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

<b>let </b><font color="Maroon" title="c1">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c2">p</font> be   <a href="newton.html#NM1" title="NEWTON:NM.1">Prime</a>;<br/>



<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b>for <font color="Olive" title="b1">p</font> being  <a href="newton.html#NM1" title="NEWTON:NM.1">Prime</a>  ex <font color="Olive" title="b2">f</font> being   <a href="trees_4.html#M1" title="TREES_4:mode.1">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 &amp; ( for <font color="Olive" title="b3">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b3">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Olive" title="b2">f</font> holds <br/><font color="Olive" title="b2">f</font> <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">.</a> <font color="Olive" title="b3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p2">(<span class="default">$1 <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Olive" title="b3">k</font></span>)</span></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span> ) &amp; <font color="Olive" title="b1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <span class="p1">(<span class="default">$1 <a href="newton.html#K12" title="NEWTON:func.12">!</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Sum</a> <font color="Olive" title="b2">f</font> );<br/>
<a NAME="E1:55"/><i><font color="Green" title="E54">A1</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" title="c3">p</font> be   <a href="newton.html#NM1" title="NEWTON:NM.1">Prime</a>;<br/>

<b>set </b><font color="Maroon" title="c4">f</font> =  <a href="wsierp_1.html#K5" title="WSIERP_1:func.5">&lt;*&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>
<b>take </b>
 <a href="wsierp_1.html#K5" title="WSIERP_1:func.5">&lt;*&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
;<br/>

<b>thus </b><a NAME="E1:55_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K5" title="WSIERP_1:func.5">&lt;*&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/>

<b>thus </b><a NAME="E2:55_1"/>
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K5" title="WSIERP_1:func.5">&lt;*&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span>)</span> holds <br/><span class="p1">(<span class="default"><a href="wsierp_1.html#K5" title="WSIERP_1:func.5">&lt;*&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span>)</span> <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p2">(<span class="default">0 <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Olive" title="b1">k</font></span>)</span></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span>
 ;<br/>

<a NAME="E3:55_1"/>
<font color="Maroon" title="c3">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 
<b>by </b><i><a class="ref" href="int_2.html#D5" title="INT_2:def.5">INT_2:def 5</a></i>;<br/>
<b>hence </b><a NAME="E4:55_1"/>
<font color="Maroon" title="c3">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <span class="p1">(<span class="default">0 <a href="newton.html#K12" title="NEWTON:func.12">!</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Sum</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K5" title="WSIERP_1:func.5">&lt;*&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span>)</span>
 <b>by </b><i><a class="ref" href="rvsum_1.html#T102" title="RVSUM_1:th.102">RVSUM_1:102</a>, <a class="ref" href="nat_3.html#T21" title="NAT_3:th.21">NAT_3:21</a>, <a class="ref" href="newton.html#T18" title="NEWTON:th.18">NEWTON:18</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:55"/><i><font color="Green" title="E55">A2</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</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" title="c3">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:55_2"/><i><font color="Green" title="E56">A3</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">n</font>]
 ;<br/>

<a NAME="E2:55_2"/>
for <font color="Olive" title="b1">p</font> being  <a href="newton.html#NM1" title="NEWTON:NM.1">Prime</a>  ex <font color="Olive" title="b2">f</font> being   <a href="trees_4.html#M1" title="TREES_4:mode.1">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 &amp; ( for <font color="Olive" title="b3">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b3">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Olive" title="b2">f</font> holds <br/><font color="Olive" title="b2">f</font> <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">.</a> <font color="Olive" title="b3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Olive" title="b3">k</font></span>)</span></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span> ) &amp; <font color="Olive" title="b1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="newton.html#K12" title="NEWTON:func.12">!</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Sum</a> <font color="Olive" title="b2">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" title="c4">p</font> be   <a href="newton.html#NM1" title="NEWTON:NM.1">Prime</a>;<br/>

<b>consider </b><font color="Maroon" title="c5">fn</font> being    <a href="trees_4.html#M1" title="TREES_4:mode.1">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E2:55_2_1"/><i><font color="Green" title="E57">A4</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">fn</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n</font> &amp; ( for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c5">fn</font> holds <br/><font color="Maroon" title="c5">fn</font> <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Olive" title="b1">k</font></span>)</span></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span> ) &amp; <font color="Maroon" title="c4">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="newton.html#K12" title="NEWTON:func.12">!</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Sum</a> <font color="Maroon" title="c5">fn</font> )
 <b>by </b><i><a class="txt" href="nat_4.html#E1:55_2"><i><font color="Green" title="E56">A3</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c6">fn1</font> being    <a href="trees_4.html#M1" title="TREES_4:mode.1">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E4:55_2_1"/><i><font color="Green" title="E58">A5</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">fn1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 &amp; ( for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">fn1</font> holds <br/>( ( <font color="Maroon" title="c6">fn1</font> <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 implies <font color="Maroon" title="c4">p</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Olive" title="b1">k</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 ) &amp; ( <font color="Maroon" title="c4">p</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Olive" title="b1">k</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 implies <font color="Maroon" title="c6">fn1</font> <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) &amp; ( <font color="Maroon" title="c6">fn1</font> <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 implies  not <font color="Maroon" title="c4">p</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Olive" title="b1">k</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 ) &amp; ( not <font color="Maroon" title="c4">p</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Olive" title="b1">k</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 implies <font color="Maroon" title="c6">fn1</font> <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 ) ) ) &amp; <font color="Maroon" title="c4">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Sum</a> <font color="Maroon" title="c6">fn1</font> )
 <b>by </b><i><a class="ref" href="nat_4.html#T47" target="_self" title="NAT_4:th.47">Th47</a></i>;<br/>
<b>set </b><font color="Maroon" title="c7">f</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c5">fn</font> <a href="wsierp_1.html#K4" title="WSIERP_1:func.4">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="rvsum_1.html#K4" title="RVSUM_1:func.4">+</a> <font color="Maroon" title="c6">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" title="c8">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:55_2_1_1"/><i><font color="Green" title="E59">A6</font></i>: 
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">fn</font> <a href="wsierp_1.html#K4" title="WSIERP_1:func.4">^</a> <span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="rvsum_1.html#K4" title="RVSUM_1:func.4">+</a> <font color="Maroon" title="c6">fn1</font></span>)</span>
 ;<br/><b>consider </b><font color="Maroon" title="c9">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:55_2_1_1"/><i><font color="Green" title="E60">A7</font></i>: 
( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">fn</font> <a href="wsierp_1.html#K4" title="WSIERP_1:func.4">^</a> <span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="rvsum_1.html#K4" title="RVSUM_1:func.4">+</a> <font color="Maroon" title="c6">fn1</font></span>)</span> &amp; <font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">fn</font> <a href="wsierp_1.html#K4" title="WSIERP_1:func.4">^</a> <span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="rvsum_1.html#K4" title="RVSUM_1:func.4">+</a> <font color="Maroon" title="c6">fn1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> )
 <b>by </b><i><a class="txt" href="nat_4.html#E1:55_2_1_1"><i><font color="Green" title="E59">A6</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">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" title="c5">fn</font> <a href="wsierp_1.html#K4" title="WSIERP_1:func.4">^</a> <span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="rvsum_1.html#K4" title="RVSUM_1:func.4">+</a> <font color="Maroon" title="c6">fn1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">fn</font> <a href="wsierp_1.html#K4" title="WSIERP_1:func.4">^</a> <span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c9">x</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">fn1</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c9">x</font></span>)</span>
 <b>by </b><i><a class="txt" href="nat_4.html#E3:55_2_1_1"><i><font color="Green" title="E60">A7</font></i></a>, <a class="ref" href="seq_1.html#D3" title="SEQ_1:def.3">SEQ_1:def 3</a></i>;<br/><b>hence </b><a NAME="E5:55_2_1_1"/>
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="txt" href="nat_4.html#E3:55_2_1_1"><i><font color="Green" title="E60">A7</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E6:55_2_1"/><b>then </b><i><font color="Green" title="E59">A8</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">fn</font> <a href="wsierp_1.html#K4" title="WSIERP_1:func.4">^</a> <span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="rvsum_1.html#K4" title="RVSUM_1:func.4">+</a> <font color="Maroon" title="c6">fn1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">f</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c5">fn</font> <a href="wsierp_1.html#K4" title="WSIERP_1:func.4">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="rvsum_1.html#K4" title="RVSUM_1:func.4">+</a> <font color="Maroon" title="c6">fn1</font> as    <a href="trees_4.html#M1" title="TREES_4:mode.1">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="nat_4.html#E6:55_2_1"><i><font color="Green" title="E59">A8</font></i></a>, <a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c8">f</font>
;<br/>

<b>set </b><font color="Maroon" title="c9">fn0</font> = <font color="Maroon" title="c5">fn</font> <a href="wsierp_1.html#K4" title="WSIERP_1:func.4">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>;<br/>
<a NAME="E8:55_2_1"/><i><font color="Green" title="E60">A9</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">fn</font> <a href="wsierp_1.html#K4" title="WSIERP_1:func.4">^</a> <span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">fn1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="nat_4.html#T41" target="_self" title="NAT_4:th.41">Th41</a></i>;<br/>
<i><font color="Green" title="E61">A10</font></i>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">fn</font> <a href="wsierp_1.html#K4" title="WSIERP_1:func.4">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">fn</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></i>
<br/>.= 
<font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
<b>by </b><i><a class="txt" href="nat_4.html#E2:55_2_1"><i><font color="Green" title="E57">A4</font></i></a>, <a class="ref" href="finseq_1.html#T56" title="FINSEQ_1:th.56">FINSEQ_1:56</a></i>
;<br/>
<a NAME="E10:55_2_1"/><b>then </b><i><font color="Green" title="E62">A11</font></i>: 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">fn</font> <a href="wsierp_1.html#K4" title="WSIERP_1:func.4">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E11:55_2_1"/><i><font color="Green" title="E63">A12</font></i>: 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">fn1</font>
 
<b>by </b><i><a class="txt" href="nat_4.html#E4:55_2_1"><i><font color="Green" title="E58">A5</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<b>thus </b><a NAME="E12:55_2_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 <b>by </b><i><a class="txt" href="nat_4.html#E8:55_2_1"><i><font color="Green" title="E60">A9</font></i></a>, <a class="txt" href="nat_4.html#E10:55_2_1"><i><font color="Green" title="E62">A11</font></i></a>, <a class="txt" href="nat_4.html#E11:55_2_1"><i><font color="Green" title="E63">A12</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>

<b>thus </b><a NAME="E13:55_2_1"/>
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">f</font> holds <br/><font color="Maroon" title="c8">f</font> <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Olive" title="b1">k</font></span>)</span></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</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" title="c10">k</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:55_2_1_3"/><i><font color="Green" title="E64">A13</font></i>: 
<font color="Maroon" title="c10">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">f</font>
 ;<br/>

<a NAME="E2:55_2_1_3"/><b>then </b><i><font color="Green" title="E65">A14</font></i>: 
<font color="Maroon" title="c8">f</font> <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">fn</font> <a href="wsierp_1.html#K4" title="WSIERP_1:func.4">^</a> <span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">fn1</font> <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">.</a> <font color="Maroon" title="c10">k</font></span>)</span>
 
<b>by </b><i><a class="ref" href="seq_1.html#D3" title="SEQ_1:def.3">SEQ_1:def 3</a></i>;<br/>
<a NAME="E3:55_2_1_3"/><i><font color="Green" title="E66">A15</font></i>: 
( <font color="Maroon" title="c10">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">fn</font> <a href="wsierp_1.html#K4" title="WSIERP_1:func.4">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> &amp; <font color="Maroon" title="c10">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">fn1</font> )
 
<b>by </b><i><a class="txt" href="nat_4.html#E8:55_2_1"><i><font color="Green" title="E60">A9</font></i></a>, <a class="txt" href="nat_4.html#E1:55_2_1_3"><i><font color="Green" title="E64">A13</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E4:55_2_1_3"/><i><font color="Green" title="E67">A16</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">fn</font> <a href="wsierp_1.html#K4" title="WSIERP_1:func.4">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</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" title="c10">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c5">fn</font> or  ex <font color="Olive" title="b1">n1</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">n1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> &amp; <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">fn</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Olive" title="b1">n1</font> ) )
 </span><b>by </b><i><a class="txt" href="nat_4.html#E3:55_2_1_3"><i><font color="Green" title="E66">A15</font></i></a>, <a class="ref" href="finseq_1.html#T38" title="FINSEQ_1:th.38">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" title="E68">A17</font></i>: 
<font color="Maroon" title="c10">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c5">fn</font>
 ;<br/><div class="add"><a NAME="E2:55_2_1_3_1_1_1"/><b>then </b>
<font color="Maroon" title="c5">fn</font> <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span>
 <b>by </b><i><a class="txt" href="nat_4.html#E2:55_2_1"><i><font color="Green" title="E57">A4</font></i></a></i>;<br/><b>hence </b><a NAME="E3:55_2_1_3_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">fn</font> <a href="wsierp_1.html#K4" title="WSIERP_1:func.4">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span>
 <b>by </b><i><a class="txt" href="nat_4.html#E1:55_2_1_3_1_1_1"><i><font color="Green" title="E68">A17</font></i></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">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" title="E68">A18</font></i>: 
 ex <font color="Olive" title="b1">n1</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">n1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> &amp; <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">fn</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Olive" title="b1">n1</font> )
 ;<br/><div class="add"><b>consider </b><font color="Maroon" title="c11">n1</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:55_2_1_3_1_1_2"/><i><font color="Green" title="E69">A19</font></i>: 
( <font color="Maroon" title="c11">n1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> &amp; <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">fn</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c11">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" title="E68">A18</font></i></a></i>;<br/><a NAME="E4:55_2_1_3_1_1_2"/>
<font color="Maroon" title="c11">n1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">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" title="E69">A19</font></i></a>, <a class="ref" href="finseq_1.html#T55" title="FINSEQ_1:th.55">FINSEQ_1:55</a></i>;<br/><a NAME="E5:55_2_1_3_1_1_2"/><b>then </b><i><font color="Green" title="E70">A20</font></i>: 
<font color="Maroon" title="c11">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a>, <a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">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" title="c5">fn</font> <a href="wsierp_1.html#K4" title="WSIERP_1:func.4">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">.</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" title="E70">A20</font></i></a>, <a class="txt" href="nat_4.html#E2:55_2_1"><i><font color="Green" title="E57">A4</font></i></a>, <a class="txt" href="nat_4.html#E3:55_2_1_3_1_1_2"><i><font color="Green" title="E69">A19</font></i></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>;<br/><a NAME="E7:55_2_1_3_1_1_2"/><b>then </b><i><font color="Green" title="E71">A21</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">fn</font> <a href="wsierp_1.html#K4" title="WSIERP_1:func.4">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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" title="E70">A20</font></i></a>, <a class="txt" href="nat_4.html#E2:55_2_1"><i><font color="Green" title="E57">A4</font></i></a>, <a class="txt" href="nat_4.html#E3:55_2_1_3_1_1_2"><i><font color="Green" title="E69">A19</font></i></a>, <a class="ref" href="finseq_1.html#T57" title="FINSEQ_1:th.57">FINSEQ_1:57</a></i>;<br/><a NAME="E8:55_2_1_3_1_1_2"/><i><font color="Green" title="E72">A22</font></i>: 
<font color="Maroon" title="c4">p</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c4">p</font>
 <b>by </b><i><a class="ref" href="newton.html#T11" title="NEWTON:th.11">NEWTON:11</a></i>;<br/><a NAME="E9:55_2_1_3_1_1_2"/><i><font color="Green" title="E73">A23</font></i>: 
<font color="Maroon" title="c4">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 <b>by </b><i><a class="ref" href="int_2.html#D5" title="INT_2:def.5">INT_2:def 5</a></i>;<br/><a NAME="E10:55_2_1_3_1_1_2"/><b>then </b><i><font color="Green" title="E74">A24</font></i>: 
<font color="Maroon" title="c4">p</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c3">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T70" title="XREAL_1:th.70">XREAL_1:70</a></i>;<br/><a NAME="E11:55_2_1_3_1_1_2"/>
<font color="Maroon" title="c4">p</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c3">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" title="E73">A23</font></i></a>, <a class="ref" href="nat_4.html#T3" target="_self" title="NAT_4:th.3">Th3</a></i>;<br/><a NAME="E12:55_2_1_3_1_1_2"/><b>then </b>
<font color="Maroon" title="c4">p</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c3">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" title="E70">A20</font></i></a>, <a class="txt" href="nat_4.html#E10:55_2_1_3_1_1_2"><i><font color="Green" title="E74">A24</font></i></a>, <a class="txt" href="nat_4.html#E8:55_2_1_3_1_1_2"><i><font color="Green" title="E72">A22</font></i></a>, <a class="txt" href="nat_4.html#E2:55_2_1"><i><font color="Green" title="E57">A4</font></i></a>, <a class="txt" href="nat_4.html#E3:55_2_1_3_1_1_2"><i><font color="Green" title="E69">A19</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E13:55_2_1_3_1_1_2"/><b>then </b>
<font color="Maroon" title="c3">n</font> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T191" title="XREAL_1:th.191">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" title="c3">n</font> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">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" title="c5">fn</font> <a href="wsierp_1.html#K4" title="WSIERP_1:func.4">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</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" title="E71">A21</font></i></a>, <a class="ref" href="int_1.html#D4" title="INT_1:def.4">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" title="c4">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 
<b>by </b><i><a class="ref" href="int_2.html#D5" title="INT_2:def.5">INT_2:def 5</a></i>;<br/>
<a NAME="E6:55_2_1_3"/><b>then </b><i><font color="Green" title="E68">A25</font></i>: 
<font color="Maroon" title="c4">p</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<b>by </b><i><a class="ref" href="prepower.html#T19" title="PREPOWER:th.19">PREPOWER:19</a></i>;<br/>
<a NAME="E7:55_2_1_3"/><i><font color="Green" title="E69">A26</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="ref" href="xcmplx_1.html#T60" title="XCMPLX_1:th.60">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" title="c4">p</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c10">k</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 or  not <font color="Maroon" title="c4">p</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c10">k</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</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" title="E70">A27</font></i>: 
<font color="Maroon" title="c4">p</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c10">k</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 ;<br/><div class="add"><b>consider </b><font color="Maroon" title="c11">r</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E3:55_2_1_3_2_1"/><i><font color="Green" title="E71">A28</font></i>: 
<font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c11">r</font>
 <b>by </b><i><a class="txt" href="nat_4.html#E1:55_2_1_3_2_1"><i><font color="Green" title="E70">A27</font></i></a>, <a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">NAT_D:def 3</a></i>;<br/><i><font color="Green" title="E72">A29</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c10">k</font></span>)</span> = 
<font color="Maroon" title="c11">r</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="nat_4.html#E3:55_2_1_3_2_1"><i><font color="Green" title="E71">A28</font></i></a>, <a class="ref" href="xcmplx_1.html#T78" title="XCMPLX_1:th.78">XCMPLX_1:78</a></i>
<br/>.= 
<font color="Maroon" title="c11">r</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T60" title="XCMPLX_1:th.60">XCMPLX_1:60</a></i>
<br/>.= 
<font color="Maroon" title="c11">r</font>

;<br/><a NAME="E5:55_2_1_3_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a>, <a class="txt" href="nat_4.html#E6:55_2_1_3"><i><font color="Green" title="E68">A25</font></i></a></i>;<br/><a NAME="E6:55_2_1_3_2_1"/><b>then </b><i><font color="Green" title="E73">A30</font></i>: 
0 <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1</span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c10">k</font></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T74" title="XREAL_1:th.74">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" title="c4">p</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> 1</span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 0
 <b>by </b><i><a class="txt" href="nat_4.html#E7:55_2_1_3"><i><font color="Green" title="E69">A26</font></i></a></i>;<br/><a NAME="E8:55_2_1_3_2_1"/><b>then </b><i><font color="Green" title="E74">A31</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p3">(<span class="default"><a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 0
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T63" title="XCMPLX_1:th.63">XCMPLX_1:63</a></i>;<br/><span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 = 
<span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> 1</span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span>
<b>by </b><i><a class="ref" href="int_1.html#T51" title="INT_1:th.51">INT_1:51</a></i>
<br/>.= 
<span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T60" title="XCMPLX_1:th.60">XCMPLX_1:60</a></i>
<br/>.= 
<span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T63" title="XCMPLX_1:th.63">XCMPLX_1:63</a></i>
<br/>.= 
<span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</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" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span>

<br/>.= 
<span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</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" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1</span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T63" title="XCMPLX_1:th.63">XCMPLX_1:63</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</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" title="c4">p</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1</span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span>
<b>by </b><i><a class="txt" href="nat_4.html#E4:55_2_1_3_2_1"><i><font color="Green" title="E72">A29</font></i></a>, <a class="ref" href="int_1.html#T51" title="INT_1:th.51">INT_1:51</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> 0
<b>by </b><i><a class="txt" href="nat_4.html#E6:55_2_1_3_2_1"><i><font color="Green" title="E73">A30</font></i></a>, <a class="txt" href="nat_4.html#E8:55_2_1_3_2_1"><i><font color="Green" title="E74">A31</font></i></a>, <a class="ref" href="int_1.html#D4" title="INT_1:def.4">INT_1:def 4</a></i>
<br/>.= 
<span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span>
<b>by </b><i><a class="ref" href="int_1.html#T47" title="INT_1:th.47">INT_1:47</a>, <a class="txt" href="nat_4.html#E4:55_2_1_3_2_1"><i><font color="Green" title="E72">A29</font></i></a></i>
;<br/><b>hence </b><a NAME="E10:55_2_1_3_2_1"/>
<font color="Maroon" title="c8">f</font> <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span>
 <b>by </b><i><a class="txt" href="nat_4.html#E1:55_2_1_3_2_1"><i><font color="Green" title="E70">A27</font></i></a>, <a class="txt" href="nat_4.html#E2:55_2_1_3"><i><font color="Green" title="E65">A14</font></i></a>, <a class="txt" href="nat_4.html#E3:55_2_1_3"><i><font color="Green" title="E66">A15</font></i></a>, <a class="txt" href="nat_4.html#E4:55_2_1_3"><i><font color="Green" title="E67">A16</font></i></a>, <a class="txt" href="nat_4.html#E4:55_2_1"><i><font color="Green" title="E58">A5</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" title="E70">A32</font></i>: 
not <font color="Maroon" title="c4">p</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c10">k</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 ;<br/><div class="add"><a NAME="E2:55_2_1_3_2_2"/><b>then </b><i><font color="Green" title="E71">A33</font></i>: 
<font color="Maroon" title="c6">fn1</font> <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="txt" href="nat_4.html#E3:55_2_1_3"><i><font color="Green" title="E66">A15</font></i></a>, <a class="txt" href="nat_4.html#E4:55_2_1"><i><font color="Green" title="E58">A5</font></i></a></i>;<br/><b>set </b><font color="Maroon" title="c11">d</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c10">k</font></span>)</span>;<br/><b>set </b><font color="Maroon" title="c12">m</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c10">k</font></span>)</span>;<br/><a NAME="E3:55_2_1_3_2_2"/><i><font color="Green" title="E72">A34</font></i>: 
<font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="nat_d.html#T2" title="NAT_D:th.2">NAT_D:2</a></i>;<br/><a NAME="E4:55_2_1_3_2_2"/>
not <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="txt" href="nat_4.html#E3:55_2_1_3_2_2"><i><font color="Green" title="E72">A34</font></i></a>, <a class="txt" href="nat_4.html#E1:55_2_1_3_2_2"><i><font color="Green" title="E70">A32</font></i></a>, <a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">NAT_D: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" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&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" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">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" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</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" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E9:55_2_1_3_2_2"/><b>then </b><i><font color="Green" title="E73">A35</font></i>: 
0 <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1</span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c10">k</font></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T74" title="XREAL_1:th.74">XREAL_1:74</a></i>;<br/><a NAME="E10:55_2_1_3_2_2"/><i><font color="Green" title="E74">A36</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c4">p</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c10">k</font>
 <b>by </b><i><a class="ref" href="nat_d.html#T1" title="NAT_D:th.1">NAT_D:1</a></i>;<br/><a NAME="E11:55_2_1_3_2_2"/>
0 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c10">k</font></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">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" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 <b>by </b><i><a class="txt" href="nat_4.html#E10:55_2_1_3_2_2"><i><font color="Green" title="E74">A36</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</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" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">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" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1</span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c10">k</font></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T76" title="XREAL_1:th.76">XREAL_1:76</a></i>;<br/><a NAME="E15:55_2_1_3_2_2"/><b>then </b><i><font color="Green" title="E75">A37</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" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1</span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1
 <b>by </b><i><a class="txt" href="nat_4.html#E7:55_2_1_3"><i><font color="Green" title="E69">A26</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">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" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c10">k</font></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T76" title="XREAL_1:th.76">XREAL_1:76</a>, <a class="txt" href="nat_4.html#E10:55_2_1_3_2_2"><i><font color="Green" title="E74">A36</font></i></a></i>;<br/><a NAME="E17:55_2_1_3_2_2"/><b>then </b><i><font color="Green" title="E76">A38</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1
 <b>by </b><i><a class="txt" href="nat_4.html#E7:55_2_1_3"><i><font color="Green" title="E69">A26</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E18:55_2_1_3_2_2"/>
<font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="nat_4.html#E3:55_2_1_3_2_2"><i><font color="Green" title="E72">A34</font></i></a></i>;<br/><b>then </b><span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span> = 
<span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</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" title="c4">p</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p0">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</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" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p0">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1</span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T63" title="XCMPLX_1:th.63">XCMPLX_1:63</a></i>
<br/>.= 
<span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</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" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</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" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p0">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1</span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T75" title="XCMPLX_1:th.75">XCMPLX_1:75</a></i>
<br/>.= 
<span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</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" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> 1</span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</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" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p0">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1</span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T60" title="XCMPLX_1:th.60">XCMPLX_1:60</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</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" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1</span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span>
<b>by </b><i><a class="ref" href="int_1.html#T51" title="INT_1:th.51">INT_1:51</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 0
<b>by </b><i><a class="txt" href="nat_4.html#E9:55_2_1_3_2_2"><i><font color="Green" title="E73">A35</font></i></a>, <a class="txt" href="nat_4.html#E15:55_2_1_3_2_2"><i><font color="Green" title="E75">A37</font></i></a>, <a class="ref" href="int_1.html#D4" title="INT_1:def.4">INT_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</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" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span>
<b>by </b><i><a class="txt" href="nat_4.html#E17:55_2_1_3_2_2"><i><font color="Green" title="E76">A38</font></i></a>, <a class="ref" href="int_1.html#D4" title="INT_1:def.4">INT_1:def 4</a></i>
<br/>.= 
<span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</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" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> 1</span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span>
<b>by </b><i><a class="ref" href="int_1.html#T51" title="INT_1:th.51">INT_1:51</a></i>
<br/>.= 
<span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</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" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T60" title="XCMPLX_1:th.60">XCMPLX_1:60</a></i>
<br/>.= 
<span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</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" title="c4">p</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p0">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T75" title="XCMPLX_1:th.75">XCMPLX_1:75</a></i>
<br/>.= 
<span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</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" title="c4">p</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p0">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T63" title="XCMPLX_1:th.63">XCMPLX_1:63</a></i>
<br/>.= 
<span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span>
<b>by </b><i><a class="ref" href="nat_d.html#T2" title="NAT_D:th.2">NAT_D:2</a></i>
;<br/><b>hence </b><a NAME="E20:55_2_1_3_2_2"/>
<font color="Maroon" title="c8">f</font> <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span>
 <b>by </b><i><a class="txt" href="nat_4.html#E2:55_2_1_3_2_2"><i><font color="Green" title="E71">A33</font></i></a>, <a class="txt" href="nat_4.html#E4:55_2_1_3"><i><font color="Green" title="E67">A16</font></i></a>, <a class="txt" href="nat_4.html#E2:55_2_1_3"><i><font color="Green" title="E65">A14</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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

<b>reconsider </b><font color="Maroon" title="c10">fn0</font> = <font color="Maroon" title="c5">fn</font> <a href="wsierp_1.html#K4" title="WSIERP_1:func.4">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <b>by </b><i><a class="txt" href="nat_4.html#E9:55_2_1"><i><font color="Green" title="E61">A10</font></i></a>, <a class="ref" href="finseq_2.html#T110" title="FINSEQ_2:th.110">FINSEQ_2:110</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">fn1</font> = <font color="Maroon" title="c6">fn1</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <b>by </b><i><a class="txt" href="nat_4.html#E4:55_2_1"><i><font color="Green" title="E58">A5</font></i></a>, <a class="ref" href="finseq_2.html#T110" title="FINSEQ_2:th.110">FINSEQ_2:110</a></i>;<br/>
<i><font color="Green" title="E64">A39</font></i>:  <a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Sum</a> <font color="Maroon" title="c8">f</font> = 
<span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Maroon" title="c10">fn0</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Maroon" title="c11">fn1</font></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T119" title="RVSUM_1:th.119">RVSUM_1:119</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Sum</a> <font color="Maroon" title="c5">fn</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 0</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="nat_4.html#E4:55_2_1"><i><font color="Green" title="E58">A5</font></i></a>, <a class="ref" href="rvsum_1.html#T104" title="RVSUM_1:th.104">RVSUM_1:104</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="newton.html#K12" title="NEWTON:func.12">!</a> </span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="nat_4.html#E2:55_2_1"><i><font color="Green" title="E57">A4</font></i></a></i>
;<br/>
<a NAME="E17:55_2_1"/>
( <font color="Maroon" title="c3">n</font> <a href="newton.html#K12" title="NEWTON:func.12">!</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 &amp; <font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 )
 
<b>by </b><i><a class="ref" href="newton.html#T23" title="NEWTON:th.23">NEWTON:23</a></i>;<br/>
<b>then </b><span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="newton.html#K12" title="NEWTON:func.12">!</a> </span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> = 
<font color="Maroon" title="c4">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="newton.html#K12" title="NEWTON:func.12">!</a> </span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="nat_3.html#T28" title="NAT_3:th.28">NAT_3:28</a></i>
<br/>.= 
<font color="Maroon" title="c4">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="newton.html#K12" title="NEWTON:func.12">!</a> </span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T21" title="NEWTON:th.21">NEWTON:21</a></i>
;<br/>
<b>hence </b><a NAME="E19:55_2_1"/>
<font color="Maroon" title="c4">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="newton.html#K12" title="NEWTON:func.12">!</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Sum</a> <font color="Maroon" title="c8">f</font>
 <b>by </b><i><a class="txt" href="nat_4.html#E16:55_2_1"><i><font color="Green" title="E64">A39</font></i></a></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" title="c3">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:55"/><i><font color="Green" title="E56">A40</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="nat_4.html#E1:55"><i><font color="Green" title="E54">A1</font></i></a>, <a class="txt" href="nat_4.html#E2:55"><i><font color="Green" title="E55">A2</font></i></a>);<br/></i>
<b>consider </b><font color="Maroon" title="c3">f</font> being    <a href="trees_4.html#M1" title="TREES_4:mode.1">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E5:55"/><i><font color="Green" title="E57">A41</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> &amp; ( for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font> holds <br/><font color="Maroon" title="c3">f</font> <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Olive" title="b1">k</font></span>)</span></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span> ) &amp; <font color="Maroon" title="c2">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="newton.html#K12" title="NEWTON:func.12">!</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Sum</a> <font color="Maroon" title="c3">f</font> )
 <b>by </b><i><a class="txt" href="nat_4.html#E3:55"><i><font color="Green" title="E56">A40</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c3">f</font>
;<br/>

<b>thus </b><a NAME="E6:55"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> &amp; ( for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font> holds <br/><font color="Maroon" title="c3">f</font> <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Olive" title="b1">k</font></span>)</span></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span> ) &amp; <font color="Maroon" title="c2">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="newton.html#K12" title="NEWTON:func.12">!</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Sum</a> <font color="Maroon" title="c3">f</font> )
 <b>by </b><i><a class="txt" href="nat_4.html#E5:55"><i><font color="Green" title="E57">A41</font></i></a></i>;<br/>


</div>
