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

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



<b>assume </b><a NAME="E1:37"/><i><font color="Green">E51</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default">0,<font color="Maroon">n</font></span><a href="domain_1.html#K7">}</a></span>
 ;<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>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="Olive">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> holds <br/><font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="stirl2_1.html#K8">.</a> <font color="Olive">k</font></span>)</span>;<br/>
<a NAME="E2:37"/><i><font color="Green">E53</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="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">p</font> holds <br/> ex <font color="Olive">x</font> being   <a href="subset_1.html#M2">Element</a> of <span class="p1"><a href="domain_1.html#K7">{</a><span class="default">0,<font color="Maroon">n</font></span><a href="domain_1.html#K7">}</a></span> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>,<font color="Olive">x</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1"><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="E1:37_1"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">p</font>
 ;<br/>

<a NAME="E2:37_1"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">p</font>
 
<b>by </b><i>, <a class="ref" href="afinsq_1.html#D1">AFINSQ_1:def 1</a></i>;<br/>
<a NAME="E3:37_1"/><b>then </b>
<font color="Maroon">p</font> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/>
<a NAME="E4:37_1"/><b>then </b>
( <font color="Maroon">p</font> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">p</font> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> )
 
<b>by </b><i>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<a NAME="E5:37_1"/><b>then </b>
( <font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default">0,<font color="Maroon">n</font></span><a href="domain_1.html#K7">}</a></span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>,<font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">k</font></span>)</span>] )
 
<b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<b>hence </b><a NAME="E6:37_1"/>
 ex <font color="Olive">x</font> being   <a href="subset_1.html#M2">Element</a> of <span class="p1"><a href="domain_1.html#K7">{</a><span class="default">0,<font color="Maroon">n</font></span><a href="domain_1.html#K7">}</a></span> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>,<font color="Olive">x</font>]
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">q</font> being   <a href="afinsq_1.html#NM2">XFinSequence</a> of <span class="p1"><a href="domain_1.html#K7">{</a><span class="default">0,<font color="Maroon">n</font></span><a href="domain_1.html#K7">}</a></span><b> such that </b><br/><a NAME="E4:37"/><i><font color="Green">E57</font></i>: 
(  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a>  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">p</font> &amp; ( 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="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">p</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>,<font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font>] ) )
 <b>from </b><i><a class="ref" href="stirl2_1.html#S6">STIRL2_1:sch 6</a>();<br/></i>
<a NAME="E5:37"/>
(  <a href="relat_1.html#K2">rng</a> <font color="Maroon">q</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default">0,<font color="Maroon">n</font></span><a href="domain_1.html#K7">}</a></span> &amp; <span class="p1"><a href="domain_1.html#K7">{</a><span class="default">0,<font color="Maroon">n</font></span><a href="domain_1.html#K7">}</a></span> is   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K5">NAT</a>  )
 
<b>by </b><i><a class="ref" href="ordinal1.html#D8">ORDINAL1:def 8</a></i>;<br/>
<a NAME="E6:37"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">q</font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">q</font> = <font color="Maroon">q</font> as   <a href="afinsq_1.html#NM2">XFinSequence</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D8">ORDINAL1:def 8</a></i>;<br/>
<b>take </b>
<font color="Maroon">q</font>
;<br/>

<b>thus </b><a NAME="E8:37"/>
 <a href="afinsq_1.html#K1">len</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">q</font>
 <b>by </b><i>, <a class="ref" href="afinsq_1.html#D1">AFINSQ_1:def 1</a></i>;<br/>

<b>thus </b><a NAME="E9:37"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">q</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default">0,<font color="Maroon">n</font></span><a href="domain_1.html#K7">}</a></span>
 <b>by </b><i><a class="ref" href="ordinal1.html#D8">ORDINAL1:def 8</a></i>;<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">p</font> implies <span class="p1">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <font color="Maroon">a<sub>1</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="card_fin.html#K4">|</a> <font color="Maroon">a<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="binop_2.html#K24">*</a> <font color="Maroon">a<sub>1</sub></font> );<br/>
<a NAME="E10:37"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:37_2"/>
0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">p</font>
 ;<br/>

<a NAME="E2:37_2"/>
 <a href="afinsq_1.html#K2">dom</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
;<br/>
<a NAME="E3:37_2"/><b>then </b>
(  <a href="afinsq_1.html#K1">len</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">p</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">q</font> <a href="card_fin.html#K4">|</a> 0 <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="binop_1.html#K3">the_unity_wrt</a> <a href="binop_2.html#K47">addnat</a>  <a href="hidden.html#R1">=</a> 0 )
 
<b>by </b><i><a class="ref" href="relat_1.html#T110">RELAT_1:110</a>, <a class="ref" href="afinsq_1.html#D1">AFINSQ_1:def 1</a>, <a class="ref" href="binop_2.html#T5">BINOP_2:5</a></i>;<br/>
<a NAME="E4:37_2"/><b>then </b>
( <a href="binop_2.html#K47">addnat</a>  <a href="stirl2_1.html#K4">"**"</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> 0</span>)</span> <a href="hidden.html#R1">=</a> 0 &amp; <a href="binop_2.html#K47">addnat</a>  <a href="stirl2_1.html#K4">"**"</a> <span class="p1">(<span class="default"><font color="Maroon">q</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="stirl2_1.html#D3">STIRL2_1:def 3</a></i>;<br/>
<b>hence </b><a NAME="E5:37_2"/>
<span class="p1">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> 0</span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="card_fin.html#K4">|</a> 0</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="binop_2.html#K24">*</a> 0
 ;<br/>


</div><b>end;</b></div>
<a NAME="E11:37"/><i><font color="Green">E59</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>2</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Maroon">S<sub>2</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="37_3"><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="E1:37_3"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">k</font>]
 ;<br/>

<b>set </b><font color="Maroon">k1</font> = <font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1;<br/>
<b>assume </b><a NAME="E2:37_3"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">k</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">p</font>
 ;<br/>

<a NAME="E3:37_3"/><b>then </b>
( <font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1 <a href="tarski.html#R1">c=</a>  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">p</font> &amp;  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">p</font> )
 
<b>by </b><i><a class="ref" href="card_1.html#T56">CARD_1:56</a>, <a class="ref" href="afinsq_1.html#D1">AFINSQ_1:def 1</a></i>;<br/>
<a NAME="E4:37_3"/><b>then </b>
(  <a href="afinsq_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1 &amp;  <a href="afinsq_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1 &amp; <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="relat_1.html#T91">RELAT_1:91</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E5:37_3"/><b>then </b>
(  <a href="afinsq_1.html#K1">len</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1 &amp;  <a href="afinsq_1.html#K1">len</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1 &amp; <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</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; <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="card_fin.html#K4">|</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; <font color="Maroon">k</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1 &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">p</font> )
 
<b>by </b><i>, <a class="ref" href="afinsq_1.html#D1">AFINSQ_1:def 1</a>, <a class="ref" href="euler_1.html#T1">EULER_1:1</a>, <a class="ref" href="card_1.html#T56">CARD_1:56</a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<a NAME="E6:37_3"/><b>then </b>
( <font color="Maroon">p</font> <a href="card_fin.html#K4">|</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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="card_fin.html#K4">|</a> <font color="Maroon">k</font></span>)</span> <a href="afinsq_1.html#K5">^</a> <span class="p1"><a href="afinsq_1.html#K9">&lt;%</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">k</font></span>)</span></span><a href="afinsq_1.html#K9">%&gt;</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">k</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="card_fin.html#K4">|</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <font color="Maroon">k</font> &amp; <font color="Maroon">q</font> <a href="card_fin.html#K4">|</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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="card_fin.html#K4">|</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="card_fin.html#K4">|</a> <font color="Maroon">k</font></span>)</span> <a href="afinsq_1.html#K5">^</a> <span class="p1"><a href="afinsq_1.html#K9">&lt;%</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="card_fin.html#K4">|</a> <span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">k</font></span>)</span></span><a href="afinsq_1.html#K9">%&gt;</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">k</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="card_fin.html#K4">|</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="card_fin.html#K4">|</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">p</font> )
 
<b>by </b><i><a class="ref" href="card_fin.html#T43">CARD_FIN:43</a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a>, <a class="ref" href="relat_1.html#T103">RELAT_1:103</a>, <a class="ref" href="euler_1.html#T1">EULER_1:1</a></i>;<br/>
<a NAME="E7:37_3"/><b>then </b>
(  <a href="stirl2_1.html#K7">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><font color="Maroon">k</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"><a href="stirl2_1.html#K7">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <span class="p2"><a href="afinsq_1.html#K9">&lt;%</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">k</font></span>)</span></span><a href="afinsq_1.html#K9">%&gt;</a></span></span>)</span> &amp; <font color="Maroon">q</font> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">k</font></span>)</span> &amp;  <a href="stirl2_1.html#K7">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><font color="Maroon">k</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"><a href="stirl2_1.html#K7">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="card_fin.html#K4">|</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <span class="p2"><a href="afinsq_1.html#K9">&lt;%</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">k</font></span>)</span></span><a href="afinsq_1.html#K9">%&gt;</a></span></span>)</span> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E8:37_3"/><b>then </b>
(  <a href="stirl2_1.html#K7">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><font color="Maroon">k</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"><a href="stirl2_1.html#K7">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</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">p</font> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">k</font></span>)</span> &amp;  <a href="stirl2_1.html#K7">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="card_fin.html#K4">|</a> <span class="p2">(<span class="default"><font color="Maroon">k</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"><a href="stirl2_1.html#K7">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="card_fin.html#K4">|</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">n</font> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E9:37_3"/>
<span class="p1">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="card_fin.html#K4">|</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>
 <b>by </b><i>, , <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E12:37"/>
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>2</sub></font>[<font color="Olive">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );<br/></i>
<b>hence </b><a NAME="E13:37"/>
( ( 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="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">p</font> holds <br/><span class="p1">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="card_fin.html#K4">|</a> <font color="Olive">k</font></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="card_fin.html#K4">|</a> <font color="Olive">k</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="binop_2.html#K24">*</a> <font color="Olive">k</font> ) &amp; ( 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="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">p</font> holds <br/><font color="Maroon">q</font> <a href="stirl2_1.html#K8">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="stirl2_1.html#K8">.</a> <font color="Olive">k</font></span>)</span> ) )
 <b>by </b><i/>;<br/>


</div>
