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

<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>let </b><font color="Maroon">p</font> be   <a href="newton.html#NM1">Prime</a>;<br/>



<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">p</font> being  <a href="newton.html#NM1">Prime</a>  ex <font color="Olive">f</font> being   <a href="trees_4.html#M1">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>(  <a href="finseq_1.html#K3">len</a> <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></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="relset_1.html#K4">dom</a> <font color="Olive">f</font> holds <br/><font color="Olive">f</font> <a href="wsierp_1.html#K3">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="binop_2.html#K12">/</a> <span class="p3">(<span class="default"><font color="Olive">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Olive">k</font></span>)</span></span>)</span></span><a href="int_1.html#K1">/]</a></span> ) &amp; <font color="Olive">p</font> <a href="nat_3.html#K11">|-count</a> <span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="newton.html#K12">!</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="wsierp_1.html#K7">Sum</a> <font color="Olive">f</font> );<br/>
<a NAME="E1:55"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</font> be   <a href="newton.html#NM1">Prime</a>;<br/>

<b>set </b><font color="Maroon">f</font> =  <a href="wsierp_1.html#K5">&lt;*&gt;</a> <a href="numbers.html#K5">NAT</a> ;<br/>
<b>take </b>
 <a href="wsierp_1.html#K5">&lt;*&gt;</a> <a href="numbers.html#K5">NAT</a> 
;<br/>

<b>thus </b><a NAME="E1:55_1"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K5">&lt;*&gt;</a> <a href="numbers.html#K5">NAT</a> </span>)</span> <a href="hidden.html#R1">=</a> 0
 ;<br/>

<b>thus </b><a NAME="E2:55_1"/>
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="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K5">&lt;*&gt;</a> <a href="numbers.html#K5">NAT</a> </span>)</span> holds <br/><span class="p1">(<span class="default"><a href="wsierp_1.html#K5">&lt;*&gt;</a> <a href="numbers.html#K5">NAT</a> </span>)</span> <a href="wsierp_1.html#K3">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default">0 <a href="binop_2.html#K12">/</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Olive">k</font></span>)</span></span>)</span></span><a href="int_1.html#K1">/]</a></span>
 ;<br/>

<a NAME="E3:55_1"/>
<font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 
<b>by </b><i><a class="ref" href="int_2.html#D5">INT_2:def 5</a></i>;<br/>
<b>hence </b><a NAME="E4:55_1"/>
<font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <span class="p1">(<span class="default">0 <a href="newton.html#K12">!</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="wsierp_1.html#K7">Sum</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K5">&lt;*&gt;</a> <a href="numbers.html#K5">NAT</a> </span>)</span>
 <b>by </b><i><a class="ref" href="rvsum_1.html#T102">RVSUM_1:102</a>, <a class="ref" href="nat_3.html#T21">NAT_3:21</a>, <a class="ref" href="newton.html#T18">NEWTON:18</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:55"/><i><font color="Green">E39</font></i>: 
for <font color="Olive">n</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">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font> <a href="binop_2.html#K23">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_2"><b>proof </b></a><div class="add">

<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="E1:55_2"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>]
 ;<br/>

<a NAME="E2:55_2"/>
for <font color="Olive">p</font> being  <a href="newton.html#NM1">Prime</a>  ex <font color="Olive">f</font> being   <a href="trees_4.html#M1">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>(  <a href="finseq_1.html#K3">len</a> <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1 &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="relset_1.html#K4">dom</a> <font color="Olive">f</font> holds <br/><font color="Olive">f</font> <a href="wsierp_1.html#K3">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p3">(<span class="default"><font color="Olive">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Olive">k</font></span>)</span></span>)</span></span><a href="int_1.html#K1">/]</a></span> ) &amp; <font color="Olive">p</font> <a href="nat_3.html#K11">|-count</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="newton.html#K12">!</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="wsierp_1.html#K7">Sum</a> <font color="Olive">f</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</font> be   <a href="newton.html#NM1">Prime</a>;<br/>

<b>consider </b><font color="Maroon">fn</font> being    <a href="trees_4.html#M1">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E2:55_2_1"/><i><font color="Green">E46</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">fn</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</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="relset_1.html#K4">dom</a> <font color="Maroon">fn</font> holds <br/><font color="Maroon">fn</font> <a href="wsierp_1.html#K3">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K12">/</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Olive">k</font></span>)</span></span>)</span></span><a href="int_1.html#K1">/]</a></span> ) &amp; <font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="newton.html#K12">!</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="wsierp_1.html#K7">Sum</a> <font color="Maroon">fn</font> )
 <b>by </b><i><a class="txt" href="nat_4.html#E9:55_2_1"><i><font color="Green">E56</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">fn1</font> being    <a href="trees_4.html#M1">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E4:55_2_1"/><i><font color="Green">E47</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">fn1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1 &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="relset_1.html#K4">dom</a> <font color="Maroon">fn1</font> holds <br/>( ( <font color="Maroon">fn1</font> <a href="wsierp_1.html#K3">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> 1 implies <font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Olive">k</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1 ) &amp; ( <font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Olive">k</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1 implies <font color="Maroon">fn1</font> <a href="wsierp_1.html#K3">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> 1 ) &amp; ( <font color="Maroon">fn1</font> <a href="wsierp_1.html#K3">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> 0 implies  not <font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Olive">k</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1 ) &amp; ( not <font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Olive">k</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1 implies <font color="Maroon">fn1</font> <a href="wsierp_1.html#K3">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> 0 ) ) ) &amp; <font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="wsierp_1.html#K7">Sum</a> <font color="Maroon">fn1</font> )
 <b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">f</font> = <span class="p1">(<span class="default"><font color="Maroon">fn</font> <a href="wsierp_1.html#K4">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="rvsum_1.html#K3">+</a> <font color="Maroon">fn1</font>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_2_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:55_2_1_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">fn</font> <a href="wsierp_1.html#K4">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="rvsum_1.html#K3">+</a> <font color="Maroon">fn1</font></span>)</span>
 ;<br/><b>consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:55_2_1_1"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">fn</font> <a href="wsierp_1.html#K4">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="rvsum_1.html#K3">+</a> <font color="Maroon">fn1</font></span>)</span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">fn</font> <a href="wsierp_1.html#K4">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="rvsum_1.html#K3">+</a> <font color="Maroon">fn1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E4:55_2_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">fn</font> <a href="wsierp_1.html#K4">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="rvsum_1.html#K3">+</a> <font color="Maroon">fn1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">fn</font> <a href="wsierp_1.html#K4">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">x</font></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><font color="Maroon">fn1</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">x</font></span>)</span>
 <b>by </b><i><a class="txt" href="nat_4.html#E11:55_2_1"><i><font color="Green">E60</font></i></a>, <a class="ref" href="rvsum_1.html#T26">RVSUM_1:26</a></i>;<br/><b>hence </b><a NAME="E5:55_2_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="txt" href="nat_4.html#E11:55_2_1"><i><font color="Green">E60</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E6:55_2_1"/><b>then </b><i><font color="Green">E54</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">fn</font> <a href="wsierp_1.html#K4">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="rvsum_1.html#K3">+</a> <font color="Maroon">fn1</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">f</font> = <span class="p1">(<span class="default"><font color="Maroon">fn</font> <a href="wsierp_1.html#K4">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="rvsum_1.html#K3">+</a> <font color="Maroon">fn1</font> as    <a href="trees_4.html#M1">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i>, <a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<b>take </b>
<font color="Maroon">f</font>
;<br/>

<b>set </b><font color="Maroon">fn0</font> = <font color="Maroon">fn</font> <a href="wsierp_1.html#K4">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12">*&gt;</a></span>;<br/>
<a NAME="E8:55_2_1"/><i><font color="Green">E55</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><font color="Maroon">fn</font> <a href="wsierp_1.html#K4">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">fn1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="nat_4.html#E2:55_2_1"><i><font color="Green">E46</font></i></a></i>;<br/>
<i><font color="Green">E56</font></i>:  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">fn</font> <a href="wsierp_1.html#K4">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">fn</font></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>
<br/>.= 
<font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1
<b>by </b><i>, <a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>
;<br/>
<a NAME="E10:55_2_1"/><b>then </b><i><font color="Green">E58</font></i>: 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">fn</font> <a href="wsierp_1.html#K4">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E11:55_2_1"/><i><font color="Green">E60</font></i>: 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">fn1</font>
 
<b>by </b><i><a class="txt" href="nat_4.html#E10:55_2_1"><i><font color="Green">E58</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>thus </b><a NAME="E12:55_2_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1
 <b>by </b><i><a class="txt" href="nat_4.html#E11:55_2_1"><i><font color="Green">E60</font></i></a>, , , <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>

<b>thus </b><a NAME="E13:55_2_1"/>
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="relset_1.html#K4">dom</a> <font color="Maroon">f</font> holds <br/><font color="Maroon">f</font> <a href="wsierp_1.html#K3">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Olive">k</font></span>)</span></span>)</span></span><a href="int_1.html#K1">/]</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_2_1_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="E1:55_2_1_3"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 ;<br/>

<a NAME="E2:55_2_1_3"/><b>then </b><i><font color="Green">E62</font></i>: 
<font color="Maroon">f</font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">fn</font> <a href="wsierp_1.html#K4">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><font color="Maroon">fn1</font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">k</font></span>)</span>
 
<b>by </b><i><a class="ref" href="rvsum_1.html#T26">RVSUM_1:26</a></i>;<br/>
<a NAME="E3:55_2_1_3"/><i><font color="Green">E63</font></i>: 
( <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">fn</font> <a href="wsierp_1.html#K4">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> &amp; <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">fn1</font> )
 
<b>by </b><i><a class="txt" href="nat_4.html#E11:55_2_1"><i><font color="Green">E60</font></i></a>, <a class="txt" href="nat_4.html#E4:55_2_1_3"><i><font color="Green">E64</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E4:55_2_1_3"/><i><font color="Green">E64</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">fn</font> <a href="wsierp_1.html#K4">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K12">/</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font></span>)</span></span>)</span></span><a href="int_1.html#K1">/]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_2_1_3_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_2_1_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:55_2_1_3_1_1"/>
( <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">fn</font> or  ex <font color="Olive">n1</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">n1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12">*&gt;</a></span> &amp; <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">fn</font></span>)</span> <a href="binop_2.html#K23">+</a> <font color="Olive">n1</font> ) )
 </span><b>by </b><i>, <a class="ref" href="finseq_1.html#T38">FINSEQ_1:38</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_2_1_3_1_1_1"><b>suppose </b></a><a NAME="E1:55_2_1_3_1_1_1"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">fn</font>
 ;<br/><div class="add"><a NAME="E2:55_2_1_3_1_1_1"/><b>then </b>
<font color="Maroon">fn</font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K12">/</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font></span>)</span></span>)</span></span><a href="int_1.html#K1">/]</a></span>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:55_2_1_3_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">fn</font> <a href="wsierp_1.html#K4">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K12">/</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font></span>)</span></span>)</span></span><a href="int_1.html#K1">/]</a></span>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_2_1_3_1_1_2"><b>suppose </b></a><a NAME="E1:55_2_1_3_1_1_2"/><i><font color="Green">E68</font></i>: 
 ex <font color="Olive">n1</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">n1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12">*&gt;</a></span> &amp; <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">fn</font></span>)</span> <a href="binop_2.html#K23">+</a> <font color="Olive">n1</font> )
 ;<br/><div class="add"><b>consider </b><font color="Maroon">n1</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:55_2_1_3_1_1_2"/><i><font color="Green">E69</font></i>: 
( <font color="Maroon">n1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12">*&gt;</a></span> &amp; <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">fn</font></span>)</span> <a href="binop_2.html#K23">+</a> <font color="Maroon">n1</font> )
 <b>by </b><i><a class="txt" href="nat_4.html#E1:55_2_1_3_1_1_2"><i><font color="Green">E68</font></i></a></i>;<br/><a NAME="E4:55_2_1_3_1_1_2"/>
<font color="Maroon">n1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 1
 <b>by </b><i><a class="txt" href="nat_4.html#E3:55_2_1_3_1_1_2"><i><font color="Green">E69</font></i></a>, <a class="ref" href="finseq_1.html#T55">FINSEQ_1:55</a></i>;<br/><a NAME="E5:55_2_1_3_1_1_2"/><b>then </b><i><font color="Green">E70</font></i>: 
<font color="Maroon">n1</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a>, <a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a></i>;<br/><a NAME="E6:55_2_1_3_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">fn</font> <a href="wsierp_1.html#K4">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="wsierp_1.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="wsierp_1.html#K3">.</a> 1
 <b>by </b><i><a class="txt" href="nat_4.html#E5:55_2_1_3_1_1_2"><i><font color="Green">E70</font></i></a>, , <a class="txt" href="nat_4.html#E3:55_2_1_3_1_1_2"><i><font color="Green">E69</font></i></a>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/><a NAME="E7:55_2_1_3_1_1_2"/><b>then </b><i><font color="Green">E71</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">fn</font> <a href="wsierp_1.html#K4">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="nat_4.html#E5:55_2_1_3_1_1_2"><i><font color="Green">E70</font></i></a>, , <a class="txt" href="nat_4.html#E3:55_2_1_3_1_1_2"><i><font color="Green">E69</font></i></a>, <a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>;<br/><a NAME="E8:55_2_1_3_1_1_2"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K24">*</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="ref" href="newton.html#T11">NEWTON:11</a></i>;<br/><a NAME="E9:55_2_1_3_1_1_2"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">p</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 <b>by </b><i><a class="ref" href="int_2.html#D5">INT_2:def 5</a></i>;<br/><a NAME="E10:55_2_1_3_1_1_2"/><b>then </b><i><font color="Green">E74</font></i>: 
<font color="Maroon">p</font> <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><a NAME="E11:55_2_1_3_1_1_2"/>
<font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="txt" href="nat_4.html#E9:55_2_1_3_1_1_2"><i><font color="Green">E73</font></i></a>, </i>;<br/><a NAME="E12:55_2_1_3_1_1_2"/><b>then </b>
<font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="txt" href="nat_4.html#E5:55_2_1_3_1_1_2"><i><font color="Green">E70</font></i></a>, <a class="txt" href="nat_4.html#E10:55_2_1_3_1_1_2"><i><font color="Green">E74</font></i></a>, <a class="txt" href="nat_4.html#E8:55_2_1_3_1_1_2"><i><font color="Green">E72</font></i></a>, , <a class="txt" href="nat_4.html#E3:55_2_1_3_1_1_2"><i><font color="Green">E69</font></i></a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E13:55_2_1_3_1_1_2"/><b>then </b>
<font color="Maroon">n</font> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T191">XREAL_1:191</a></i>;<br/><a NAME="E14:55_2_1_3_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K12">/</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> 1 <a href="xxreal_0.html#R1">&lt;</a> 1 <a href="binop_2.html#K10">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><b>hence </b><a NAME="E15:55_2_1_3_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">fn</font> <a href="wsierp_1.html#K4">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K12">/</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font></span>)</span></span>)</span></span><a href="int_1.html#K1">/]</a></span>
 <b>by </b><i><a class="txt" href="nat_4.html#E7:55_2_1_3_1_1_2"><i><font color="Green">E71</font></i></a>, <a class="ref" href="int_1.html#D4">INT_1:def 4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E5:55_2_1_3"/>
<font color="Maroon">p</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 
<b>by </b><i><a class="ref" href="int_2.html#D5">INT_2:def 5</a></i>;<br/>
<a NAME="E6:55_2_1_3"/><b>then </b><i><font color="Green">E75</font></i>: 
<font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<b>by </b><i><a class="ref" href="prepower.html#T19">PREPOWER:19</a></i>;<br/>
<a NAME="E7:55_2_1_3"/><i><font color="Green">E76</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="xcmplx_1.html#T60">XCMPLX_1:60</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_2_1_3_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:55_2_1_3_2"/>
( <font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1 or  not <font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_2_1_3_2_1"><b>suppose </b></a><a NAME="E1:55_2_1_3_2_1"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1
 ;<br/><div class="add"><b>consider </b><font color="Maroon">r</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:55_2_1_3_2_1"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/><i><font color="Green">E79</font></i>: <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font></span>)</span> = 
<font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="xcmplx_1.html#T78">XCMPLX_1:78</a></i>
<br/>.= 
<font color="Maroon">r</font> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K12">/</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T60">XCMPLX_1:60</a></i>
<br/>.= 
<font color="Maroon">r</font>

;<br/><a NAME="E5:55_2_1_3_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K10">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="binop_2.html#K10">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a>, </i>;<br/><a NAME="E6:55_2_1_3_2_1"/><b>then </b><i><font color="Green">E80</font></i>: 
0 <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K10">-</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T74">XREAL_1:74</a></i>;<br/><a NAME="E7:55_2_1_3_2_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="binop_2.html#K7">-</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> 1 <a href="xxreal_0.html#R1">&lt;</a> 0
 <b>by </b><i/>;<br/><a NAME="E8:55_2_1_3_2_1"/><b>then </b><i><font color="Green">E81</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p3">(<span class="default"><a href="binop_2.html#K7">-</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> 1 <a href="xxreal_0.html#R1">&lt;</a> 0
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T63">XCMPLX_1:63</a></i>;<br/><span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K12">/</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font></span>)</span></span>)</span></span><a href="int_1.html#K1">/]</a></span> <a href="xcmplx_0.html#K2">+</a> 1 = 
<span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K12">/</a> <span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> 1</span>)</span></span><a href="int_1.html#K1">/]</a></span>
<b>by </b><i><a class="ref" href="int_1.html#T51">INT_1:51</a></i>
<br/>.= 
<span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K12">/</a> <span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span></span><a href="int_1.html#K1">/]</a></span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T60">XCMPLX_1:60</a></i>
<br/>.= 
<span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> <span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font></span>)</span></span>)</span></span><a href="int_1.html#K1">/]</a></span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T63">XCMPLX_1:63</a></i>
<br/>.= 
<span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K9">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K10">-</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font></span>)</span></span>)</span></span><a href="int_1.html#K1">/]</a></span>

<br/>.= 
<span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K10">-</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span></span><a href="int_1.html#K1">/]</a></span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T63">XCMPLX_1:63</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K10">-</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font></span>)</span></span>)</span></span><a href="int_1.html#K1">/]</a></span>
<b>by </b><i>, <a class="ref" href="int_1.html#T51">INT_1:51</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> 0
<b>by </b><i>, , <a class="ref" href="int_1.html#D4">INT_1:def 4</a></i>
<br/>.= 
<span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font></span>)</span></span>)</span></span><a href="int_1.html#K1">/]</a></span>
<b>by </b><i><a class="ref" href="int_1.html#T47">INT_1:47</a>, </i>
;<br/><b>hence </b><a NAME="E10:55_2_1_3_2_1"/>
<font color="Maroon">f</font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font></span>)</span></span>)</span></span><a href="int_1.html#K1">/]</a></span>
 <b>by </b><i>, , , , <a class="txt" href="nat_4.html#E10:55_2_1"><i><font color="Green">E58</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_2_1_3_2_2"><b>suppose </b></a><a NAME="E1:55_2_1_3_2_2"/><i><font color="Green">E82</font></i>: 
not <font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1
 ;<br/><div class="add"><a NAME="E2:55_2_1_3_2_2"/><b>then </b><i><font color="Green">E83</font></i>: 
<font color="Maroon">fn1</font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, <a class="txt" href="nat_4.html#E10:55_2_1"><i><font color="Green">E58</font></i></a></i>;<br/><b>set </b><font color="Maroon">d</font> = <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font></span>)</span>;<br/><b>set </b><font color="Maroon">m</font> = <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font></span>)</span>;<br/><a NAME="E3:55_2_1_3_2_2"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1 <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="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K24">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T47">NAT_1:47</a></i>;<br/><a NAME="E4:55_2_1_3_2_2"/>
not <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, , <a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/><a NAME="E5:55_2_1_3_2_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><a NAME="E6:55_2_1_3_2_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 <a href="binop_2.html#K23">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E7:55_2_1_3_2_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E8:55_2_1_3_2_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="binop_2.html#K10">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E9:55_2_1_3_2_2"/><b>then </b><i><font color="Green">E85</font></i>: 
0 <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T74">XREAL_1:74</a></i>;<br/><a NAME="E10:55_2_1_3_2_2"/><i><font color="Green">E86</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T46">NAT_1:46</a></i>;<br/><a NAME="E11:55_2_1_3_2_2"/>
0 <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> 1 <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E12:55_2_1_3_2_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K23">+</a> 1
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E13:55_2_1_3_2_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K10">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E14:55_2_1_3_2_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T76">XREAL_1:76</a></i>;<br/><a NAME="E15:55_2_1_3_2_2"/><b>then </b><i><font color="Green">E87</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> 1 <a href="xxreal_0.html#R1">&lt;</a> 1 <a href="binop_2.html#K10">-</a> 1
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E16:55_2_1_3_2_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T76">XREAL_1:76</a>, </i>;<br/><a NAME="E17:55_2_1_3_2_2"/><b>then </b><i><font color="Green">E88</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> 1 <a href="xxreal_0.html#R1">&lt;</a> 1 <a href="binop_2.html#K10">-</a> 1
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E18:55_2_1_3_2_2"/>
<font color="Maroon">n</font> <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="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K24">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> 1</span>)</span>
 <b>by </b><i/>;<br/><b>then </b><span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K12">/</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font></span>)</span></span>)</span></span><a href="int_1.html#K1">/]</a></span> = 
<span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K24">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <span class="p0">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p0">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span></span><a href="int_1.html#K1">/]</a></span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T63">XCMPLX_1:63</a></i>
<br/>.= 
<span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <span class="p5">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p5">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p0">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span></span><a href="int_1.html#K1">/]</a></span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T75">XCMPLX_1:75</a></i>
<br/>.= 
<span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <span class="p5">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binop_2.html#K24">*</a> 1</span>)</span> <a href="binop_2.html#K9">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p0">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span></span><a href="int_1.html#K1">/]</a></span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T60">XCMPLX_1:60</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p5">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font></span>)</span></span>)</span></span><a href="int_1.html#K1">/]</a></span>
<b>by </b><i><a class="ref" href="int_1.html#T51">INT_1:51</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> 0
<b>by </b><i>, <a class="txt" href="nat_4.html#E6:55_2_1_3"><i><font color="Green">E75</font></i></a>, <a class="ref" href="int_1.html#D4">INT_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font></span>)</span></span>)</span></span><a href="int_1.html#K1">/]</a></span>
<b>by </b><i><a class="txt" href="nat_4.html#E7:55_2_1_3"><i><font color="Green">E76</font></i></a>, <a class="ref" href="int_1.html#D4">INT_1:def 4</a></i>
<br/>.= 
<span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <span class="p5">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binop_2.html#K24">*</a> 1</span>)</span> <a href="binop_2.html#K9">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p5">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span></span><a href="int_1.html#K1">/]</a></span>
<b>by </b><i><a class="ref" href="int_1.html#T51">INT_1:51</a></i>
<br/>.= 
<span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <span class="p5">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p5">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p5">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span></span><a href="int_1.html#K1">/]</a></span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T60">XCMPLX_1:60</a></i>
<br/>.= 
<span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K24">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <span class="p0">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p5">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span></span><a href="int_1.html#K1">/]</a></span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T75">XCMPLX_1:75</a></i>
<br/>.= 
<span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K24">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <span class="p0">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p5">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font></span>)</span></span>)</span></span><a href="int_1.html#K1">/]</a></span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T63">XCMPLX_1:63</a></i>
<br/>.= 
<span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font></span>)</span></span>)</span></span><a href="int_1.html#K1">/]</a></span>
<b>by </b><i><a class="ref" href="nat_1.html#T47">NAT_1:47</a></i>
;<br/><b>hence </b><a NAME="E20:55_2_1_3_2_2"/>
<font color="Maroon">f</font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font></span>)</span></span>)</span></span><a href="int_1.html#K1">/]</a></span>
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>

<b>reconsider </b><font color="Maroon">fn0</font> = <font color="Maroon">fn</font> <a href="wsierp_1.html#K4">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12">*&gt;</a></span> as    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K1">REAL</a>  <b>by </b><i>, <a class="ref" href="finseq_2.html#T110">FINSEQ_2:110</a></i>;<br/>
<b>reconsider </b><font color="Maroon">fn1</font> = <font color="Maroon">fn1</font> as    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="txt" href="nat_4.html#E10:55_2_1"><i><font color="Green">E58</font></i></a>, <a class="ref" href="finseq_2.html#T110">FINSEQ_2:110</a></i>;<br/>
<i><font color="Green">E90</font></i>:  <a href="wsierp_1.html#K7">Sum</a> <font color="Maroon">f</font> = 
<span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">fn0</font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">fn1</font></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T119">RVSUM_1:119</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="wsierp_1.html#K7">Sum</a> <font color="Maroon">fn</font></span>)</span> <a href="binop_2.html#K23">+</a> 0</span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="nat_4.html#E10:55_2_1"><i><font color="Green">E58</font></i></a>, <a class="ref" href="rvsum_1.html#T104">RVSUM_1:104</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="newton.html#K12">!</a> </span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>
<b>by </b><i/>
;<br/>
<a NAME="E17:55_2_1"/>
( <font color="Maroon">n</font> <a href="newton.html#K12">!</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 
<b>by </b><i><a class="ref" href="newton.html#T23">NEWTON:23</a></i>;<br/>
<b>then </b><span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="newton.html#K12">!</a> </span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> = 
<font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="newton.html#K12">!</a> </span>)</span> <a href="binop_2.html#K24">*</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="nat_3.html#T28">NAT_3:28</a></i>
<br/>.= 
<font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="newton.html#K12">!</a> </span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T21">NEWTON:21</a></i>
;<br/>
<b>hence </b><a NAME="E19:55_2_1"/>
<font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="newton.html#K12">!</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="wsierp_1.html#K7">Sum</a> <font color="Maroon">f</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:55_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:55"/><i><font color="Green">E91</font></i>: 
for <font color="Olive">n</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">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );<br/></i>
<b>consider </b><font color="Maroon">f</font> being    <a href="trees_4.html#M1">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E5:55"/><i><font color="Green">E92</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</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="relset_1.html#K4">dom</a> <font color="Maroon">f</font> holds <br/><font color="Maroon">f</font> <a href="wsierp_1.html#K3">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K12">/</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Olive">k</font></span>)</span></span>)</span></span><a href="int_1.html#K1">/]</a></span> ) &amp; <font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="newton.html#K12">!</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="wsierp_1.html#K7">Sum</a> <font color="Maroon">f</font> )
 <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">f</font>
;<br/>

<b>thus </b><a NAME="E6:55"/>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</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="relset_1.html#K4">dom</a> <font color="Maroon">f</font> holds <br/><font color="Maroon">f</font> <a href="wsierp_1.html#K3">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K12">/</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Olive">k</font></span>)</span></span>)</span></span><a href="int_1.html#K1">/]</a></span> ) &amp; <font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="newton.html#K12">!</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="wsierp_1.html#K7">Sum</a> <font color="Maroon">f</font> )
 <b>by </b><i/>;<br/>


</div>
