <a href="stirl2_1.html#K7">Sum</a> <font color="Maroon">Fn</font> )
 
<b>by </b><i>, <a class="ref" href="binop_2.html#T4">BINOP_2:4</a>, <a class="ref" href="binop_2.html#T5">BINOP_2:5</a>, <a class="ref" href="stirl2_1.html#D3">STIRL2_1:def 3</a>, <a class="ref" href="stirl2_1.html#D4">STIRL2_1:def 4</a></i>;<br/>
<b>hence </b><a NAME="E4:78_1"/>
 <a href="card_fin.html#K6" target="_self">Sum</a> <font color="Maroon">Fi</font> <a href="hidden.html#R1">=</a>  <a href="stirl2_1.html#K7">Sum</a> <font color="Maroon">Fn</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E2:78"/><i><font color="Green">E51</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font> <a href="binop_2.html#K23">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="78_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:78_2"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>]
 ;<br/>

<b>let </b><font color="Maroon">Fn</font> be   <a href="afinsq_1.html#NM2">XFinSequence</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">Fi</font> be   <a href="afinsq_1.html#NM2">XFinSequence</a> of  <a href="numbers.html#K4">INT</a> ;<br/>



<b>assume </b><a NAME="E2:78_2"/><i><font color="Green">E57</font></i>: 
( <font color="Maroon">Fi</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Fn</font> &amp;  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">Fn</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1 )
 ;<br/>

<a NAME="E3:78_2"/><i><font color="Green">E58</font></i>: 
( <a href="binop_2.html#K44">addint</a>  <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="binop_2.html#K44">addint</a>  <a href="stirl2_1.html#K4">"**"</a> <span class="p2">(<span class="default"><font color="Maroon">Fi</font> <a href="card_fin.html#K4" target="_self">|</a> <font color="Maroon">k</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">Fi</font> <a href="card_fin.html#K7" target="_self">.</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="binop_2.html#K44">addint</a>  <a href="stirl2_1.html#K4">"**"</a> <span class="p2">(<span class="default"><font color="Maroon">Fi</font> <a href="card_fin.html#K4" target="_self">|</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">Fi</font> <a href="card_fin.html#K7" target="_self">.</a> <font color="Maroon">k</font></span>)</span> &amp; <a href="binop_2.html#K47">addnat</a>  <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="binop_2.html#K47">addnat</a>  <a href="stirl2_1.html#K4">"**"</a> <span class="p2">(<span class="default"><font color="Maroon">Fn</font> <a href="card_fin.html#K4" target="_self">|</a> <font color="Maroon">k</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">Fn</font> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="binop_2.html#K47">addnat</a>  <a href="stirl2_1.html#K4">"**"</a> <span class="p2">(<span class="default"><font color="Maroon">Fn</font> <a href="card_fin.html#K4" target="_self">|</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><font color="Maroon">Fn</font> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">k</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="gr_cy_1.html#T14">GR_CY_1:14</a>, <a class="ref" href="binop_2.html#D23">BINOP_2:def 23</a></i>;<br/>
<a NAME="E4:78_2"/>
<font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E5:78_2"/><b>then </b>
( <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">Fn</font> &amp;  <a href="binop_2.html#K47">addnat</a>  <a href="setwiseo.html#NR1" target="_self">has_a_unity</a>  &amp;  <a href="binop_2.html#K44">addint</a>  <a href="setwiseo.html#NR1" target="_self">has_a_unity</a>  )
 
<b>by </b><i>, <a class="ref" href="euler_1.html#T1">EULER_1:1</a></i>;<br/>
<a NAME="E6:78_2"/><b>then </b><i><font color="Green">E59</font></i>: 
( <a href="binop_2.html#K47">addnat</a>  <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="binop_2.html#K47">addnat</a>  <a href="stirl2_1.html#K4">"**"</a> <span class="p2">(<span class="default"><font color="Maroon">Fn</font> <a href="card_fin.html#K4" target="_self">|</a> <font color="Maroon">k</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">Fn</font> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <a href="binop_2.html#K47">addnat</a>  <a href="stirl2_1.html#K4">"**"</a> <span class="p1">(<span class="default"><font color="Maroon">Fn</font> <a href="card_fin.html#K4" target="_self">|</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> &amp; <a href="binop_2.html#K44">addint</a>  <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="binop_2.html#K44">addint</a>  <a href="stirl2_1.html#K4">"**"</a> <span class="p2">(<span class="default"><font color="Maroon">Fi</font> <a href="card_fin.html#K4" target="_self">|</a> <font color="Maroon">k</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">Fi</font> <a href="card_fin.html#K7" target="_self">.</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <a href="binop_2.html#K44">addint</a>  <a href="stirl2_1.html#K4">"**"</a> <span class="p1">(<span class="default"><font color="Maroon">Fi</font> <a href="card_fin.html#K4" target="_self">|</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E7:78_2"/>
( <font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1 &amp;  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">Fn</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1 )
 
<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="E8:78_2"/><b>then </b>
(  <a href="afinsq_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Maroon">Fn</font> <a href="card_fin.html#K4" target="_self">|</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> &amp;  <a href="afinsq_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Maroon">Fi</font> <a href="card_fin.html#K4" target="_self">|</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> )
 
<b>by </b><i>, <a class="ref" href="afinsq_1.html#T14">AFINSQ_1:14</a></i>;<br/>
<a NAME="E9:78_2"/><b>then </b>
(  <a href="stirl2_1.html#K7">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">Fn</font> <a href="card_fin.html#K4" target="_self">|</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_fin.html#K6" target="_self">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">Fi</font> <a href="card_fin.html#K4" target="_self">|</a> <font color="Maroon">k</font></span>)</span> &amp;  <a href="card_fin.html#K6" target="_self">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">Fi</font> <a href="card_fin.html#K4" target="_self">|</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <a href="binop_2.html#K44">addint</a>  <a href="stirl2_1.html#K4">"**"</a> <span class="p1">(<span class="default"><font color="Maroon">Fi</font> <a href="card_fin.html#K4" target="_self">|</a> <font color="Maroon">k</font></span>)</span> &amp; <font color="Maroon">Fn</font> <a href="card_fin.html#K4" target="_self">|</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">Fn</font> &amp; <font color="Maroon">Fi</font> <a href="card_fin.html#K4" target="_self">|</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">Fi</font> &amp;  <a href="stirl2_1.html#K7">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">Fn</font> <a href="card_fin.html#K4" target="_self">|</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <a href="binop_2.html#K47">addnat</a>  <a href="stirl2_1.html#K4">"**"</a> <span class="p1">(<span class="default"><font color="Maroon">Fn</font> <a href="card_fin.html#K4" target="_self">|</a> <font color="Maroon">k</font></span>)</span> )
 
<b>by </b><i>, , <a class="ref" href="relat_1.html#T98">RELAT_1:98</a>, <a class="ref" href="stirl2_1.html#D4">STIRL2_1:def 4</a></i>;<br/>
<a NAME="E10:78_2"/><b>then </b>
 <a href="stirl2_1.html#K7">Sum</a> <font color="Maroon">Fn</font> <a href="hidden.html#R1">=</a> <a href="binop_2.html#K44">addint</a>  <a href="stirl2_1.html#K4">"**"</a> <font color="Maroon">Fi</font>
 
<b>by </b><i>, <a class="ref" href="card_fin.html#T3" target="_self">Th3</a>, <a class="ref" href="card_fin.html#T4" target="_self">Th4</a>, <a class="ref" href="stirl2_1.html#D4">STIRL2_1:def 4</a></i>;<br/>
<b>hence </b><a NAME="E11:78_2"/>
 <a href="card_fin.html#K6" target="_self">Sum</a> <font color="Maroon">Fi</font> <a href="hidden.html#R1">=</a>  <a href="stirl2_1.html#K7">Sum</a> <font color="Maroon">Fn</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:78"/><i><font color="Green">E60</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="card_fin.html#E4:78"><i><font color="Green">E61</font></i></a>, );<br/></i>
<b>let </b><font color="Maroon">Fn</font> be   <a href="afinsq_1.html#NM2">XFinSequence</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">Fi</font> be   <a href="afinsq_1.html#NM2">XFinSequence</a> of  <a href="numbers.html#K4">INT</a> ;<br/>



<b>assume </b><a NAME="E4:78"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">Fi</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Fn</font>
 ;<br/>

<a NAME="E5:78"/>
 <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">Fn</font> <a href="hidden.html#R1">=</a>  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">Fn</font>
 
<b>by </b><i><a class="ref" href="afinsq_1.html#D1">AFINSQ_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E6:78"/>
 <a href="card_fin.html#K6" target="_self">Sum</a> <font color="Maroon">Fi</font> <a href="hidden.html#R1">=</a>  <a href="stirl2_1.html#K7">Sum</a> <font color="Maroon">Fn</font>
 <b>by </b><i>, </i>;<br/>


</div>
