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

<b>reconsider </b><font color="Maroon" title="c3">b1</font> = <font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 as    <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>by </b><i><a class="txt" href="numeral1.html#E1:9"><i><font color="Green" title="E9">A1</font></i></a>, <a class="ref" href="nat_1.html#T20" title="NAT_1:th.20">NAT_1:20</a></i>;<br/>
<b>let </b><font color="Maroon" title="c4">d</font>, <font color="Maroon" title="c5">e</font> be   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>thus </b><a NAME="E2:9_1_2"/>
( <font color="Maroon" title="c1">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="numeral1.html#K1" title="NUMERAL1:func.1">value</a> <font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c4">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c4">d</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; ( for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c4">d</font> holds <br/>(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Maroon" title="c4">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">b</font> ) ) &amp;  <a href="numeral1.html#K1" title="NUMERAL1:func.1">value</a> <font color="Maroon" title="c5">e</font>,<font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c5">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c5">e</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; ( for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c5">e</font> holds <br/>(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Maroon" title="c5">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">b</font> ) ) implies <font color="Maroon" title="c4">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">e</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:9_1_2_1"/>
<font color="Maroon" title="c1">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/>

<a NAME="E2:9_1_2_1"/><b>then </b><i><font color="Green" title="E10">A35</font></i>: 
<font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
;<br/>
<b>assume </b><a NAME="E3:9_1_2_1"/><i><font color="Green" title="E11">A36</font></i>: 
(  <a href="numeral1.html#K1" title="NUMERAL1:func.1">value</a> <font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c4">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c4">d</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; ( for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c4">d</font> holds <br/>(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Maroon" title="c4">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">b</font> ) ) )
 ;<br/>

<b>assume </b><a NAME="E4:9_1_2_1"/><i><font color="Green" title="E12">A37</font></i>: 
(  <a href="numeral1.html#K1" title="NUMERAL1:func.1">value</a> <font color="Maroon" title="c5">e</font>,<font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c5">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c5">e</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; ( for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c5">e</font> holds <br/>(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Maroon" title="c5">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">b</font> ) ) )
 ;<br/>

<b>consider </b><font color="Maroon" title="c6">D</font> being   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E6:9_1_2_1"/><i><font color="Green" title="E13">A38</font></i>: 
(  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c6">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c4">d</font> &amp; ( for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c6">D</font> holds <br/><font color="Maroon" title="c6">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Olive" title="b1">i</font></span>)</span> ) &amp; <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <font color="Maroon" title="c6">D</font> )
 <b>by </b><i><a class="txt" href="numeral1.html#E3:9_1_2_1"><i><font color="Green" title="E11">A36</font></i></a>, <a class="ref" href="numeral1.html#D1" target="_self" title="NUMERAL1:def.1">Def1</a></i>;<br/>
<a NAME="E7:9_1_2_1"/><i><font color="Green" title="E14">A39</font></i>: 
<span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c4">d</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c4">d</font>
 
<b>by </b><i><a class="txt" href="numeral1.html#E3:9_1_2_1"><i><font color="Green" title="E11">A36</font></i></a>, <a class="ref" href="funct_1.html#D4" title="FUNCT_1:def.4">FUNCT_1:def 4</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c7">k</font> = <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c4">d</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 as    <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/>
<a NAME="E9:9_1_2_1"/>
<font color="Maroon" title="c4">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c7">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<b>by </b><i><a class="txt" href="numeral1.html#E3:9_1_2_1"><i><font color="Green" title="E11">A36</font></i></a>, <a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></i>;<br/>
<a NAME="E10:9_1_2_1"/><b>then </b><i><font color="Green" title="E15">A40</font></i>: 
1 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c7">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">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c7">k</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c7">k</font></span>)</span>
 
<b>by </b><i><a class="ref" href="nat_1.html#T4" title="NAT_1:th.4">NAT_1:4</a></i>;<br/>
<a NAME="E11:9_1_2_1"/>
<font color="Maroon" title="c7">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c6">D</font>
 
<b>by </b><i><a class="txt" href="numeral1.html#E3:9_1_2_1"><i><font color="Green" title="E11">A36</font></i></a>, <a class="txt" href="numeral1.html#E6:9_1_2_1"><i><font color="Green" title="E13">A38</font></i></a>, <a class="ref" href="funct_1.html#D4" title="FUNCT_1:def.4">FUNCT_1:def 4</a></i>;<br/>
<a NAME="E12:9_1_2_1"/><b>then </b><i><font color="Green" title="E16">A41</font></i>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c6">D</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="numeral1.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a></i>;<br/>
<a NAME="E13:9_1_2_1"/><i><font color="Green" title="E17">A42</font></i>: 
( <font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c7">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
<b>by </b><i><a class="txt" href="numeral1.html#E1:9"><i><font color="Green" title="E9">A1</font></i></a>, <a class="ref" href="prepower.html#T13" title="PREPOWER:th.13">PREPOWER:13</a></i>;<br/>
<a NAME="E14:9_1_2_1"/>
 <a href="power.html#K5" title="POWER:func.5">log</a> 2,<font color="Maroon" title="c2">b</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="power.html#K6" title="POWER:func.6">log</a> 2,1
 
<b>by </b><i><a class="txt" href="numeral1.html#E1:9"><i><font color="Green" title="E9">A1</font></i></a>, <a class="ref" href="power.html#T65" title="POWER:th.65">POWER:65</a></i>;<br/>
<a NAME="E15:9_1_2_1"/><b>then </b><i><font color="Green" title="E18">A43</font></i>: 
 <a href="power.html#K5" title="POWER:func.5">log</a> 2,<font color="Maroon" title="c2">b</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="power.html#T59" title="POWER:th.59">POWER:59</a></i>;<br/>
<a NAME="E16:9_1_2_1"/>
<font color="Maroon" title="c6">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c7">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c7">k</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c7">k</font></span>)</span>
 
<b>by </b><i><a class="txt" href="numeral1.html#E6:9_1_2_1"><i><font color="Green" title="E13">A38</font></i></a>, <a class="txt" href="numeral1.html#E7:9_1_2_1"><i><font color="Green" title="E14">A39</font></i></a></i>;<br/>
<a NAME="E17:9_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c7">k</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c7">k</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="txt" href="numeral1.html#E6:9_1_2_1"><i><font color="Green" title="E13">A38</font></i></a>, <a class="txt" href="numeral1.html#E7:9_1_2_1"><i><font color="Green" title="E14">A39</font></i></a>, <a class="txt" href="numeral1.html#E12:9_1_2_1"><i><font color="Green" title="E16">A41</font></i></a>, <a class="ref" href="stirl2_1.html#T52" title="STIRL2_1:th.52">STIRL2_1:52</a></i>;<br/>
<a NAME="E18:9_1_2_1"/><b>then </b>
<font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c7">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="txt" href="numeral1.html#E10:9_1_2_1"><i><font color="Green" title="E15">A40</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E19:9_1_2_1"/><b>then </b>
 <a href="power.html#K5" title="POWER:func.5">log</a> 2,<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c7">k</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="power.html#K5" title="POWER:func.5">log</a> 2,<font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="txt" href="numeral1.html#E13:9_1_2_1"><i><font color="Green" title="E17">A42</font></i></a>, <a class="ref" href="pre_ff.html#T12" title="PRE_FF:th.12">PRE_FF:12</a></i>;<br/>
<a NAME="E20:9_1_2_1"/><b>then </b>
 <a href="power.html#K5" title="POWER:func.5">log</a> 2,<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c7">k</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="power.html#K5" title="POWER:func.5">log</a> 2,<font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="txt" href="numeral1.html#E1:9"><i><font color="Green" title="E9">A1</font></i></a>, <a class="ref" href="power.html#T46" title="POWER:th.46">POWER:46</a></i>;<br/>
<a NAME="E21:9_1_2_1"/><b>then </b>
<font color="Maroon" title="c7">k</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="power.html#K5" title="POWER:func.5">log</a> 2,<font color="Maroon" title="c2">b</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="power.html#K5" title="POWER:func.5">log</a> 2,<font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="txt" href="numeral1.html#E1:9"><i><font color="Green" title="E9">A1</font></i></a>, <a class="ref" href="power.html#T63" title="POWER:th.63">POWER:63</a></i>;<br/>
<a NAME="E22:9_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><a href="power.html#K5" title="POWER:func.5">log</a> 2,<font color="Maroon" title="c2">b</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><a href="power.html#K5" title="POWER:func.5">log</a> 2,<font color="Maroon" title="c2">b</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="power.html#K5" title="POWER:func.5">log</a> 2,<font color="Maroon" title="c1">n</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><a href="power.html#K5" title="POWER:func.5">log</a> 2,<font color="Maroon" title="c2">b</font></span>)</span>
 
<b>by </b><i><a class="txt" href="numeral1.html#E15:9_1_2_1"><i><font color="Green" title="E18">A43</font></i></a>, <a class="ref" href="xreal_1.html#T74" title="XREAL_1:th.74">XREAL_1:74</a></i>;<br/>
<a NAME="E23:9_1_2_1"/><b>then </b><i><font color="Green" title="E19">A44</font></i>: 
<font color="Maroon" title="c7">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="power.html#K5" title="POWER:func.5">log</a> 2,<font color="Maroon" title="c1">n</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><a href="power.html#K5" title="POWER:func.5">log</a> 2,<font color="Maroon" title="c2">b</font></span>)</span>
 
<b>by </b><i><a class="txt" href="numeral1.html#E15:9_1_2_1"><i><font color="Green" title="E18">A43</font></i></a>, <a class="ref" href="xcmplx_1.html#T90" title="XCMPLX_1:th.90">XCMPLX_1:90</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">E</font> being   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E25:9_1_2_1"/><i><font color="Green" title="E20">A45</font></i>: 
(  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c8">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c5">e</font> &amp; ( for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c8">E</font> holds <br/><font color="Maroon" title="c8">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Olive" title="b1">i</font></span>)</span> ) &amp; <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <font color="Maroon" title="c8">E</font> )
 <b>by </b><i><a class="txt" href="numeral1.html#E4:9_1_2_1"><i><font color="Green" title="E12">A37</font></i></a>, <a class="ref" href="numeral1.html#D1" target="_self" title="NUMERAL1:def.1">Def1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">g</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c4">d</font></span>)</span> as   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="numeral1.html#T3" target="_self" title="NUMERAL1:th.3">Th3</a></i>;<br/>
<a NAME="E27:9_1_2_1"/>
 <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="c2">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E28:9_1_2_1"/><b>then </b>
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c4">d</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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="c2">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="ordinal1.html#D2" title="ORDINAL1:def.2">ORDINAL1:def 2</a></i>;<br/>
<a NAME="E29:9_1_2_1"/><b>then </b><i><font color="Green" title="E21">A46</font></i>: 
 <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c9">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c4">d</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a></i>;<br/>
<a NAME="E30:9_1_2_1"/><i><font color="Green" title="E22">A47</font></i>: 
1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="numeral1.html#E1:9"><i><font color="Green" title="E9">A1</font></i></a></i>;<br/>
<a NAME="E31:9_1_2_1"/>
<font color="Maroon" title="c9">g</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="c2">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>
 
;<br/>
<b>then </b><i><font color="Green" title="E23">A48</font></i>:  <a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <font color="Maroon" title="c9">g</font> = 
<span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">k</font>
<b>by </b><i><a class="ref" href="numeral1.html#T2" target="_self" title="NUMERAL1:th.2">Th2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">k</font>
<b>by </b><i><a class="ref" href="series_1.html#T12" title="SERIES_1:th.12">SERIES_1:12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">k</font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T13" title="SEQ_1:th.13">SEQ_1:13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="numeral1.html#E1:9"><i><font color="Green" title="E9">A1</font></i></a>, <a class="ref" href="series_1.html#T26" title="SERIES_1:th.26">SERIES_1:26</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="numeral1.html#E1:9"><i><font color="Green" title="E9">A1</font></i></a>, <a class="ref" href="power.html#T46" title="POWER:th.46">POWER:46</a></i>
<br/>.= 
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p5">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span></span>)</span>

<br/>.= 
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="numeral1.html#E30:9_1_2_1"><i><font color="Green" title="E22">A47</font></i></a>, <a class="ref" href="xcmplx_1.html#T88" title="XCMPLX_1:th.88">XCMPLX_1:88</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1

;<br/>
<a NAME="E33:9_1_2_1"/><i><font color="Green" title="E24">A49</font></i>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c6">D</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="txt" href="numeral1.html#E12:9_1_2_1"><i><font color="Green" title="E16">A41</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<i><font color="Green" title="E25">A50</font></i>:  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c6">D</font> = 
 <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c6">D</font>
<b>by </b><i><a class="ref" href="afinsq_1.html#D1" title="AFINSQ_1:def.1">AFINSQ_1:def 1</a></i>
<br/>.= 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c4">d</font>
<b>by </b><i><a class="txt" href="numeral1.html#E6:9_1_2_1"><i><font color="Green" title="E13">A38</font></i></a>, <a class="ref" href="afinsq_1.html#D1" title="AFINSQ_1:def.1">AFINSQ_1:def 1</a></i>
<br/>.= 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c9">g</font>
<b>by </b><i><a class="txt" href="numeral1.html#E29:9_1_2_1"><i><font color="Green" title="E21">A46</font></i></a>, <a class="ref" href="afinsq_1.html#D1" title="AFINSQ_1:def.1">AFINSQ_1:def 1</a></i>
;<br/>
<a NAME="E35:9_1_2_1"/>
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c6">D</font> holds <br/><font color="Maroon" title="c6">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">g</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_2_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>reconsider </b><font color="Maroon" title="c11">I</font> = <font color="Maroon" title="c10">i</font> as    <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>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<b>assume </b><a NAME="E2:9_1_2_1_3"/><i><font color="Green" title="E26">A51</font></i>: 
<font color="Maroon" title="c10">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c6">D</font>
 ;<br/>

<a NAME="E3:9_1_2_1_3"/><b>then </b>
<font color="Maroon" title="c10">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c4">d</font>
 
<b>by </b><i><a class="txt" href="numeral1.html#E6:9_1_2_1"><i><font color="Green" title="E13">A38</font></i></a>, <a class="ref" href="afinsq_1.html#D1" title="AFINSQ_1:def.1">AFINSQ_1:def 1</a></i>;<br/>
<b>then </b><i><font color="Green" title="E27">A52</font></i>: <font color="Maroon" title="c9">g</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c11">I</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">I</font>
<b>by </b><i><a class="txt" href="numeral1.html#E29:9_1_2_1"><i><font color="Green" title="E21">A46</font></i></a>, <a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">I</font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T13" title="SEQ_1:th.13">SEQ_1:13</a></i>
<br/>.= 
<font color="Maroon" title="c3">b1</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c11">I</font></span>)</span>
<b>by </b><i><a class="ref" href="prepower.html#D1" title="PREPOWER:def.1">PREPOWER:def 1</a></i>
;<br/>
<a NAME="E5:9_1_2_1_3"/>
<font color="Maroon" title="c4">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c10">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">b1</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 
<b>by </b><i><a class="txt" href="numeral1.html#E3:9_1_2_1"><i><font color="Green" title="E11">A36</font></i></a>, <a class="txt" href="numeral1.html#E6:9_1_2_1"><i><font color="Green" title="E13">A38</font></i></a>, <a class="txt" href="numeral1.html#E2:9_1_2_1_3"><i><font color="Green" title="E26">A51</font></i></a></i>;<br/>
<a NAME="E6:9_1_2_1_3"/><b>then </b><i><font color="Green" title="E28">A53</font></i>: 
<font color="Maroon" title="c4">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c10">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">b1</font>
 
<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="E7:9_1_2_1_3"/>
<font color="Maroon" title="c6">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c10">i</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c10">i</font></span>)</span>
 
<b>by </b><i><a class="txt" href="numeral1.html#E6:9_1_2_1"><i><font color="Green" title="E13">A38</font></i></a>, <a class="txt" href="numeral1.html#E2:9_1_2_1_3"><i><font color="Green" title="E26">A51</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:9_1_2_1_3"/>
<font color="Maroon" title="c6">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c10">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">g</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c10">i</font>
 <b>by </b><i><a class="txt" href="numeral1.html#E4:9_1_2_1_3"><i><font color="Green" title="E27">A52</font></i></a>, <a class="txt" href="numeral1.html#E6:9_1_2_1_3"><i><font color="Green" title="E28">A53</font></i></a>, <a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E36:9_1_2_1"/><b>then </b>
<font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 
<b>by </b><i><a class="txt" href="numeral1.html#E6:9_1_2_1"><i><font color="Green" title="E13">A38</font></i></a>, <a class="txt" href="numeral1.html#E32:9_1_2_1"><i><font color="Green" title="E23">A48</font></i></a>, <a class="txt" href="numeral1.html#E33:9_1_2_1"><i><font color="Green" title="E24">A49</font></i></a>, <a class="txt" href="numeral1.html#E34:9_1_2_1"><i><font color="Green" title="E25">A50</font></i></a>, <a class="ref" href="numeral1.html#T4" target="_self" title="NUMERAL1:th.4">Th4</a></i>;<br/>
<a NAME="E37:9_1_2_1"/><b>then </b>
<font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T41" title="XREAL_1:th.41">XREAL_1:41</a></i>;<br/>
<a NAME="E38:9_1_2_1"/><b>then </b>
 <a href="power.html#K5" title="POWER:func.5">log</a> 2,<font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="power.html#K5" title="POWER:func.5">log</a> 2,<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">k</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="numeral1.html#E2:9_1_2_1"><i><font color="Green" title="E10">A35</font></i></a>, <a class="ref" href="power.html#T65" title="POWER:th.65">POWER:65</a></i>;<br/>
<a NAME="E39:9_1_2_1"/><b>then </b>
 <a href="power.html#K5" title="POWER:func.5">log</a> 2,<font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="power.html#K5" title="POWER:func.5">log</a> 2,<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">k</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="numeral1.html#E1:9"><i><font color="Green" title="E9">A1</font></i></a>, <a class="ref" href="power.html#T46" title="POWER:th.46">POWER:46</a></i>;<br/>
<a NAME="E40:9_1_2_1"/><b>then </b>
 <a href="power.html#K5" title="POWER:func.5">log</a> 2,<font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="power.html#K5" title="POWER:func.5">log</a> 2,<font color="Maroon" title="c2">b</font></span>)</span>
 
<b>by </b><i><a class="txt" href="numeral1.html#E1:9"><i><font color="Green" title="E9">A1</font></i></a>, <a class="ref" href="power.html#T63" title="POWER:th.63">POWER:63</a></i>;<br/>
<a NAME="E41:9_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="power.html#K5" title="POWER:func.5">log</a> 2,<font color="Maroon" title="c1">n</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><a href="power.html#K5" title="POWER:func.5">log</a> 2,<font color="Maroon" title="c2">b</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><a href="power.html#K5" title="POWER:func.5">log</a> 2,<font color="Maroon" title="c2">b</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><a href="power.html#K5" title="POWER:func.5">log</a> 2,<font color="Maroon" title="c2">b</font></span>)</span>
 
<b>by </b><i><a class="txt" href="numeral1.html#E15:9_1_2_1"><i><font color="Green" title="E18">A43</font></i></a>, <a class="ref" href="xreal_1.html#T76" title="XREAL_1:th.76">XREAL_1:76</a></i>;<br/>
<a NAME="E42:9_1_2_1"/><b>then </b><i><font color="Green" title="E26">A54</font></i>: 
<span class="p1">(<span class="default"><a href="power.html#K5" title="POWER:func.5">log</a> 2,<font color="Maroon" title="c1">n</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><a href="power.html#K5" title="POWER:func.5">log</a> 2,<font color="Maroon" title="c2">b</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 
<b>by </b><i><a class="txt" href="numeral1.html#E15:9_1_2_1"><i><font color="Green" title="E18">A43</font></i></a>, <a class="ref" href="xcmplx_1.html#T90" title="XCMPLX_1:th.90">XCMPLX_1:90</a></i>;<br/>
<a NAME="E43:9_1_2_1"/><i><font color="Green" title="E27">A55</font></i>: 
<span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c5">e</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c5">e</font>
 
<b>by </b><i><a class="txt" href="numeral1.html#E4:9_1_2_1"><i><font color="Green" title="E12">A37</font></i></a>, <a class="ref" href="funct_1.html#D4" title="FUNCT_1:def.4">FUNCT_1:def 4</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c10">l</font> = <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c5">e</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 as    <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/>
<a NAME="E45:9_1_2_1"/>
<font color="Maroon" title="c5">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c10">l</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<b>by </b><i><a class="txt" href="numeral1.html#E4:9_1_2_1"><i><font color="Green" title="E12">A37</font></i></a>, <a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></i>;<br/>
<a NAME="E46:9_1_2_1"/><b>then </b><i><font color="Green" title="E28">A56</font></i>: 
1 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c10">l</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="c5">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c10">l</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c10">l</font></span>)</span>
 
<b>by </b><i><a class="ref" href="nat_1.html#T4" title="NAT_1:th.4">NAT_1:4</a></i>;<br/>
<a NAME="E47:9_1_2_1"/>
<font color="Maroon" title="c10">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c8">E</font>
 
<b>by </b><i><a class="txt" href="numeral1.html#E4:9_1_2_1"><i><font color="Green" title="E12">A37</font></i></a>, <a class="txt" href="numeral1.html#E25:9_1_2_1"><i><font color="Green" title="E20">A45</font></i></a>, <a class="ref" href="funct_1.html#D4" title="FUNCT_1:def.4">FUNCT_1:def 4</a></i>;<br/>
<a NAME="E48:9_1_2_1"/><b>then </b><i><font color="Green" title="E29">A57</font></i>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c8">E</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="numeral1.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a></i>;<br/>
<a NAME="E49:9_1_2_1"/><i><font color="Green" title="E30">A58</font></i>: 
( <font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c10">l</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">l</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
<b>by </b><i><a class="txt" href="numeral1.html#E1:9"><i><font color="Green" title="E9">A1</font></i></a>, <a class="ref" href="prepower.html#T13" title="PREPOWER:th.13">PREPOWER:13</a></i>;<br/>
<a NAME="E50:9_1_2_1"/>
 <a href="power.html#K5" title="POWER:func.5">log</a> 2,<font color="Maroon" title="c2">b</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="power.html#K6" title="POWER:func.6">log</a> 2,1
 
<b>by </b><i><a class="txt" href="numeral1.html#E1:9"><i><font color="Green" title="E9">A1</font></i></a>, <a class="ref" href="power.html#T65" title="POWER:th.65">POWER:65</a></i>;<br/>
<a NAME="E51:9_1_2_1"/><b>then </b><i><font color="Green" title="E31">A59</font></i>: 
 <a href="power.html#K5" title="POWER:func.5">log</a> 2,<font color="Maroon" title="c2">b</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="power.html#T59" title="POWER:th.59">POWER:59</a></i>;<br/>
<a NAME="E52:9_1_2_1"/>
<font color="Maroon" title="c8">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c10">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c10">l</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c10">l</font></span>)</span>
 
<b>by </b><i><a class="txt" href="numeral1.html#E25:9_1_2_1"><i><font color="Green" title="E20">A45</font></i></a>, <a class="txt" href="numeral1.html#E43:9_1_2_1"><i><font color="Green" title="E27">A55</font></i></a></i>;<br/>
<a NAME="E53:9_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c10">l</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c10">l</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="txt" href="numeral1.html#E25:9_1_2_1"><i><font color="Green" title="E20">A45</font></i></a>, <a class="txt" href="numeral1.html#E43:9_1_2_1"><i><font color="Green" title="E27">A55</font></i></a>, <a class="txt" href="numeral1.html#E48:9_1_2_1"><i><font color="Green" title="E29">A57</font></i></a>, <a class="ref" href="stirl2_1.html#T52" title="STIRL2_1:th.52">STIRL2_1:52</a></i>;<br/>
<a NAME="E54:9_1_2_1"/><b>then </b>
<font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c10">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="txt" href="numeral1.html#E46:9_1_2_1"><i><font color="Green" title="E28">A56</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E55:9_1_2_1"/><b>then </b>
 <a href="power.html#K5" title="POWER:func.5">log</a> 2,<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c10">l</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="power.html#K5" title="POWER:func.5">log</a> 2,<font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="txt" href="numeral1.html#E49:9_1_2_1"><i><font color="Green" title="E30">A58</font></i></a>, <a class="ref" href="pre_ff.html#T12" title="PRE_FF:th.12">PRE_FF:12</a></i>;<br/>
<a NAME="E56:9_1_2_1"/><b>then </b>
 <a href="power.html#K5" title="POWER:func.5">log</a> 2,<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c10">l</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="power.html#K5" title="POWER:func.5">log</a> 2,<font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="txt" href="numeral1.html#E1:9"><i><font color="Green" title="E9">A1</font></i></a>, <a class="ref" href="power.html#T46" title="POWER:th.46">POWER:46</a></i>;<br/>
<a NAME="E57:9_1_2_1"/><b>then </b>
<font color="Maroon" title="c10">l</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="power.html#K5" title="POWER:func.5">log</a> 2,<font color="Maroon" title="c2">b</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="power.html#K5" title="POWER:func.5">log</a> 2,<font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="txt" href="numeral1.html#E1:9"><i><font color="Green" title="E9">A1</font></i></a>, <a class="ref" href="power.html#T63" title="POWER:th.63">POWER:63</a></i>;<br/>
<a NAME="E58:9_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c10">l</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><a href="power.html#K5" title="POWER:func.5">log</a> 2,<font color="Maroon" title="c2">b</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><a href="power.html#K5" title="POWER:func.5">log</a> 2,<font color="Maroon" title="c2">b</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="power.html#K5" title="POWER:func.5">log</a> 2,<font color="Maroon" title="c1">n</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><a href="power.html#K5" title="POWER:func.5">log</a> 2,<font color="Maroon" title="c2">b</font></span>)</span>
 
<b>by </b><i><a class="txt" href="numeral1.html#E51:9_1_2_1"><i><font color="Green" title="E31">A59</font></i></a>, <a class="ref" href="xreal_1.html#T74" title="XREAL_1:th.74">XREAL_1:74</a></i>;<br/>
<a NAME="E59:9_1_2_1"/><b>then </b><i><font color="Green" title="E32">A60</font></i>: 
<font color="Maroon" title="c10">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="power.html#K5" title="POWER:func.5">log</a> 2,<font color="Maroon" title="c1">n</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><a href="power.html#K5" title="POWER:func.5">log</a> 2,<font color="Maroon" title="c2">b</font></span>)</span>
 
<b>by </b><i><a class="txt" href="numeral1.html#E51:9_1_2_1"><i><font color="Green" title="E31">A59</font></i></a>, <a class="ref" href="xcmplx_1.html#T90" title="XCMPLX_1:th.90">XCMPLX_1:90</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">g</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c5">e</font></span>)</span> as   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="numeral1.html#T3" target="_self" title="NUMERAL1:th.3">Th3</a></i>;<br/>
<a NAME="E61:9_1_2_1"/>
 <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="c2">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E62:9_1_2_1"/><b>then </b>
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c5">e</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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="c2">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="ordinal1.html#D2" title="ORDINAL1:def.2">ORDINAL1:def 2</a></i>;<br/>
<a NAME="E63:9_1_2_1"/><b>then </b><i><font color="Green" title="E33">A61</font></i>: 
 <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c11">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c5">e</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a></i>;<br/>
<a NAME="E64:9_1_2_1"/><i><font color="Green" title="E34">A62</font></i>: 
1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="numeral1.html#E1:9"><i><font color="Green" title="E9">A1</font></i></a></i>;<br/>
<a NAME="E65:9_1_2_1"/>
<font color="Maroon" title="c11">g</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="c2">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">l</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>
 
;<br/>
<b>then </b><i><font color="Green" title="E35">A63</font></i>:  <a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <font color="Maroon" title="c11">g</font> = 
<span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">l</font>
<b>by </b><i><a class="ref" href="numeral1.html#T2" target="_self" title="NUMERAL1:th.2">Th2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">l</font>
<b>by </b><i><a class="ref" href="series_1.html#T12" title="SERIES_1:th.12">SERIES_1:12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">l</font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T13" title="SEQ_1:th.13">SEQ_1:13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p4">(<span class="default"><font color="Maroon" title="c10">l</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="numeral1.html#E1:9"><i><font color="Green" title="E9">A1</font></i></a>, <a class="ref" href="series_1.html#T26" title="SERIES_1:th.26">SERIES_1:26</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p4">(<span class="default"><font color="Maroon" title="c10">l</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="numeral1.html#E1:9"><i><font color="Green" title="E9">A1</font></i></a>, <a class="ref" href="power.html#T46" title="POWER:th.46">POWER:46</a></i>
<br/>.= 
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p5">(<span class="default"><font color="Maroon" title="c10">l</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span></span>)</span>

<br/>.= 
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">l</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="numeral1.html#E64:9_1_2_1"><i><font color="Green" title="E34">A62</font></i></a>, <a class="ref" href="xcmplx_1.html#T88" title="XCMPLX_1:th.88">XCMPLX_1:88</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">l</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1

;<br/>
<a NAME="E67:9_1_2_1"/><i><font color="Green" title="E36">A64</font></i>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c8">E</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="txt" href="numeral1.html#E48:9_1_2_1"><i><font color="Green" title="E29">A57</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<i><font color="Green" title="E37">A65</font></i>:  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c8">E</font> = 
 <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c8">E</font>
<b>by </b><i><a class="ref" href="afinsq_1.html#D1" title="AFINSQ_1:def.1">AFINSQ_1:def 1</a></i>
<br/>.= 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c5">e</font>
<b>by </b><i><a class="txt" href="numeral1.html#E25:9_1_2_1"><i><font color="Green" title="E20">A45</font></i></a>, <a class="ref" href="afinsq_1.html#D1" title="AFINSQ_1:def.1">AFINSQ_1:def 1</a></i>
<br/>.= 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c11">g</font>
<b>by </b><i><a class="txt" href="numeral1.html#E63:9_1_2_1"><i><font color="Green" title="E33">A61</font></i></a>, <a class="ref" href="afinsq_1.html#D1" title="AFINSQ_1:def.1">AFINSQ_1:def 1</a></i>
;<br/>
<a NAME="E69:9_1_2_1"/>
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c8">E</font> holds <br/><font color="Maroon" title="c8">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">g</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_2_1_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c12">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>reconsider </b><font color="Maroon" title="c13">I</font> = <font color="Maroon" title="c12">i</font> as    <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>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<b>assume </b><a NAME="E2:9_1_2_1_6"/><i><font color="Green" title="E38">A66</font></i>: 
<font color="Maroon" title="c12">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c8">E</font>
 ;<br/>

<a NAME="E3:9_1_2_1_6"/><b>then </b>
<font color="Maroon" title="c12">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c5">e</font>
 
<b>by </b><i><a class="txt" href="numeral1.html#E25:9_1_2_1"><i><font color="Green" title="E20">A45</font></i></a>, <a class="ref" href="afinsq_1.html#D1" title="AFINSQ_1:def.1">AFINSQ_1:def 1</a></i>;<br/>
<b>then </b><i><font color="Green" title="E39">A67</font></i>: <font color="Maroon" title="c11">g</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">I</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">I</font>
<b>by </b><i><a class="txt" href="numeral1.html#E63:9_1_2_1"><i><font color="Green" title="E33">A61</font></i></a>, <a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">I</font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T13" title="SEQ_1:th.13">SEQ_1:13</a></i>
<br/>.= 
<font color="Maroon" title="c3">b1</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">I</font></span>)</span>
<b>by </b><i><a class="ref" href="prepower.html#D1" title="PREPOWER:def.1">PREPOWER:def 1</a></i>
;<br/>
<a NAME="E5:9_1_2_1_6"/>
<font color="Maroon" title="c5">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c12">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">b1</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 
<b>by </b><i><a class="txt" href="numeral1.html#E4:9_1_2_1"><i><font color="Green" title="E12">A37</font></i></a>, <a class="txt" href="numeral1.html#E25:9_1_2_1"><i><font color="Green" title="E20">A45</font></i></a>, <a class="txt" href="numeral1.html#E2:9_1_2_1_6"><i><font color="Green" title="E38">A66</font></i></a></i>;<br/>
<a NAME="E6:9_1_2_1_6"/><b>then </b><i><font color="Green" title="E40">A68</font></i>: 
<font color="Maroon" title="c5">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c12">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">b1</font>
 
<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="E7:9_1_2_1_6"/>
<font color="Maroon" title="c8">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c12">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c12">i</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c12">i</font></span>)</span>
 
<b>by </b><i><a class="txt" href="numeral1.html#E25:9_1_2_1"><i><font color="Green" title="E20">A45</font></i></a>, <a class="txt" href="numeral1.html#E2:9_1_2_1_6"><i><font color="Green" title="E38">A66</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:9_1_2_1_6"/>
<font color="Maroon" title="c8">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c12">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">g</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c12">i</font>
 <b>by </b><i><a class="txt" href="numeral1.html#E4:9_1_2_1_6"><i><font color="Green" title="E39">A67</font></i></a>, <a class="txt" href="numeral1.html#E6:9_1_2_1_6"><i><font color="Green" title="E40">A68</font></i></a>, <a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E70:9_1_2_1"/><b>then </b>
<font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">l</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 
<b>by </b><i><a class="txt" href="numeral1.html#E25:9_1_2_1"><i><font color="Green" title="E20">A45</font></i></a>, <a class="txt" href="numeral1.html#E66:9_1_2_1"><i><font color="Green" title="E35">A63</font></i></a>, <a class="txt" href="numeral1.html#E67:9_1_2_1"><i><font color="Green" title="E36">A64</font></i></a>, <a class="txt" href="numeral1.html#E68:9_1_2_1"><i><font color="Green" title="E37">A65</font></i></a>, <a class="ref" href="numeral1.html#T4" target="_self" title="NUMERAL1:th.4">Th4</a></i>;<br/>
<a NAME="E71:9_1_2_1"/><b>then </b>
<font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">l</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T41" title="XREAL_1:th.41">XREAL_1:41</a></i>;<br/>
<a NAME="E72:9_1_2_1"/><b>then </b>
 <a href="power.html#K5" title="POWER:func.5">log</a> 2,<font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="power.html#K5" title="POWER:func.5">log</a> 2,<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">l</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="numeral1.html#E2:9_1_2_1"><i><font color="Green" title="E10">A35</font></i></a>, <a class="ref" href="power.html#T65" title="POWER:th.65">POWER:65</a></i>;<br/>
<a NAME="E73:9_1_2_1"/><b>then </b>
 <a href="power.html#K5" title="POWER:func.5">log</a> 2,<font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="power.html#K5" title="POWER:func.5">log</a> 2,<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">l</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="numeral1.html#E1:9"><i><font color="Green" title="E9">A1</font></i></a>, <a class="ref" href="power.html#T46" title="POWER:th.46">POWER:46</a></i>;<br/>
<a NAME="E74:9_1_2_1"/><b>then </b>
 <a href="power.html#K5" title="POWER:func.5">log</a> 2,<font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">l</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="power.html#K5" title="POWER:func.5">log</a> 2,<font color="Maroon" title="c2">b</font></span>)</span>
 
<b>by </b><i><a class="txt" href="numeral1.html#E1:9"><i><font color="Green" title="E9">A1</font></i></a>, <a class="ref" href="power.html#T63" title="POWER:th.63">POWER:63</a></i>;<br/>
<a NAME="E75:9_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="power.html#K5" title="POWER:func.5">log</a> 2,<font color="Maroon" title="c1">n</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><a href="power.html#K5" title="POWER:func.5">log</a> 2,<font color="Maroon" title="c2">b</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">l</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><a href="power.html#K5" title="POWER:func.5">log</a> 2,<font color="Maroon" title="c2">b</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><a href="power.html#K5" title="POWER:func.5">log</a> 2,<font color="Maroon" title="c2">b</font></span>)</span>
 
<b>by </b><i><a class="txt" href="numeral1.html#E51:9_1_2_1"><i><font color="Green" title="E31">A59</font></i></a>, <a class="ref" href="xreal_1.html#T76" title="XREAL_1:th.76">XREAL_1:76</a></i>;<br/>
<a NAME="E76:9_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="power.html#K5" title="POWER:func.5">log</a> 2,<font color="Maroon" title="c1">n</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><a href="power.html#K5" title="POWER:func.5">log</a> 2,<font color="Maroon" title="c2">b</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">l</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 
<b>by </b><i><a class="txt" href="numeral1.html#E51:9_1_2_1"><i><font color="Green" title="E31">A59</font></i></a>, <a class="ref" href="xcmplx_1.html#T90" title="XCMPLX_1:th.90">XCMPLX_1:90</a></i>;<br/>
<a NAME="E77:9_1_2_1"/><b>then </b>
<font color="Maroon" title="c7">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">l</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 
<b>by </b><i><a class="txt" href="numeral1.html#E23:9_1_2_1"><i><font color="Green" title="E19">A44</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E78:9_1_2_1"/><b>then </b><i><font color="Green" title="E38">A69</font></i>: 
<font color="Maroon" title="c7">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">l</font>
 
<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="E79:9_1_2_1"/>
<font color="Maroon" title="c10">l</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 
<b>by </b><i><a class="txt" href="numeral1.html#E42:9_1_2_1"><i><font color="Green" title="E26">A54</font></i></a>, <a class="txt" href="numeral1.html#E59:9_1_2_1"><i><font color="Green" title="E32">A60</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E80:9_1_2_1"/><b>then </b>
<font color="Maroon" title="c10">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">k</font>
 
<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="E81:9_1_2_1"/><b>then </b><i><font color="Green" title="E39">A70</font></i>: 
<font color="Maroon" title="c7">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">l</font>
 
<b>by </b><i><a class="txt" href="numeral1.html#E78:9_1_2_1"><i><font color="Green" title="E38">A69</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/>
<i><font color="Green" title="E40">A71</font></i>:  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c4">d</font> = 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c4">d</font>
<b>by </b><i><a class="ref" href="afinsq_1.html#D1" title="AFINSQ_1:def.1">AFINSQ_1:def 1</a></i>
<br/>.= 
 <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c5">e</font>
<b>by </b><i><a class="txt" href="numeral1.html#E81:9_1_2_1"><i><font color="Green" title="E39">A70</font></i></a>, <a class="ref" href="afinsq_1.html#D1" title="AFINSQ_1:def.1">AFINSQ_1:def 1</a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_2_1_8"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c12">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:9_1_2_1_8"/><i><font color="Green" title="E41">A72</font></i>: 
<font color="Maroon" title="c12">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c4">d</font>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c13">o</font> = <font color="Maroon" title="c12">a</font> as    <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/><a NAME="E3:9_1_2_1_8"/>
<font color="Maroon" title="c12">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c5">e</font>
 <b>by </b><i><a class="txt" href="numeral1.html#E81:9_1_2_1"><i><font color="Green" title="E39">A70</font></i></a>, <a class="txt" href="numeral1.html#E1:9_1_2_1_8"><i><font color="Green" title="E41">A72</font></i></a>, <a class="ref" href="afinsq_1.html#D1" title="AFINSQ_1:def.1">AFINSQ_1:def 1</a></i>;<br/><a NAME="E4:9_1_2_1_8"/><b>then </b><i><font color="Green" title="E42">A73</font></i>: 
<font color="Maroon" title="c12">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c5">e</font>
 <b>by </b><i><a class="ref" href="afinsq_1.html#D1" title="AFINSQ_1:def.1">AFINSQ_1:def 1</a></i>;<br/><a NAME="E5:9_1_2_1_8"/><i><font color="Green" title="E43">A74</font></i>: 
<font color="Maroon" title="c13">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c4">d</font>
 <b>by </b><i><a class="txt" href="numeral1.html#E1:9_1_2_1_8"><i><font color="Green" title="E41">A72</font></i></a>, <a class="ref" href="afinsq_1.html#D1" title="AFINSQ_1:def.1">AFINSQ_1:def 1</a></i>;<br/><a NAME="E6:9_1_2_1_8"/><b>then </b><i><font color="Green" title="E44">A75</font></i>: 
<font color="Maroon" title="c13">o</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c4">d</font>
 <b>by </b><i><a class="ref" href="prgcor_2.html#T1" title="PRGCOR_2:th.1">PRGCOR_2:1</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c14">oo</font> = <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c4">d</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c13">o</font> as    <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>by </b><i><a class="ref" href="nat_1.html#T21" title="NAT_1:th.21">NAT_1:21</a></i>;<br/><a NAME="E8:9_1_2_1_8"/>
<font color="Maroon" title="c13">o</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 <b>by </b><i><a class="txt" href="numeral1.html#E5:9_1_2_1_8"><i><font color="Green" title="E43">A74</font></i></a>, <a class="ref" href="prgcor_2.html#T1" title="PRGCOR_2:th.1">PRGCOR_2:1</a></i>;<br/><a NAME="E9:9_1_2_1_8"/><b>then </b><i><font color="Green" title="E45">A76</font></i>: 
<font color="Maroon" title="c13">o</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">k</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1_2_1_8_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:9_1_2_1_8_1"/>
( ( <font color="Maroon" title="c13">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c13">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">k</font> ) or ( <font color="Maroon" title="c13">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c13">o</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">k</font> ) or ( <font color="Maroon" title="c13">o</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c13">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">k</font> ) or ( <font color="Maroon" title="c13">o</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c13">o</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">k</font> ) )
 </span><b>by </b><i><a class="txt" href="numeral1.html#E9:9_1_2_1_8"><i><font color="Green" title="E45">A76</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1_2_1_8_1_1"><b>suppose </b></a><a NAME="E1:9_1_2_1_8_1_1"/><i><font color="Green" title="E46">A77</font></i>: 
( <font color="Maroon" title="c13">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c13">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">k</font> )
 ;<br/><div class="add"><a NAME="E2:9_1_2_1_8_1_1"/><b>then </b>
 <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c4">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="ref" href="afinsq_1.html#D1" title="AFINSQ_1:def.1">AFINSQ_1:def 1</a></i>;<br/><a NAME="E3:9_1_2_1_8_1_1"/><b>then </b>
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c6">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="numeral1.html#E6:9_1_2_1"><i><font color="Green" title="E13">A38</font></i></a>, <a class="ref" href="afinsq_1.html#D1" title="AFINSQ_1:def.1">AFINSQ_1:def 1</a></i>;<br/><b>then </b><i><font color="Green" title="E47">A78</font></i>: <font color="Maroon" title="c6">D</font> = 
<span class="p1"><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">%&gt;</a></span>
<b>by </b><i><a class="ref" href="afinsq_1.html#T38" title="AFINSQ_1:th.38">AFINSQ_1:38</a></i>
<br/>.= 
<span class="p1"><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">&lt;%</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">%&gt;</a></span>
<b>by </b><i><a class="txt" href="numeral1.html#E6:9_1_2_1"><i><font color="Green" title="E13">A38</font></i></a>, <a class="txt" href="numeral1.html#E1:9_1_2_1_8"><i><font color="Green" title="E41">A72</font></i></a>, <a class="txt" href="numeral1.html#E1:9_1_2_1_8_1_1"><i><font color="Green" title="E46">A77</font></i></a></i>
<br/>.= 
<span class="p1"><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">&lt;%</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> 1</span>)</span></span><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">%&gt;</a></span>
<b>by </b><i><a class="ref" href="newton.html#T9" title="NEWTON:th.9">NEWTON:9</a></i>
;<br/><a NAME="E5:9_1_2_1_8_1_1"/>
 <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c5">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="numeral1.html#E81:9_1_2_1"><i><font color="Green" title="E39">A70</font></i></a>, <a class="txt" href="numeral1.html#E1:9_1_2_1_8_1_1"><i><font color="Green" title="E46">A77</font></i></a>, <a class="ref" href="afinsq_1.html#D1" title="AFINSQ_1:def.1">AFINSQ_1:def 1</a></i>;<br/><a NAME="E6:9_1_2_1_8_1_1"/><b>then </b>
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c8">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="numeral1.html#E25:9_1_2_1"><i><font color="Green" title="E20">A45</font></i></a>, <a class="ref" href="afinsq_1.html#D1" title="AFINSQ_1:def.1">AFINSQ_1:def 1</a></i>;<br/><b>then </b><i><font color="Green" title="E48">A79</font></i>: <font color="Maroon" title="c8">E</font> = 
<span class="p1"><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">%&gt;</a></span>
<b>by </b><i><a class="ref" href="afinsq_1.html#T38" title="AFINSQ_1:th.38">AFINSQ_1:38</a></i>
<br/>.= 
<span class="p1"><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">&lt;%</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">%&gt;</a></span>
<b>by </b><i><a class="txt" href="numeral1.html#E25:9_1_2_1"><i><font color="Green" title="E20">A45</font></i></a>, <a class="txt" href="numeral1.html#E4:9_1_2_1_8"><i><font color="Green" title="E42">A73</font></i></a>, <a class="txt" href="numeral1.html#E1:9_1_2_1_8_1_1"><i><font color="Green" title="E46">A77</font></i></a></i>
<br/>.= 
<span class="p1"><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">&lt;%</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> 1</span>)</span></span><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">%&gt;</a></span>
<b>by </b><i><a class="ref" href="newton.html#T9" title="NEWTON:th.9">NEWTON:9</a></i>
;<br/><b>thus </b><font color="Maroon" title="c4">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c12">a</font> = 
<font color="Maroon" title="c1">n</font>
<b>by </b><i><a class="txt" href="numeral1.html#E6:9_1_2_1"><i><font color="Green" title="E13">A38</font></i></a>, <a class="txt" href="numeral1.html#E1:9_1_2_1_8_1_1"><i><font color="Green" title="E46">A77</font></i></a>, <a class="txt" href="numeral1.html#E4:9_1_2_1_8_1_1"><i><font color="Green" title="E47">A78</font></i></a>, <a class="ref" href="stirl2_1.html#T44" title="STIRL2_1:th.44">STIRL2_1:44</a></i>
<br/>.= 
<font color="Maroon" title="c5">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c12">a</font>
<b>by </b><i><a class="txt" href="numeral1.html#E25:9_1_2_1"><i><font color="Green" title="E20">A45</font></i></a>, <a class="txt" href="numeral1.html#E1:9_1_2_1_8_1_1"><i><font color="Green" title="E46">A77</font></i></a>, <a class="txt" href="numeral1.html#E7:9_1_2_1_8_1_1"><i><font color="Green" title="E48">A79</font></i></a>, <a class="ref" href="stirl2_1.html#T44" title="STIRL2_1:th.44">STIRL2_1:44</a></i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1_2_1_8_1_2"><b>suppose </b></a><a NAME="E1:9_1_2_1_8_1_2"/><i><font color="Green" title="E46">A80</font></i>: 
( <font color="Maroon" title="c13">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c13">o</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">k</font> )
 ;<br/><div class="add"><b>set </b><font color="Maroon" title="c15">d'</font> = <font color="Maroon" title="c6">D</font>;<br/><a NAME="E2:9_1_2_1_8_1_2"/><i><font color="Green" title="E47">A81</font></i>: 
 <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c6">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">oo</font>
 <b>by </b><i><a class="txt" href="numeral1.html#E6:9_1_2_1"><i><font color="Green" title="E13">A38</font></i></a>, <a class="txt" href="numeral1.html#E1:9_1_2_1_8_1_2"><i><font color="Green" title="E46">A80</font></i></a>, <a class="ref" href="afinsq_1.html#D1" title="AFINSQ_1:def.1">AFINSQ_1:def 1</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c16">do</font> =  <a href="afinsq_1.html#K4" title="AFINSQ_1:func.4">&lt;%&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  as   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/><a NAME="E4:9_1_2_1_8_1_2"/><i><font color="Green" title="E48">A82</font></i>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c6">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">oo</font>
 <b>by </b><i><a class="txt" href="numeral1.html#E2:9_1_2_1_8_1_2"><i><font color="Green" title="E47">A81</font></i></a>, <a class="ref" href="afinsq_1.html#D1" title="AFINSQ_1:def.1">AFINSQ_1:def 1</a></i>;<br/><a NAME="E5:9_1_2_1_8_1_2"/><i><font color="Green" title="E49">A83</font></i>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c16">do</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">o</font>
 <b>by </b><i><a class="txt" href="numeral1.html#E1:9_1_2_1_8_1_2"><i><font color="Green" title="E46">A80</font></i></a>, <a class="ref" href="afinsq_1.html#T18" title="AFINSQ_1:th.18">AFINSQ_1:18</a></i>;<br/><a NAME="E6:9_1_2_1_8_1_2"/><b>then </b><i><font color="Green" title="E50">A84</font></i>: 
 <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c6">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c16">do</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c6">D</font></span>)</span>
 <b>by </b><i><a class="txt" href="numeral1.html#E6:9_1_2_1"><i><font color="Green" title="E13">A38</font></i></a>, <a class="txt" href="numeral1.html#E4:9_1_2_1_8_1_2"><i><font color="Green" title="E48">A82</font></i></a>, <a class="ref" href="afinsq_1.html#D1" title="AFINSQ_1:def.1">AFINSQ_1:def 1</a></i>;<br/><a NAME="E7:9_1_2_1_8_1_2"/>
 <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><a NAME="E8:9_1_2_1_8_1_2"/><b>then </b><i><font color="Green" title="E51">A85</font></i>: 
for <font color="Olive" title="b1">x</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">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c16">do</font> holds <br/><font color="Maroon" title="c6">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">do</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 ;<br/><a NAME="E9:9_1_2_1_8_1_2"/>
for <font color="Olive" title="b1">x</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">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c6">D</font> holds <br/><font color="Maroon" title="c6">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c16">do</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 <b>by </b><i><a class="txt" href="numeral1.html#E1:9_1_2_1_8_1_2"><i><font color="Green" title="E46">A80</font></i></a>, <a class="txt" href="numeral1.html#E5:9_1_2_1_8_1_2"><i><font color="Green" title="E49">A83</font></i></a></i>;<br/><a NAME="E10:9_1_2_1_8_1_2"/><b>then </b>
<font color="Maroon" title="c6">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">do</font> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <font color="Maroon" title="c6">D</font>
 <b>by </b><i><a class="txt" href="numeral1.html#E6:9_1_2_1_8_1_2"><i><font color="Green" title="E50">A84</font></i></a>, <a class="txt" href="numeral1.html#E8:9_1_2_1_8_1_2"><i><font color="Green" title="E51">A85</font></i></a>, <a class="ref" href="afinsq_1.html#D4" title="AFINSQ_1:def.4">AFINSQ_1:def 4</a></i>;<br/><a NAME="E11:9_1_2_1_8_1_2"/><b>then </b><i><font color="Green" title="E52">A86</font></i>: 
<font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <font color="Maroon" title="c16">do</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <font color="Maroon" title="c6">D</font></span>)</span>
 <b>by </b><i><a class="txt" href="numeral1.html#E6:9_1_2_1"><i><font color="Green" title="E13">A38</font></i></a>, <a class="ref" href="numeral1.html#T1" target="_self" title="NUMERAL1:th.1">Th1</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c17">bo</font> = <font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font> as    <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>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><a NAME="E13:9_1_2_1_8_1_2"/><i><font color="Green" title="E53">A87</font></i>: 
<font color="Maroon" title="c17">bo</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="numeral1.html#E1:9"><i><font color="Green" title="E9">A1</font></i></a>, <a class="ref" href="prepower.html#T12" title="PREPOWER:th.12">PREPOWER:12</a></i>;<br/><a NAME="E14:9_1_2_1_8_1_2"/>
 <a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <font color="Maroon" title="c16">do</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="numeral1.html#E1:9_1_2_1_8_1_2"><i><font color="Green" title="E46">A80</font></i></a>, <a class="txt" href="numeral1.html#E5:9_1_2_1_8_1_2"><i><font color="Green" title="E49">A83</font></i></a>, <a class="ref" href="stirl2_1.html#T53" title="STIRL2_1:th.53">STIRL2_1:53</a></i>;<br/><a NAME="E15:9_1_2_1_8_1_2"/><b>then </b><i><font color="Green" title="E54">A88</font></i>: 
<span class="p1">(<span class="default"><a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <font color="Maroon" title="c16">do</font></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="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="nat_2.html#T4" title="NAT_2:th.4">NAT_2:4</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_2_1_8_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c18">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>reconsider </b><font color="Maroon" title="c19">K</font> = <font color="Maroon" title="c18">k</font> as    <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>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><b>assume </b><a NAME="E2:9_1_2_1_8_1_2_1"/>
<font color="Maroon" title="c18">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c6">D</font>
 ;<br/><a NAME="E3:9_1_2_1_8_1_2_1"/><b>then </b>
<font color="Maroon" title="c18">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c6">D</font>
 <b>by </b><i><a class="ref" href="afinsq_1.html#D1" title="AFINSQ_1:def.1">AFINSQ_1:def 1</a></i>;<br/><a NAME="E4:9_1_2_1_8_1_2_1"/><b>then </b>
<font color="Maroon" title="c18">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c6">D</font>
 <b>by </b><i><a class="ref" href="prgcor_2.html#T1" title="PRGCOR_2:th.1">PRGCOR_2:1</a></i>;<br/><a NAME="E5:9_1_2_1_8_1_2_1"/><b>then </b>
<font color="Maroon" title="c13">o</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c18">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c6">D</font>
 <b>by </b><i><a class="txt" href="numeral1.html#E1:9_1_2_1_8_1_2"><i><font color="Green" title="E46">A80</font></i></a>, <a class="ref" href="afinsq_1.html#D1" title="AFINSQ_1:def.1">AFINSQ_1:def 1</a></i>;<br/><a NAME="E6:9_1_2_1_8_1_2_1"/><b>then </b>
<font color="Maroon" title="c13">o</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c18">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c6">D</font>
 <b>by </b><i><a class="ref" href="prgcor_2.html#T1" title="PRGCOR_2:th.1">PRGCOR_2:1</a></i>;<br/><b>then </b><font color="Maroon" title="c6">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c19">K</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">o</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c18">k</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">o</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c18">k</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="numeral1.html#E6:9_1_2_1"><i><font color="Green" title="E13">A38</font></i></a>, <a class="txt" href="numeral1.html#E1:9_1_2_1_8_1_2"><i><font color="Green" title="E46">A80</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">o</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c18">k</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c18">k</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T13" title="NEWTON:th.13">NEWTON:13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">o</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c18">k</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c18">k</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span>

;<br/><b>hence </b><a NAME="E8:9_1_2_1_8_1_2_1"/>
<font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c6">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c18">k</font>
 <b>by </b><i><a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">NAT_D:def 3</a></i>;<br/></div><b>end;</b></div><a NAME="E17:9_1_2_1_8_1_2"/><b>then </b><i><font color="Green" title="E55">A89</font></i>: 
<font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a>  <a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <font color="Maroon" title="c6">D</font>
 <b>by </b><i><a class="ref" href="numeral1.html#T5" target="_self" title="NUMERAL1:th.5">Th5</a></i>;<br/><a NAME="E18:9_1_2_1_8_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <font color="Maroon" title="c6">D</font></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="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="numeral1.html#E13:9_1_2_1_8_1_2"><i><font color="Green" title="E53">A87</font></i></a>, <a class="ref" href="pepin.html#T6" title="PEPIN:th.6">PEPIN:6</a></i>;<br/><a NAME="E19:9_1_2_1_8_1_2"/><b>then </b>
<font color="Maroon" title="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <font color="Maroon" title="c6">D</font></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="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></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"><a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <font color="Maroon" title="c16">do</font></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="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="numeral1.html#E11:9_1_2_1_8_1_2"><i><font color="Green" title="E52">A86</font></i></a>, <a class="ref" href="nat_d.html#T19" title="NAT_D:th.19">NAT_D:19</a></i>;<br/><a NAME="E20:9_1_2_1_8_1_2"/><b>then </b><i><font color="Green" title="E56">A90</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <font color="Maroon" title="c6">D</font>
 <b>by </b><i><a class="txt" href="numeral1.html#E15:9_1_2_1_8_1_2"><i><font color="Green" title="E54">A88</font></i></a>, <a class="txt" href="numeral1.html#E17:9_1_2_1_8_1_2"><i><font color="Green" title="E55">A89</font></i></a>, <a class="ref" href="nat_d.html#T3" title="NAT_D:th.3">NAT_D:3</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c18">o1</font> = <font color="Maroon" title="c13">o</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 as    <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/><a NAME="E22:9_1_2_1_8_1_2"/>
( <font color="Maroon" title="c18">o1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">k</font> &amp; <font color="Maroon" title="c7">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c4">d</font> )
 <b>by </b><i><a class="txt" href="numeral1.html#E1:9_1_2_1_8_1_2"><i><font color="Green" title="E46">A80</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a>, <a class="ref" href="xreal_1.html#T149" title="XREAL_1:th.149">XREAL_1:149</a></i>;<br/><a NAME="E23:9_1_2_1_8_1_2"/><b>then </b><i><font color="Green" title="E57">A91</font></i>: 
<font color="Maroon" title="c18">o1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c4">d</font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c19">oo1</font> = <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c4">d</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c18">o1</font> as    <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>by </b><i><a class="txt" href="numeral1.html#E1:9_1_2_1_8_1_2"><i><font color="Green" title="E46">A80</font></i></a></i>;<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> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>$2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default">$1 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c18">o1</font></span>)</span>;<br/><a NAME="E25:9_1_2_1_8_1_2"/><i><font color="Green" title="E58">A92</font></i>: 
for <font color="Olive" title="b1">u</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">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">oo1</font> holds <br/> ex <font color="Olive" title="b2">x</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">u</font>,<font color="Olive" title="b2">x</font>]
 ;<br/><b>consider </b><font color="Maroon" title="c20">d'1</font> being   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E27:9_1_2_1_8_1_2"/><i><font color="Green" title="E59">A93</font></i>: 
(  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c20">d'1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">oo1</font> &amp; ( for <font color="Olive" title="b1">x</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">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">oo1</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c20">d'1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">x</font>] ) )
 <b>from </b><i><a class="ref" href="stirl2_1.html#S6" title="STIRL2_1:sch.6">STIRL2_1:sch 6</a>(<a class="txt" href="numeral1.html#E25:9_1_2_1_8_1_2"><i><font color="Green" title="E58">A92</font></i></a>);<br/></i><b>reconsider </b><font color="Maroon" title="c21">do1</font> = <font color="Maroon" title="c6">D</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c18">o1</font> as   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="afinsq_1.html#T16" title="AFINSQ_1:th.16">AFINSQ_1:16</a></i>;<br/><a NAME="E29:9_1_2_1_8_1_2"/><i><font color="Green" title="E60">A94</font></i>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c20">d'1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">oo1</font>
 <b>by </b><i><a class="txt" href="numeral1.html#E27:9_1_2_1_8_1_2"><i><font color="Green" title="E59">A93</font></i></a>, <a class="ref" href="afinsq_1.html#D1" title="AFINSQ_1:def.1">AFINSQ_1:def 1</a></i>;<br/><i><font color="Green" title="E61">A95</font></i>:  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c4">d</font> = 
 <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c6">D</font>
<b>by </b><i><a class="txt" href="numeral1.html#E6:9_1_2_1"><i><font color="Green" title="E13">A38</font></i></a>, <a class="ref" href="afinsq_1.html#D1" title="AFINSQ_1:def.1">AFINSQ_1:def 1</a></i>
<br/>.= 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c6">D</font>
<b>by </b><i><a class="ref" href="afinsq_1.html#D1" title="AFINSQ_1:def.1">AFINSQ_1:def 1</a></i>
;<br/><a NAME="E31:9_1_2_1_8_1_2"/><b>then </b><i><font color="Green" title="E62">A96</font></i>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c21">do1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">o1</font>
 <b>by </b><i><a class="txt" href="numeral1.html#E23:9_1_2_1_8_1_2"><i><font color="Green" title="E57">A91</font></i></a>, <a class="ref" href="afinsq_1.html#T14" title="AFINSQ_1:th.14">AFINSQ_1:14</a></i>;<br/><a NAME="E32:9_1_2_1_8_1_2"/><b>then </b><i><font color="Green" title="E63">A97</font></i>: 
 <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c6">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c21">do1</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c20">d'1</font></span>)</span>
 <b>by </b><i><a class="txt" href="numeral1.html#E6:9_1_2_1"><i><font color="Green" title="E13">A38</font></i></a>, <a class="txt" href="numeral1.html#E29:9_1_2_1_8_1_2"><i><font color="Green" title="E60">A94</font></i></a>, <a class="ref" href="afinsq_1.html#D1" title="AFINSQ_1:def.1">AFINSQ_1:def 1</a></i>;<br/><a NAME="E33:9_1_2_1_8_1_2"/><i><font color="Green" title="E64">A98</font></i>: 
for <font color="Olive" title="b1">x</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">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c21">do1</font> holds <br/><font color="Maroon" title="c6">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">do1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 <b>by </b><i><a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>;<br/><a NAME="E34:9_1_2_1_8_1_2"/>
for <font color="Olive" title="b1">x</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">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c20">d'1</font> holds <br/><font color="Maroon" title="c6">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c21">do1</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">d'1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 <b>by </b><i><a class="txt" href="numeral1.html#E27:9_1_2_1_8_1_2"><i><font color="Green" title="E59">A93</font></i></a>, <a class="txt" href="numeral1.html#E31:9_1_2_1_8_1_2"><i><font color="Green" title="E62">A96</font></i></a></i>;<br/><a NAME="E35:9_1_2_1_8_1_2"/><b>then </b>
<font color="Maroon" title="c6">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">do1</font> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <font color="Maroon" title="c20">d'1</font>
 <b>by </b><i><a class="txt" href="numeral1.html#E32:9_1_2_1_8_1_2"><i><font color="Green" title="E63">A97</font></i></a>, <a class="txt" href="numeral1.html#E33:9_1_2_1_8_1_2"><i><font color="Green" title="E64">A98</font></i></a>, <a class="ref" href="afinsq_1.html#D4" title="AFINSQ_1:def.4">AFINSQ_1:def 4</a></i>;<br/><a NAME="E36:9_1_2_1_8_1_2"/><b>then </b><i><font color="Green" title="E65">A99</font></i>: 
<font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <font color="Maroon" title="c21">do1</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <font color="Maroon" title="c20">d'1</font></span>)</span>
 <b>by </b><i><a class="txt" href="numeral1.html#E6:9_1_2_1"><i><font color="Green" title="E13">A38</font></i></a>, <a class="ref" href="numeral1.html#T1" target="_self" title="NUMERAL1:th.1">Th1</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c22">bo1</font> = <font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c18">o1</font> as    <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>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><a NAME="E38:9_1_2_1_8_1_2"/><i><font color="Green" title="E66">A100</font></i>: 
<font color="Maroon" title="c22">bo1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="numeral1.html#E1:9"><i><font color="Green" title="E9">A1</font></i></a>, <a class="ref" href="prepower.html#T12" title="PREPOWER:th.12">PREPOWER:12</a></i>;<br/><b>consider </b><font color="Maroon" title="c23">ok1</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E40:9_1_2_1_8_1_2"/><i><font color="Green" title="E67">A101</font></i>: 
<font color="Maroon" title="c23">ok1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">o1</font>
 ;<br/><b>reconsider </b><font color="Maroon" title="c24">ok1</font> = <font color="Maroon" title="c23">ok1</font> as    <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>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c25">g1</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">ok1</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> as   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="numeral1.html#T3" target="_self" title="NUMERAL1:th.3">Th3</a></i>;<br/><a NAME="E43:9_1_2_1_8_1_2"/>
 <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="c2">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><a NAME="E44:9_1_2_1_8_1_2"/><b>then </b>
<font color="Maroon" title="c18">o1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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="c2">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="ordinal1.html#D2" title="ORDINAL1:def.2">ORDINAL1:def 2</a></i>;<br/><a NAME="E45:9_1_2_1_8_1_2"/><b>then </b><i><font color="Green" title="E68">A102</font></i>: 
 <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c25">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">o1</font>
 <b>by </b><i><a class="txt" href="numeral1.html#E40:9_1_2_1_8_1_2"><i><font color="Green" title="E67">A101</font></i></a>, <a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a></i>;<br/><a NAME="E46:9_1_2_1_8_1_2"/><i><font color="Green" title="E69">A103</font></i>: 
1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="numeral1.html#E1:9"><i><font color="Green" title="E9">A1</font></i></a></i>;<br/><i><font color="Green" title="E70">A104</font></i>:  <a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <font color="Maroon" title="c25">g1</font> = 
<span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">ok1</font>
<b>by </b><i><a class="ref" href="numeral1.html#T2" target="_self" title="NUMERAL1:th.2">Th2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">ok1</font>
<b>by </b><i><a class="ref" href="series_1.html#T12" title="SERIES_1:th.12">SERIES_1:12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">ok1</font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T13" title="SEQ_1:th.13">SEQ_1:13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p4">(<span class="default"><font color="Maroon" title="c24">ok1</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="numeral1.html#E1:9"><i><font color="Green" title="E9">A1</font></i></a>, <a class="ref" href="series_1.html#T26" title="SERIES_1:th.26">SERIES_1:26</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p4">(<span class="default"><font color="Maroon" title="c24">ok1</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="numeral1.html#E1:9"><i><font color="Green" title="E9">A1</font></i></a>, <a class="ref" href="power.html#T46" title="POWER:th.46">POWER:46</a></i>
<br/>.= 
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c18">o1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="numeral1.html#E40:9_1_2_1_8_1_2"><i><font color="Green" title="E67">A101</font></i></a></i>
<br/>.= 
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c18">o1</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="numeral1.html#E46:9_1_2_1_8_1_2"><i><font color="Green" title="E69">A103</font></i></a>, <a class="ref" href="xcmplx_1.html#T88" title="XCMPLX_1:th.88">XCMPLX_1:88</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c18">o1</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1

;<br/><a NAME="E48:9_1_2_1_8_1_2"/><i><font color="Green" title="E71">A105</font></i>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c21">do1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="numeral1.html#E31:9_1_2_1_8_1_2"><i><font color="Green" title="E62">A96</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E49:9_1_2_1_8_1_2"/><i><font color="Green" title="E72">A106</font></i>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c21">do1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c25">g1</font>
 <b>by </b><i><a class="txt" href="numeral1.html#E31:9_1_2_1_8_1_2"><i><font color="Green" title="E62">A96</font></i></a>, <a class="txt" href="numeral1.html#E45:9_1_2_1_8_1_2"><i><font color="Green" title="E68">A102</font></i></a>, <a class="ref" href="afinsq_1.html#D1" title="AFINSQ_1:def.1">AFINSQ_1:def 1</a></i>;<br/><a NAME="E50:9_1_2_1_8_1_2"/>
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c21">do1</font> holds <br/><font color="Maroon" title="c21">do1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c25">g1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">i</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_2_1_8_1_2_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c26">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>reconsider </b><font color="Maroon" title="c27">I</font> = <font color="Maroon" title="c26">i</font> as    <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>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<b>assume </b><a NAME="E2:9_1_2_1_8_1_2_4"/><i><font color="Green" title="E73">A107</font></i>: 
<font color="Maroon" title="c26">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c21">do1</font>
 ;<br/>

<a NAME="E3:9_1_2_1_8_1_2_4"/><b>then </b>
<font color="Maroon" title="c26">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">o1</font>
 
<b>by </b><i><a class="txt" href="numeral1.html#E23:9_1_2_1_8_1_2"><i><font color="Green" title="E57">A91</font></i></a>, <a class="txt" href="numeral1.html#E30:9_1_2_1_8_1_2"><i><font color="Green" title="E61">A95</font></i></a>, <a class="ref" href="afinsq_1.html#T14" title="AFINSQ_1:th.14">AFINSQ_1:14</a></i>;<br/>
<b>then </b><i><font color="Green" title="E74">A108</font></i>: <font color="Maroon" title="c25">g1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c27">I</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c27">I</font>
<b>by </b><i><a class="txt" href="numeral1.html#E45:9_1_2_1_8_1_2"><i><font color="Green" title="E68">A102</font></i></a>, <a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c27">I</font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T13" title="SEQ_1:th.13">SEQ_1:13</a></i>
<br/>.= 
<font color="Maroon" title="c3">b1</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c27">I</font></span>)</span>
<b>by </b><i><a class="ref" href="prepower.html#D1" title="PREPOWER:def.1">PREPOWER:def 1</a></i>
;<br/>
<a NAME="E5:9_1_2_1_8_1_2_4"/><i><font color="Green" title="E75">A109</font></i>: 
 <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c21">do1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c6">D</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T89" title="RELAT_1:th.89">RELAT_1:89</a></i>;<br/>
<a NAME="E6:9_1_2_1_8_1_2_4"/><b>then </b>
<font color="Maroon" title="c4">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c26">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">b1</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 
<b>by </b><i><a class="txt" href="numeral1.html#E3:9_1_2_1"><i><font color="Green" title="E11">A36</font></i></a>, <a class="txt" href="numeral1.html#E6:9_1_2_1"><i><font color="Green" title="E13">A38</font></i></a>, <a class="txt" href="numeral1.html#E2:9_1_2_1_8_1_2_4"><i><font color="Green" title="E73">A107</font></i></a></i>;<br/>
<a NAME="E7:9_1_2_1_8_1_2_4"/><b>then </b><i><font color="Green" title="E76">A110</font></i>: 
<font color="Maroon" title="c4">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c26">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">b1</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<font color="Maroon" title="c21">do1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c26">i</font> = 
<font color="Maroon" title="c6">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c26">i</font>
<b>by </b><i><a class="txt" href="numeral1.html#E2:9_1_2_1_8_1_2_4"><i><font color="Green" title="E73">A107</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c26">i</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c26">i</font></span>)</span>
<b>by </b><i><a class="txt" href="numeral1.html#E6:9_1_2_1"><i><font color="Green" title="E13">A38</font></i></a>, <a class="txt" href="numeral1.html#E2:9_1_2_1_8_1_2_4"><i><font color="Green" title="E73">A107</font></i></a>, <a class="txt" href="numeral1.html#E5:9_1_2_1_8_1_2_4"><i><font color="Green" title="E75">A109</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E9:9_1_2_1_8_1_2_4"/>
<font color="Maroon" title="c21">do1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c26">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c25">g1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c26">i</font>
 <b>by </b><i><a class="txt" href="numeral1.html#E4:9_1_2_1_8_1_2_4"><i><font color="Green" title="E74">A108</font></i></a>, <a class="txt" href="numeral1.html#E7:9_1_2_1_8_1_2_4"><i><font color="Green" title="E76">A110</font></i></a>, <a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a></i>;<br/>


</div><b>end;</b></div><a NAME="E51:9_1_2_1_8_1_2"/><b>then </b>
 <a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <font color="Maroon" title="c21">do1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c18">o1</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <b>by </b><i><a class="txt" href="numeral1.html#E47:9_1_2_1_8_1_2"><i><font color="Green" title="E70">A104</font></i></a>, <a class="txt" href="numeral1.html#E48:9_1_2_1_8_1_2"><i><font color="Green" title="E71">A105</font></i></a>, <a class="txt" href="numeral1.html#E49:9_1_2_1_8_1_2"><i><font color="Green" title="E72">A106</font></i></a>, <a class="ref" href="numeral1.html#T4" target="_self" title="NUMERAL1:th.4">Th4</a></i>;<br/><a NAME="E52:9_1_2_1_8_1_2"/><b>then </b>
 <a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <font color="Maroon" title="c21">do1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c18">o1</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T147" title="XREAL_1:th.147">XREAL_1:147</a></i>;<br/><a NAME="E53:9_1_2_1_8_1_2"/><b>then </b><i><font color="Green" title="E73">A111</font></i>: 
<span class="p1">(<span class="default"><a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <font color="Maroon" title="c21">do1</font></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="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c18">o1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="nat_d.html#T27" title="NAT_D:th.27">NAT_D:27</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_2_1_8_1_2_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c26">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>reconsider </b><font color="Maroon" title="c27">K</font> = <font color="Maroon" title="c26">k</font> as    <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>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><b>assume </b><a NAME="E2:9_1_2_1_8_1_2_5"/><i><font color="Green" title="E74">A112</font></i>: 
<font color="Maroon" title="c26">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c20">d'1</font>
 ;<br/><a NAME="E3:9_1_2_1_8_1_2_5"/><b>then </b>
<font color="Maroon" title="c26">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c20">d'1</font>
 <b>by </b><i><a class="ref" href="afinsq_1.html#D1" title="AFINSQ_1:def.1">AFINSQ_1:def 1</a></i>;<br/><a NAME="E4:9_1_2_1_8_1_2_5"/><b>then </b>
<font color="Maroon" title="c26">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c20">d'1</font>
 <b>by </b><i><a class="ref" href="prgcor_2.html#T1" title="PRGCOR_2:th.1">PRGCOR_2:1</a></i>;<br/><a NAME="E5:9_1_2_1_8_1_2_5"/><b>then </b>
<font color="Maroon" title="c18">o1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c26">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c6">D</font>
 <b>by </b><i><a class="txt" href="numeral1.html#E31:9_1_2_1_8_1_2"><i><font color="Green" title="E62">A96</font></i></a>, <a class="txt" href="numeral1.html#E32:9_1_2_1_8_1_2"><i><font color="Green" title="E63">A97</font></i></a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/><a NAME="E6:9_1_2_1_8_1_2_5"/><b>then </b>
<font color="Maroon" title="c18">o1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c26">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c6">D</font>
 <b>by </b><i><a class="ref" href="prgcor_2.html#T1" title="PRGCOR_2:th.1">PRGCOR_2:1</a></i>;<br/><a NAME="E7:9_1_2_1_8_1_2_5"/><b>then </b>
<font color="Maroon" title="c6">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">o1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c26">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">o1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c26">k</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">o1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c26">k</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="numeral1.html#E6:9_1_2_1"><i><font color="Green" title="E13">A38</font></i></a></i>;<br/><b>then </b><font color="Maroon" title="c20">d'1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c27">K</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">o1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c26">k</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">o1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c26">k</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="numeral1.html#E27:9_1_2_1_8_1_2"><i><font color="Green" title="E59">A93</font></i></a>, <a class="txt" href="numeral1.html#E2:9_1_2_1_8_1_2_5"><i><font color="Green" title="E74">A112</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">o1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c26">k</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c18">o1</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c26">k</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T13" title="NEWTON:th.13">NEWTON:13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c18">o1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c26">k</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c26">k</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c18">o1</font></span>)</span>

;<br/><b>hence </b><a NAME="E9:9_1_2_1_8_1_2_5"/>
<font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c18">o1</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c20">d'1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c26">k</font>
 <b>by </b><i><a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">NAT_D:def 3</a></i>;<br/></div><b>end;</b></div><a NAME="E55:9_1_2_1_8_1_2"/><b>then </b><i><font color="Green" title="E74">A113</font></i>: 
<font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c18">o1</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a>  <a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <font color="Maroon" title="c20">d'1</font>
 <b>by </b><i><a class="ref" href="numeral1.html#T5" target="_self" title="NUMERAL1:th.5">Th5</a></i>;<br/><a NAME="E56:9_1_2_1_8_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <font color="Maroon" title="c20">d'1</font></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="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c18">o1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="numeral1.html#E38:9_1_2_1_8_1_2"><i><font color="Green" title="E66">A100</font></i></a>, <a class="ref" href="pepin.html#T6" title="PEPIN:th.6">PEPIN:6</a></i>;<br/><a NAME="E57:9_1_2_1_8_1_2"/><b>then </b>
<font color="Maroon" title="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c18">o1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <font color="Maroon" title="c20">d'1</font></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="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c18">o1</font></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"><a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <font color="Maroon" title="c21">do1</font></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="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c18">o1</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="numeral1.html#E36:9_1_2_1_8_1_2"><i><font color="Green" title="E65">A99</font></i></a>, <a class="ref" href="nat_d.html#T19" title="NAT_D:th.19">NAT_D:19</a></i>;<br/><a NAME="E58:9_1_2_1_8_1_2"/><b>then </b><i><font color="Green" title="E75">A114</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c18">o1</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c18">o1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <font color="Maroon" title="c20">d'1</font>
 <b>by </b><i><a class="txt" href="numeral1.html#E53:9_1_2_1_8_1_2"><i><font color="Green" title="E73">A111</font></i></a>, <a class="txt" href="numeral1.html#E55:9_1_2_1_8_1_2"><i><font color="Green" title="E74">A113</font></i></a>, <a class="ref" href="nat_d.html#T3" title="NAT_D:th.3">NAT_D:3</a></i>;<br/><i><font color="Green" title="E76">A115</font></i>:  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c6">D</font> = 
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c20">d'1</font></span>)</span>
<b>by </b><i><a class="txt" href="numeral1.html#E6:9_1_2_1"><i><font color="Green" title="E13">A38</font></i></a>, <a class="txt" href="numeral1.html#E1:9_1_2_1_8_1_2"><i><font color="Green" title="E46">A80</font></i></a>, <a class="txt" href="numeral1.html#E27:9_1_2_1_8_1_2"><i><font color="Green" title="E59">A93</font></i></a>, <a class="ref" href="afinsq_1.html#D1" title="AFINSQ_1:def.1">AFINSQ_1:def 1</a></i>
<br/>.= 
1 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c20">d'1</font></span>)</span>
<b>by </b><i><a class="ref" href="afinsq_1.html#D1" title="AFINSQ_1:def.1">AFINSQ_1:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p2"><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">&lt;%</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">%&gt;</a></span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c20">d'1</font></span>)</span>
<b>by </b><i><a class="ref" href="afinsq_1.html#T38" title="AFINSQ_1:th.38">AFINSQ_1:38</a></i>
;<br/><div><i><font color="Green" title="E77">A116</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_2_1_8_1_2_7"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c26">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="E1:9_1_2_1_8_1_2_7"/>
<font color="Maroon" title="c26">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <span class="p1"><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">%&gt;</a></span>
 ;<br/><a NAME="E2:9_1_2_1_8_1_2_7"/><b>then </b>
<font color="Maroon" title="c26">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 1
 <b>by </b><i><a class="ref" href="afinsq_1.html#T36" title="AFINSQ_1:th.36">AFINSQ_1:36</a></i>;<br/><a NAME="E3:9_1_2_1_8_1_2_7"/><b>then </b>
<font color="Maroon" title="c26">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1
 <b>by </b><i><a class="ref" href="prgcor_2.html#T1" title="PRGCOR_2:th.1">PRGCOR_2:1</a></i>;<br/><a NAME="E4:9_1_2_1_8_1_2_7"/><b>then </b>
<font color="Maroon" title="c26">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></i>;<br/><b>hence </b><a NAME="E5:9_1_2_1_8_1_2_7"/>
<font color="Maroon" title="c6">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c26">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">%&gt;</a></span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c26">k</font>
 <b>by </b><i><a class="txt" href="numeral1.html#E1:9_1_2_1_8_1_2"><i><font color="Green" title="E46">A80</font></i></a>, <a class="ref" href="afinsq_1.html#D5" title="AFINSQ_1:def.5">AFINSQ_1:def 5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_2_1_8_1_2_8"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c26">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="E1:9_1_2_1_8_1_2_8"/><i><font color="Green" title="E78">A117</font></i>: 
<font color="Maroon" title="c26">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c20">d'1</font>
 ;<br/><b>thus </b><font color="Maroon" title="c6">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p3"><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">&lt;%</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">%&gt;</a></span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c26">k</font></span>)</span> = 
<font color="Maroon" title="c6">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c21">do1</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c26">k</font></span>)</span>
<b>by </b><i><a class="txt" href="numeral1.html#E1:9_1_2_1_8_1_2"><i><font color="Green" title="E46">A80</font></i></a>, <a class="txt" href="numeral1.html#E31:9_1_2_1_8_1_2"><i><font color="Green" title="E62">A96</font></i></a>, <a class="ref" href="afinsq_1.html#T37" title="AFINSQ_1:th.37">AFINSQ_1:37</a></i>
<br/>.= 
<font color="Maroon" title="c20">d'1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c26">k</font>
<b>by </b><i><a class="txt" href="numeral1.html#E27:9_1_2_1_8_1_2"><i><font color="Green" title="E59">A93</font></i></a>, <a class="txt" href="numeral1.html#E31:9_1_2_1_8_1_2"><i><font color="Green" title="E62">A96</font></i></a>, <a class="txt" href="numeral1.html#E1:9_1_2_1_8_1_2_8"><i><font color="Green" title="E78">A117</font></i></a></i>
;<br/></div><b>end;</b></div><a NAME="E62:9_1_2_1_8_1_2"/><b>then </b>
<font color="Maroon" title="c6">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">%&gt;</a></span> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <font color="Maroon" title="c20">d'1</font>
 <b>by </b><i><a class="txt" href="numeral1.html#E59:9_1_2_1_8_1_2"><i><font color="Green" title="E76">A115</font></i></a>, <a class="txt" href="numeral1.html#E60:9_1_2_1_8_1_2"><i><font color="Green" title="E77">A116</font></i></a>, <a class="ref" href="afinsq_1.html#D4" title="AFINSQ_1:def.4">AFINSQ_1:def 4</a></i>;<br/><a NAME="E63:9_1_2_1_8_1_2"/><b>then </b>
 <a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <font color="Maroon" title="c6">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <span class="p2"><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">&lt;%</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">%&gt;</a></span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <font color="Maroon" title="c20">d'1</font></span>)</span>
 <b>by </b><i><a class="ref" href="numeral1.html#T1" target="_self" title="NUMERAL1:th.1">Th1</a></i>;<br/><a NAME="E64:9_1_2_1_8_1_2"/><b>then </b>
<font color="Maroon" title="c6">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</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="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c18">o1</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c18">o1</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="numeral1.html#E20:9_1_2_1_8_1_2"><i><font color="Green" title="E56">A90</font></i></a>, <a class="txt" href="numeral1.html#E58:9_1_2_1_8_1_2"><i><font color="Green" title="E75">A114</font></i></a>, <a class="ref" href="stirl2_1.html#T44" title="STIRL2_1:th.44">STIRL2_1:44</a></i>;<br/><a NAME="E65:9_1_2_1_8_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <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="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c18">o1</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c18">o1</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="numeral1.html#E6:9_1_2_1"><i><font color="Green" title="E13">A38</font></i></a>, <a class="txt" href="numeral1.html#E1:9_1_2_1_8"><i><font color="Green" title="E41">A72</font></i></a></i>;<br/><a NAME="E66:9_1_2_1_8_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <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="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c18">o1</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> 1</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="newton.html#T13" title="NEWTON:th.13">NEWTON:13</a></i>;<br/><a NAME="E67:9_1_2_1_8_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c18">o1</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> 1</span>)</span></span>)</span></span>)</span>
 ;<br/><a NAME="E68:9_1_2_1_8_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <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="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c18">o1</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span>
 <b>by </b><i><a class="ref" href="newton.html#T10" title="NEWTON:th.10">NEWTON:10</a></i>;<br/><a NAME="E69:9_1_2_1_8_1_2"/><b>then </b><i><font color="Green" title="E78">A118</font></i>: 
<font color="Maroon" title="c4">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</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="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c18">o1</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span>
 <b>by </b><i><a class="txt" href="numeral1.html#E13:9_1_2_1_8_1_2"><i><font color="Green" title="E53">A87</font></i></a>, <a class="ref" href="xcmplx_1.html#T90" title="XCMPLX_1:th.90">XCMPLX_1:90</a></i>;<br/><b>set </b><font color="Maroon" title="c26">d'</font> = <font color="Maroon" title="c8">E</font>;<br/><a NAME="E70:9_1_2_1_8_1_2"/><i><font color="Green" title="E79">A119</font></i>: 
 <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c8">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">oo</font>
 <b>by </b><i><a class="txt" href="numeral1.html#E25:9_1_2_1"><i><font color="Green" title="E20">A45</font></i></a>, <a class="txt" href="numeral1.html#E81:9_1_2_1"><i><font color="Green" title="E39">A70</font></i></a>, <a class="txt" href="numeral1.html#E1:9_1_2_1_8_1_2"><i><font color="Green" title="E46">A80</font></i></a>, <a class="ref" href="afinsq_1.html#D1" title="AFINSQ_1:def.1">AFINSQ_1:def 1</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c27">do</font> =  <a href="afinsq_1.html#K4" title="AFINSQ_1:func.4">&lt;%&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  as   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/><a NAME="E72:9_1_2_1_8_1_2"/><i><font color="Green" title="E80">A120</font></i>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c8">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">oo</font>
 <b>by </b><i><a class="txt" href="numeral1.html#E70:9_1_2_1_8_1_2"><i><font color="Green" title="E79">A119</font></i></a>, <a class="ref" href="afinsq_1.html#D1" title="AFINSQ_1:def.1">AFINSQ_1:def 1</a></i>;<br/><a NAME="E73:9_1_2_1_8_1_2"/><i><font color="Green" title="E81">A121</font></i>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c27">do</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">o</font>
 <b>by </b><i><a class="txt" href="numeral1.html#E1:9_1_2_1_8_1_2"><i><font color="Green" title="E46">A80</font></i></a>, <a class="ref" href="afinsq_1.html#T18" title="AFINSQ_1:th.18">AFINSQ_1:18</a></i>;<br/><a NAME="E74:9_1_2_1_8_1_2"/><b>then </b><i><font color="Green" title="E82">A122</font></i>: 
 <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c8">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c27">do</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c8">E</font></span>)</span>
 <b>by </b><i><a class="txt" href="numeral1.html#E25:9_1_2_1"><i><font color="Green" title="E20">A45</font></i></a>, <a class="txt" href="numeral1.html#E81:9_1_2_1"><i><font color="Green" title="E39">A70</font></i></a>, <a class="txt" href="numeral1.html#E72:9_1_2_1_8_1_2"><i><font color="Green" title="E80">A120</font></i></a>, <a class="ref" href="afinsq_1.html#D1" title="AFINSQ_1:def.1">AFINSQ_1:def 1</a></i>;<br/><a NAME="E75:9_1_2_1_8_1_2"/>
 <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><a NAME="E76:9_1_2_1_8_1_2"/><b>then </b><i><font color="Green" title="E83">A123</font></i>: 
for <font color="Olive" title="b1">x</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">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c27">do</font> holds <br/><font color="Maroon" title="c8">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">do</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 ;<br/><a NAME="E77:9_1_2_1_8_1_2"/>
for <font color="Olive" title="b1">x</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">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c8">E</font> holds <br/><font color="Maroon" title="c8">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c27">do</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 <b>by </b><i><a class="txt" href="numeral1.html#E1:9_1_2_1_8_1_2"><i><font color="Green" title="E46">A80</font></i></a>, <a class="txt" href="numeral1.html#E73:9_1_2_1_8_1_2"><i><font color="Green" title="E81">A121</font></i></a></i>;<br/><a NAME="E78:9_1_2_1_8_1_2"/><b>then </b>
<font color="Maroon" title="c8">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">do</font> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <font color="Maroon" title="c8">E</font>
 <b>by </b><i><a class="txt" href="numeral1.html#E74:9_1_2_1_8_1_2"><i><font color="Green" title="E82">A122</font></i></a>, <a class="txt" href="numeral1.html#E76:9_1_2_1_8_1_2"><i><font color="Green" title="E83">A123</font></i></a>, <a class="ref" href="afinsq_1.html#D4" title="AFINSQ_1:def.4">AFINSQ_1:def 4</a></i>;<br/><a NAME="E79:9_1_2_1_8_1_2"/><b>then </b><i><font color="Green" title="E84">A124</font></i>: 
<font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <font color="Maroon" title="c27">do</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <font color="Maroon" title="c8">E</font></span>)</span>
 <b>by </b><i><a class="txt" href="numeral1.html#E25:9_1_2_1"><i><font color="Green" title="E20">A45</font></i></a>, <a class="ref" href="numeral1.html#T1" target="_self" title="NUMERAL1:th.1">Th1</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c28">bo</font> = <font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font> as    <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>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><a NAME="E81:9_1_2_1_8_1_2"/><i><font color="Green" title="E85">A125</font></i>: 
<font color="Maroon" title="c28">bo</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="numeral1.html#E1:9"><i><font color="Green" title="E9">A1</font></i></a>, <a class="ref" href="prepower.html#T12" title="PREPOWER:th.12">PREPOWER:12</a></i>;<br/><a NAME="E82:9_1_2_1_8_1_2"/>
 <a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <font color="Maroon" title="c27">do</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="numeral1.html#E1:9_1_2_1_8_1_2"><i><font color="Green" title="E46">A80</font></i></a>, <a class="txt" href="numeral1.html#E73:9_1_2_1_8_1_2"><i><font color="Green" title="E81">A121</font></i></a>, <a class="ref" href="stirl2_1.html#T53" title="STIRL2_1:th.53">STIRL2_1:53</a></i>;<br/><a NAME="E83:9_1_2_1_8_1_2"/><b>then </b><i><font color="Green" title="E86">A126</font></i>: 
<span class="p1">(<span class="default"><a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <font color="Maroon" title="c27">do</font></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="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="nat_2.html#T4" title="NAT_2:th.4">NAT_2:4</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_2_1_8_1_2_9"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c29">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>reconsider </b><font color="Maroon" title="c30">K</font> = <font color="Maroon" title="c29">k</font> as    <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>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><b>assume </b><a NAME="E2:9_1_2_1_8_1_2_9"/>
<font color="Maroon" title="c29">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c8">E</font>
 ;<br/><a NAME="E3:9_1_2_1_8_1_2_9"/><b>then </b>
<font color="Maroon" title="c29">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c8">E</font>
 <b>by </b><i><a class="ref" href="afinsq_1.html#D1" title="AFINSQ_1:def.1">AFINSQ_1:def 1</a></i>;<br/><a NAME="E4:9_1_2_1_8_1_2_9"/><b>then </b>
<font color="Maroon" title="c29">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c8">E</font>
 <b>by </b><i><a class="ref" href="prgcor_2.html#T1" title="PRGCOR_2:th.1">PRGCOR_2:1</a></i>;<br/><a NAME="E5:9_1_2_1_8_1_2_9"/><b>then </b>
<font color="Maroon" title="c13">o</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c29">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c8">E</font>
 <b>by </b><i><a class="txt" href="numeral1.html#E1:9_1_2_1_8_1_2"><i><font color="Green" title="E46">A80</font></i></a>, <a class="ref" href="afinsq_1.html#D1" title="AFINSQ_1:def.1">AFINSQ_1:def 1</a></i>;<br/><a NAME="E6:9_1_2_1_8_1_2_9"/><b>then </b>
<font color="Maroon" title="c13">o</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c29">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c8">E</font>
 <b>by </b><i><a class="ref" href="prgcor_2.html#T1" title="PRGCOR_2:th.1">PRGCOR_2:1</a></i>;<br/><b>then </b><font color="Maroon" title="c8">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c30">K</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">o</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c29">k</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">o</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c29">k</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="numeral1.html#E25:9_1_2_1"><i><font color="Green" title="E20">A45</font></i></a>, <a class="txt" href="numeral1.html#E1:9_1_2_1_8_1_2"><i><font color="Green" title="E46">A80</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">o</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c29">k</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c29">k</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T13" title="NEWTON:th.13">NEWTON:13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">o</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c29">k</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c29">k</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span>

;<br/><b>hence </b><a NAME="E8:9_1_2_1_8_1_2_9"/>
<font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c8">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c29">k</font>
 <b>by </b><i><a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">NAT_D:def 3</a></i>;<br/></div><b>end;</b></div><a NAME="E85:9_1_2_1_8_1_2"/><b>then </b><i><font color="Green" title="E87">A127</font></i>: 
<font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a>  <a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <font color="Maroon" title="c8">E</font>
 <b>by </b><i><a class="ref" href="numeral1.html#T5" target="_self" title="NUMERAL1:th.5">Th5</a></i>;<br/><a NAME="E86:9_1_2_1_8_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <font color="Maroon" title="c8">E</font></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="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="numeral1.html#E81:9_1_2_1_8_1_2"><i><font color="Green" title="E85">A125</font></i></a>, <a class="ref" href="pepin.html#T6" title="PEPIN:th.6">PEPIN:6</a></i>;<br/><a NAME="E87:9_1_2_1_8_1_2"/><b>then </b>
<font color="Maroon" title="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <font color="Maroon" title="c8">E</font></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="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></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"><a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <font color="Maroon" title="c27">do</font></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="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="numeral1.html#E79:9_1_2_1_8_1_2"><i><font color="Green" title="E84">A124</font></i></a>, <a class="ref" href="nat_d.html#T19" title="NAT_D:th.19">NAT_D:19</a></i>;<br/><a NAME="E88:9_1_2_1_8_1_2"/><b>then </b><i><font color="Green" title="E88">A128</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <font color="Maroon" title="c8">E</font>
 <b>by </b><i><a class="txt" href="numeral1.html#E83:9_1_2_1_8_1_2"><i><font color="Green" title="E86">A126</font></i></a>, <a class="txt" href="numeral1.html#E85:9_1_2_1_8_1_2"><i><font color="Green" title="E87">A127</font></i></a>, <a class="ref" href="nat_d.html#T3" title="NAT_D:th.3">NAT_D:3</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c29">o1</font> = <font color="Maroon" title="c13">o</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 as    <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/><a NAME="E90:9_1_2_1_8_1_2"/>
( <font color="Maroon" title="c29">o1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">k</font> &amp; <font color="Maroon" title="c7">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c4">d</font> )
 <b>by </b><i><a class="txt" href="numeral1.html#E1:9_1_2_1_8_1_2"><i><font color="Green" title="E46">A80</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a>, <a class="ref" href="xreal_1.html#T149" title="XREAL_1:th.149">XREAL_1:149</a></i>;<br/><a NAME="E91:9_1_2_1_8_1_2"/><b>then </b><i><font color="Green" title="E89">A129</font></i>: 
<font color="Maroon" title="c29">o1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c4">d</font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c30">oo1</font> = <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c4">d</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c29">o1</font> as    <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>by </b><i><a class="txt" href="numeral1.html#E1:9_1_2_1_8_1_2"><i><font color="Green" title="E46">A80</font></i></a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>2</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> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>$2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default">$1 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c29">o1</font></span>)</span>;<br/><a NAME="E93:9_1_2_1_8_1_2"/><i><font color="Green" title="E90">A130</font></i>: 
for <font color="Olive" title="b1">u</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">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c30">oo1</font> holds <br/> ex <font color="Olive" title="b2">x</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>2</sub></font>[<font color="Olive" title="b1">u</font>,<font color="Olive" title="b2">x</font>]
 ;<br/><b>consider </b><font color="Maroon" title="c31">d'1</font> being   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E95:9_1_2_1_8_1_2"/><i><font color="Green" title="E91">A131</font></i>: 
(  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c31">d'1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">oo1</font> &amp; ( for <font color="Olive" title="b1">x</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">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c30">oo1</font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c31">d'1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">x</font>] ) )
 <b>from </b><i><a class="ref" href="stirl2_1.html#S6" title="STIRL2_1:sch.6">STIRL2_1:sch 6</a>(<a class="txt" href="numeral1.html#E93:9_1_2_1_8_1_2"><i><font color="Green" title="E90">A130</font></i></a>);<br/></i><b>reconsider </b><font color="Maroon" title="c32">do1</font> = <font color="Maroon" title="c8">E</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c29">o1</font> as   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="afinsq_1.html#T16" title="AFINSQ_1:th.16">AFINSQ_1:16</a></i>;<br/><a NAME="E97:9_1_2_1_8_1_2"/><i><font color="Green" title="E92">A132</font></i>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c31">d'1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">oo1</font>
 <b>by </b><i><a class="txt" href="numeral1.html#E95:9_1_2_1_8_1_2"><i><font color="Green" title="E91">A131</font></i></a>, <a class="ref" href="afinsq_1.html#D1" title="AFINSQ_1:def.1">AFINSQ_1:def 1</a></i>;<br/><i><font color="Green" title="E93">A133</font></i>:  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c5">e</font> = 
 <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c8">E</font>
<b>by </b><i><a class="txt" href="numeral1.html#E25:9_1_2_1"><i><font color="Green" title="E20">A45</font></i></a>, <a class="ref" href="afinsq_1.html#D1" title="AFINSQ_1:def.1">AFINSQ_1:def 1</a></i>
<br/>.= 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c8">E</font>
<b>by </b><i><a class="ref" href="afinsq_1.html#D1" title="AFINSQ_1:def.1">AFINSQ_1:def 1</a></i>
;<br/><a NAME="E99:9_1_2_1_8_1_2"/><b>then </b><i><font color="Green" title="E94">A134</font></i>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c32">do1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">o1</font>
 <b>by </b><i><a class="txt" href="numeral1.html#E81:9_1_2_1"><i><font color="Green" title="E39">A70</font></i></a>, <a class="txt" href="numeral1.html#E91:9_1_2_1_8_1_2"><i><font color="Green" title="E89">A129</font></i></a>, <a class="ref" href="afinsq_1.html#T14" title="AFINSQ_1:th.14">AFINSQ_1:14</a></i>;<br/><a NAME="E100:9_1_2_1_8_1_2"/><b>then </b><i><font color="Green" title="E95">A135</font></i>: 
 <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c8">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c32">do1</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c31">d'1</font></span>)</span>
 <b>by </b><i><a class="txt" href="numeral1.html#E25:9_1_2_1"><i><font color="Green" title="E20">A45</font></i></a>, <a class="txt" href="numeral1.html#E81:9_1_2_1"><i><font color="Green" title="E39">A70</font></i></a>, <a class="txt" href="numeral1.html#E97:9_1_2_1_8_1_2"><i><font color="Green" title="E92">A132</font></i></a>, <a class="ref" href="afinsq_1.html#D1" title="AFINSQ_1:def.1">AFINSQ_1:def 1</a></i>;<br/><a NAME="E101:9_1_2_1_8_1_2"/><i><font color="Green" title="E96">A136</font></i>: 
for <font color="Olive" title="b1">x</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">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c32">do1</font> holds <br/><font color="Maroon" title="c8">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c32">do1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 <b>by </b><i><a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>;<br/><a NAME="E102:9_1_2_1_8_1_2"/>
for <font color="Olive" title="b1">x</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">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c31">d'1</font> holds <br/><font color="Maroon" title="c8">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c32">do1</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c31">d'1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 <b>by </b><i><a class="txt" href="numeral1.html#E95:9_1_2_1_8_1_2"><i><font color="Green" title="E91">A131</font></i></a>, <a class="txt" href="numeral1.html#E99:9_1_2_1_8_1_2"><i><font color="Green" title="E94">A134</font></i></a></i>;<br/><a NAME="E103:9_1_2_1_8_1_2"/><b>then </b>
<font color="Maroon" title="c8">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c32">do1</font> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <font color="Maroon" title="c31">d'1</font>
 <b>by </b><i><a class="txt" href="numeral1.html#E100:9_1_2_1_8_1_2"><i><font color="Green" title="E95">A135</font></i></a>, <a class="txt" href="numeral1.html#E101:9_1_2_1_8_1_2"><i><font color="Green" title="E96">A136</font></i></a>, <a class="ref" href="afinsq_1.html#D4" title="AFINSQ_1:def.4">AFINSQ_1:def 4</a></i>;<br/><a NAME="E104:9_1_2_1_8_1_2"/><b>then </b><i><font color="Green" title="E97">A137</font></i>: 
<font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <font color="Maroon" title="c32">do1</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <font color="Maroon" title="c31">d'1</font></span>)</span>
 <b>by </b><i><a class="txt" href="numeral1.html#E25:9_1_2_1"><i><font color="Green" title="E20">A45</font></i></a>, <a class="ref" href="numeral1.html#T1" target="_self" title="NUMERAL1:th.1">Th1</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c33">bo1</font> = <font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c29">o1</font> as    <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>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><a NAME="E106:9_1_2_1_8_1_2"/><i><font color="Green" title="E98">A138</font></i>: 
<font color="Maroon" title="c33">bo1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="numeral1.html#E1:9"><i><font color="Green" title="E9">A1</font></i></a>, <a class="ref" href="prepower.html#T12" title="PREPOWER:th.12">PREPOWER:12</a></i>;<br/><b>consider </b><font color="Maroon" title="c34">ok1</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E108:9_1_2_1_8_1_2"/><i><font color="Green" title="E99">A139</font></i>: 
<font color="Maroon" title="c34">ok1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">o1</font>
 ;<br/><b>reconsider </b><font color="Maroon" title="c35">ok1</font> = <font color="Maroon" title="c34">ok1</font> as    <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>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c36">g1</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c35">ok1</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> as   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="numeral1.html#T3" target="_self" title="NUMERAL1:th.3">Th3</a></i>;<br/><a NAME="E111:9_1_2_1_8_1_2"/>
 <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="c2">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><a NAME="E112:9_1_2_1_8_1_2"/><b>then </b>
<font color="Maroon" title="c29">o1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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="c2">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="ordinal1.html#D2" title="ORDINAL1:def.2">ORDINAL1:def 2</a></i>;<br/><a NAME="E113:9_1_2_1_8_1_2"/><b>then </b><i><font color="Green" title="E100">A140</font></i>: 
 <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c36">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">o1</font>
 <b>by </b><i><a class="txt" href="numeral1.html#E108:9_1_2_1_8_1_2"><i><font color="Green" title="E99">A139</font></i></a>, <a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a></i>;<br/><a NAME="E114:9_1_2_1_8_1_2"/><i><font color="Green" title="E101">A141</font></i>: 
1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="numeral1.html#E1:9"><i><font color="Green" title="E9">A1</font></i></a></i>;<br/><i><font color="Green" title="E102">A142</font></i>:  <a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <font color="Maroon" title="c36">g1</font> = 
<span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c35">ok1</font>
<b>by </b><i><a class="ref" href="numeral1.html#T2" target="_self" title="NUMERAL1:th.2">Th2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c35">ok1</font>
<b>by </b><i><a class="ref" href="series_1.html#T12" title="SERIES_1:th.12">SERIES_1:12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c35">ok1</font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T13" title="SEQ_1:th.13">SEQ_1:13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p4">(<span class="default"><font color="Maroon" title="c35">ok1</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="numeral1.html#E1:9"><i><font color="Green" title="E9">A1</font></i></a>, <a class="ref" href="series_1.html#T26" title="SERIES_1:th.26">SERIES_1:26</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p4">(<span class="default"><font color="Maroon" title="c35">ok1</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="numeral1.html#E1:9"><i><font color="Green" title="E9">A1</font></i></a>, <a class="ref" href="power.html#T46" title="POWER:th.46">POWER:46</a></i>
<br/>.= 
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c29">o1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="numeral1.html#E108:9_1_2_1_8_1_2"><i><font color="Green" title="E99">A139</font></i></a></i>
<br/>.= 
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c29">o1</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="numeral1.html#E114:9_1_2_1_8_1_2"><i><font color="Green" title="E101">A141</font></i></a>, <a class="ref" href="xcmplx_1.html#T88" title="XCMPLX_1:th.88">XCMPLX_1:88</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c29">o1</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1

;<br/><a NAME="E116:9_1_2_1_8_1_2"/><i><font color="Green" title="E103">A143</font></i>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c32">do1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="numeral1.html#E99:9_1_2_1_8_1_2"><i><font color="Green" title="E94">A134</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E117:9_1_2_1_8_1_2"/><i><font color="Green" title="E104">A144</font></i>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c32">do1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c36">g1</font>
 <b>by </b><i><a class="txt" href="numeral1.html#E99:9_1_2_1_8_1_2"><i><font color="Green" title="E94">A134</font></i></a>, <a class="txt" href="numeral1.html#E113:9_1_2_1_8_1_2"><i><font color="Green" title="E100">A140</font></i></a>, <a class="ref" href="afinsq_1.html#D1" title="AFINSQ_1:def.1">AFINSQ_1:def 1</a></i>;<br/><a NAME="E118:9_1_2_1_8_1_2"/>
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c32">do1</font> holds <br/><font color="Maroon" title="c32">do1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c36">g1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">i</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_2_1_8_1_2_12"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c37">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>reconsider </b><font color="Maroon" title="c38">I</font> = <font color="Maroon" title="c37">i</font> as    <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>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<b>assume </b><a NAME="E2:9_1_2_1_8_1_2_12"/><i><font color="Green" title="E105">A145</font></i>: 
<font color="Maroon" title="c37">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c32">do1</font>
 ;<br/>

<a NAME="E3:9_1_2_1_8_1_2_12"/><b>then </b>
<font color="Maroon" title="c37">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c29">o1</font>
 
<b>by </b><i><a class="txt" href="numeral1.html#E81:9_1_2_1"><i><font color="Green" title="E39">A70</font></i></a>, <a class="txt" href="numeral1.html#E91:9_1_2_1_8_1_2"><i><font color="Green" title="E89">A129</font></i></a>, <a class="txt" href="numeral1.html#E98:9_1_2_1_8_1_2"><i><font color="Green" title="E93">A133</font></i></a>, <a class="ref" href="afinsq_1.html#T14" title="AFINSQ_1:th.14">AFINSQ_1:14</a></i>;<br/>
<b>then </b><i><font color="Green" title="E106">A146</font></i>: <font color="Maroon" title="c36">g1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c38">I</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c38">I</font>
<b>by </b><i><a class="txt" href="numeral1.html#E113:9_1_2_1_8_1_2"><i><font color="Green" title="E100">A140</font></i></a>, <a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c38">I</font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T13" title="SEQ_1:th.13">SEQ_1:13</a></i>
<br/>.= 
<font color="Maroon" title="c3">b1</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c38">I</font></span>)</span>
<b>by </b><i><a class="ref" href="prepower.html#D1" title="PREPOWER:def.1">PREPOWER:def 1</a></i>
;<br/>
<a NAME="E5:9_1_2_1_8_1_2_12"/><i><font color="Green" title="E107">A147</font></i>: 
 <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c32">do1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c8">E</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T89" title="RELAT_1:th.89">RELAT_1:89</a></i>;<br/>
<a NAME="E6:9_1_2_1_8_1_2_12"/><b>then </b>
<font color="Maroon" title="c5">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c37">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">b1</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 
<b>by </b><i><a class="txt" href="numeral1.html#E4:9_1_2_1"><i><font color="Green" title="E12">A37</font></i></a>, <a class="txt" href="numeral1.html#E25:9_1_2_1"><i><font color="Green" title="E20">A45</font></i></a>, <a class="txt" href="numeral1.html#E2:9_1_2_1_8_1_2_12"><i><font color="Green" title="E105">A145</font></i></a></i>;<br/>
<a NAME="E7:9_1_2_1_8_1_2_12"/><b>then </b><i><font color="Green" title="E108">A148</font></i>: 
<font color="Maroon" title="c5">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c37">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">b1</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<font color="Maroon" title="c32">do1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c37">i</font> = 
<font color="Maroon" title="c8">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c37">i</font>
<b>by </b><i><a class="txt" href="numeral1.html#E2:9_1_2_1_8_1_2_12"><i><font color="Green" title="E105">A145</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c37">i</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c37">i</font></span>)</span>
<b>by </b><i><a class="txt" href="numeral1.html#E25:9_1_2_1"><i><font color="Green" title="E20">A45</font></i></a>, <a class="txt" href="numeral1.html#E2:9_1_2_1_8_1_2_12"><i><font color="Green" title="E105">A145</font></i></a>, <a class="txt" href="numeral1.html#E5:9_1_2_1_8_1_2_12"><i><font color="Green" title="E107">A147</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E9:9_1_2_1_8_1_2_12"/>
<font color="Maroon" title="c32">do1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c37">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c36">g1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c37">i</font>
 <b>by </b><i><a class="txt" href="numeral1.html#E4:9_1_2_1_8_1_2_12"><i><font color="Green" title="E106">A146</font></i></a>, <a class="txt" href="numeral1.html#E7:9_1_2_1_8_1_2_12"><i><font color="Green" title="E108">A148</font></i></a>, <a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a></i>;<br/>


</div><b>end;</b></div><a NAME="E119:9_1_2_1_8_1_2"/><b>then </b>
 <a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <font color="Maroon" title="c32">do1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c29">o1</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <b>by </b><i><a class="txt" href="numeral1.html#E115:9_1_2_1_8_1_2"><i><font color="Green" title="E102">A142</font></i></a>, <a class="txt" href="numeral1.html#E116:9_1_2_1_8_1_2"><i><font color="Green" title="E103">A143</font></i></a>, <a class="txt" href="numeral1.html#E117:9_1_2_1_8_1_2"><i><font color="Green" title="E104">A144</font></i></a>, <a class="ref" href="numeral1.html#T4" target="_self" title="NUMERAL1:th.4">Th4</a></i>;<br/><a NAME="E120:9_1_2_1_8_1_2"/><b>then </b>
 <a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <font color="Maroon" title="c32">do1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c29">o1</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T147" title="XREAL_1:th.147">XREAL_1:147</a></i>;<br/><a NAME="E121:9_1_2_1_8_1_2"/><b>then </b><i><font color="Green" title="E105">A149</font></i>: 
<span class="p1">(<span class="default"><a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <font color="Maroon" title="c32">do1</font></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="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c29">o1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="nat_d.html#T27" title="NAT_D:th.27">NAT_D:27</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_2_1_8_1_2_13"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c37">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>reconsider </b><font color="Maroon" title="c38">K</font> = <font color="Maroon" title="c37">k</font> as    <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>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><b>assume </b><a NAME="E2:9_1_2_1_8_1_2_13"/><i><font color="Green" title="E106">A150</font></i>: 
<font color="Maroon" title="c37">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c31">d'1</font>
 ;<br/><a NAME="E3:9_1_2_1_8_1_2_13"/><b>then </b>
<font color="Maroon" title="c37">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c31">d'1</font>
 <b>by </b><i><a class="ref" href="afinsq_1.html#D1" title="AFINSQ_1:def.1">AFINSQ_1:def 1</a></i>;<br/><a NAME="E4:9_1_2_1_8_1_2_13"/><b>then </b>
<font color="Maroon" title="c37">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c31">d'1</font>
 <b>by </b><i><a class="ref" href="prgcor_2.html#T1" title="PRGCOR_2:th.1">PRGCOR_2:1</a></i>;<br/><a NAME="E5:9_1_2_1_8_1_2_13"/><b>then </b>
<font color="Maroon" title="c29">o1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c37">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c8">E</font>
 <b>by </b><i><a class="txt" href="numeral1.html#E99:9_1_2_1_8_1_2"><i><font color="Green" title="E94">A134</font></i></a>, <a class="txt" href="numeral1.html#E100:9_1_2_1_8_1_2"><i><font color="Green" title="E95">A135</font></i></a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/><a NAME="E6:9_1_2_1_8_1_2_13"/><b>then </b>
<font color="Maroon" title="c29">o1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c37">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c8">E</font>
 <b>by </b><i><a class="ref" href="prgcor_2.html#T1" title="PRGCOR_2:th.1">PRGCOR_2:1</a></i>;<br/><a NAME="E7:9_1_2_1_8_1_2_13"/><b>then </b>
<font color="Maroon" title="c8">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c29">o1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c37">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c29">o1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c37">k</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c29">o1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c37">k</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="numeral1.html#E25:9_1_2_1"><i><font color="Green" title="E20">A45</font></i></a></i>;<br/><b>then </b><font color="Maroon" title="c31">d'1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c38">K</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c29">o1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c37">k</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c29">o1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c37">k</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="numeral1.html#E95:9_1_2_1_8_1_2"><i><font color="Green" title="E91">A131</font></i></a>, <a class="txt" href="numeral1.html#E2:9_1_2_1_8_1_2_13"><i><font color="Green" title="E106">A150</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c29">o1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c37">k</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c29">o1</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c37">k</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T13" title="NEWTON:th.13">NEWTON:13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c29">o1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c37">k</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c37">k</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c29">o1</font></span>)</span>

;<br/><b>hence </b><a NAME="E9:9_1_2_1_8_1_2_13"/>
<font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c29">o1</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c31">d'1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c37">k</font>
 <b>by </b><i><a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">NAT_D:def 3</a></i>;<br/></div><b>end;</b></div><a NAME="E123:9_1_2_1_8_1_2"/><b>then </b><i><font color="Green" title="E106">A151</font></i>: 
<font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c29">o1</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a>  <a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <font color="Maroon" title="c31">d'1</font>
 <b>by </b><i><a class="ref" href="numeral1.html#T5" target="_self" title="NUMERAL1:th.5">Th5</a></i>;<br/><a NAME="E124:9_1_2_1_8_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <font color="Maroon" title="c31">d'1</font></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="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c29">o1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="numeral1.html#E106:9_1_2_1_8_1_2"><i><font color="Green" title="E98">A138</font></i></a>, <a class="ref" href="pepin.html#T6" title="PEPIN:th.6">PEPIN:6</a></i>;<br/><a NAME="E125:9_1_2_1_8_1_2"/><b>then </b>
<font color="Maroon" title="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c29">o1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <font color="Maroon" title="c31">d'1</font></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="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c29">o1</font></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"><a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <font color="Maroon" title="c32">do1</font></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="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c29">o1</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="numeral1.html#E104:9_1_2_1_8_1_2"><i><font color="Green" title="E97">A137</font></i></a>, <a class="ref" href="nat_d.html#T19" title="NAT_D:th.19">NAT_D:19</a></i>;<br/><a NAME="E126:9_1_2_1_8_1_2"/><b>then </b><i><font color="Green" title="E107">A152</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c29">o1</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c29">o1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <font color="Maroon" title="c31">d'1</font>
 <b>by </b><i><a class="txt" href="numeral1.html#E121:9_1_2_1_8_1_2"><i><font color="Green" title="E105">A149</font></i></a>, <a class="txt" href="numeral1.html#E123:9_1_2_1_8_1_2"><i><font color="Green" title="E106">A151</font></i></a>, <a class="ref" href="nat_d.html#T3" title="NAT_D:th.3">NAT_D:3</a></i>;<br/><a NAME="E127:9_1_2_1_8_1_2"/>
<font color="Maroon" title="c13">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c5">e</font>
 <b>by </b><i><a class="txt" href="numeral1.html#E81:9_1_2_1"><i><font color="Green" title="E39">A70</font></i></a>, <a class="txt" href="numeral1.html#E1:9_1_2_1_8"><i><font color="Green" title="E41">A72</font></i></a>, <a class="ref" href="afinsq_1.html#D1" title="AFINSQ_1:def.1">AFINSQ_1:def 1</a></i>;<br/><a NAME="E128:9_1_2_1_8_1_2"/><b>then </b><i><font color="Green" title="E108">A153</font></i>: 
<font color="Maroon" title="c13">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c8">E</font>
 <b>by </b><i><a class="txt" href="numeral1.html#E25:9_1_2_1"><i><font color="Green" title="E20">A45</font></i></a>, <a class="ref" href="afinsq_1.html#D1" title="AFINSQ_1:def.1">AFINSQ_1:def 1</a></i>;<br/><i><font color="Green" title="E109">A154</font></i>:  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c8">E</font> = 
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c31">d'1</font></span>)</span>
<b>by </b><i><a class="txt" href="numeral1.html#E25:9_1_2_1"><i><font color="Green" title="E20">A45</font></i></a>, <a class="txt" href="numeral1.html#E81:9_1_2_1"><i><font color="Green" title="E39">A70</font></i></a>, <a class="txt" href="numeral1.html#E1:9_1_2_1_8_1_2"><i><font color="Green" title="E46">A80</font></i></a>, <a class="txt" href="numeral1.html#E95:9_1_2_1_8_1_2"><i><font color="Green" title="E91">A131</font></i></a>, <a class="ref" href="afinsq_1.html#D1" title="AFINSQ_1:def.1">AFINSQ_1:def 1</a></i>
<br/>.= 
1 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c31">d'1</font></span>)</span>
<b>by </b><i><a class="ref" href="afinsq_1.html#D1" title="AFINSQ_1:def.1">AFINSQ_1:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p2"><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">&lt;%</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">%&gt;</a></span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c31">d'1</font></span>)</span>
<b>by </b><i><a class="ref" href="afinsq_1.html#T38" title="AFINSQ_1:th.38">AFINSQ_1:38</a></i>
;<br/><div><i><font color="Green" title="E110">A155</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_2_1_8_1_2_15"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c37">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="E1:9_1_2_1_8_1_2_15"/>
<font color="Maroon" title="c37">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <span class="p1"><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">%&gt;</a></span>
 ;<br/><a NAME="E2:9_1_2_1_8_1_2_15"/><b>then </b>
<font color="Maroon" title="c37">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 1
 <b>by </b><i><a class="ref" href="afinsq_1.html#T36" title="AFINSQ_1:th.36">AFINSQ_1:36</a></i>;<br/><a NAME="E3:9_1_2_1_8_1_2_15"/><b>then </b>
<font color="Maroon" title="c37">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1
 <b>by </b><i><a class="ref" href="prgcor_2.html#T1" title="PRGCOR_2:th.1">PRGCOR_2:1</a></i>;<br/><a NAME="E4:9_1_2_1_8_1_2_15"/><b>then </b>
<font color="Maroon" title="c37">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></i>;<br/><b>hence </b><a NAME="E5:9_1_2_1_8_1_2_15"/>
<font color="Maroon" title="c8">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c37">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">%&gt;</a></span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c37">k</font>
 <b>by </b><i><a class="txt" href="numeral1.html#E1:9_1_2_1_8_1_2"><i><font color="Green" title="E46">A80</font></i></a>, <a class="ref" href="afinsq_1.html#D5" title="AFINSQ_1:def.5">AFINSQ_1:def 5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_2_1_8_1_2_16"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c37">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="E1:9_1_2_1_8_1_2_16"/><i><font color="Green" title="E111">A156</font></i>: 
<font color="Maroon" title="c37">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c31">d'1</font>
 ;<br/><b>thus </b><font color="Maroon" title="c8">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p3"><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">&lt;%</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">%&gt;</a></span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c37">k</font></span>)</span> = 
<font color="Maroon" title="c8">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c32">do1</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c37">k</font></span>)</span>
<b>by </b><i><a class="txt" href="numeral1.html#E1:9_1_2_1_8_1_2"><i><font color="Green" title="E46">A80</font></i></a>, <a class="txt" href="numeral1.html#E99:9_1_2_1_8_1_2"><i><font color="Green" title="E94">A134</font></i></a>, <a class="ref" href="afinsq_1.html#T37" title="AFINSQ_1:th.37">AFINSQ_1:37</a></i>
<br/>.= 
<font color="Maroon" title="c31">d'1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c37">k</font>
<b>by </b><i><a class="txt" href="numeral1.html#E95:9_1_2_1_8_1_2"><i><font color="Green" title="E91">A131</font></i></a>, <a class="txt" href="numeral1.html#E99:9_1_2_1_8_1_2"><i><font color="Green" title="E94">A134</font></i></a>, <a class="txt" href="numeral1.html#E1:9_1_2_1_8_1_2_16"><i><font color="Green" title="E111">A156</font></i></a></i>
;<br/></div><b>end;</b></div><a NAME="E132:9_1_2_1_8_1_2"/><b>then </b>
<font color="Maroon" title="c8">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">%&gt;</a></span> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <font color="Maroon" title="c31">d'1</font>
 <b>by </b><i><a class="txt" href="numeral1.html#E129:9_1_2_1_8_1_2"><i><font color="Green" title="E109">A154</font></i></a>, <a class="txt" href="numeral1.html#E130:9_1_2_1_8_1_2"><i><font color="Green" title="E110">A155</font></i></a>, <a class="ref" href="afinsq_1.html#D4" title="AFINSQ_1:def.4">AFINSQ_1:def 4</a></i>;<br/><a NAME="E133:9_1_2_1_8_1_2"/><b>then </b>
 <a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <font color="Maroon" title="c8">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <span class="p2"><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">&lt;%</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">%&gt;</a></span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <font color="Maroon" title="c31">d'1</font></span>)</span>
 <b>by </b><i><a class="ref" href="numeral1.html#T1" target="_self" title="NUMERAL1:th.1">Th1</a></i>;<br/><a NAME="E134:9_1_2_1_8_1_2"/><b>then </b>
<font color="Maroon" title="c8">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</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="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c29">o1</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c29">o1</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="numeral1.html#E88:9_1_2_1_8_1_2"><i><font color="Green" title="E88">A128</font></i></a>, <a class="txt" href="numeral1.html#E126:9_1_2_1_8_1_2"><i><font color="Green" title="E107">A152</font></i></a>, <a class="ref" href="stirl2_1.html#T44" title="STIRL2_1:th.44">STIRL2_1:44</a></i>;<br/><a NAME="E135:9_1_2_1_8_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <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="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c29">o1</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c29">o1</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="numeral1.html#E25:9_1_2_1"><i><font color="Green" title="E20">A45</font></i></a>, <a class="txt" href="numeral1.html#E128:9_1_2_1_8_1_2"><i><font color="Green" title="E108">A153</font></i></a></i>;<br/><a NAME="E136:9_1_2_1_8_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <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="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c29">o1</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> 1</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="newton.html#T13" title="NEWTON:th.13">NEWTON:13</a></i>;<br/><a NAME="E137:9_1_2_1_8_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c29">o1</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> 1</span>)</span></span>)</span></span>)</span>
 ;<br/><a NAME="E138:9_1_2_1_8_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <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="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c29">o1</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span>
 <b>by </b><i><a class="ref" href="newton.html#T10" title="NEWTON:th.10">NEWTON:10</a></i>;<br/><b>hence </b><a NAME="E139:9_1_2_1_8_1_2"/>
<font color="Maroon" title="c4">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c12">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c12">a</font>
 <b>by </b><i><a class="txt" href="numeral1.html#E69:9_1_2_1_8_1_2"><i><font color="Green" title="E78">A118</font></i></a>, <a class="txt" href="numeral1.html#E81:9_1_2_1_8_1_2"><i><font color="Green" title="E85">A125</font></i></a>, <a class="ref" href="xcmplx_1.html#T90" title="XCMPLX_1:th.90">XCMPLX_1:90</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1_2_1_8_1_3"><b>suppose </b></a><a NAME="E1:9_1_2_1_8_1_3"/><i><font color="Green" title="E46">A157</font></i>: 
( <font color="Maroon" title="c13">o</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c13">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">k</font> )
 ;<br/><div class="add"><b>set </b><font color="Maroon" title="c15">d'</font> = <span class="p1"><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">%&gt;</a></span>;<br/><b>reconsider </b><font color="Maroon" title="c16">do</font> = <font color="Maroon" title="c6">D</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c13">o</font> as   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="afinsq_1.html#T16" title="AFINSQ_1:th.16">AFINSQ_1:16</a></i>;<br/><a NAME="E3:9_1_2_1_8_1_3"/><i><font color="Green" title="E47">A158</font></i>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1"><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">%&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">oo</font>
 <b>by </b><i><a class="txt" href="numeral1.html#E1:9_1_2_1_8_1_3"><i><font color="Green" title="E46">A157</font></i></a>, <a class="ref" href="afinsq_1.html#T38" title="AFINSQ_1:th.38">AFINSQ_1:38</a></i>;<br/><i><font color="Green" title="E48">A159</font></i>:  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c4">d</font> = 
 <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c6">D</font>
<b>by </b><i><a class="txt" href="numeral1.html#E6:9_1_2_1"><i><font color="Green" title="E13">A38</font></i></a>, <a class="ref" href="afinsq_1.html#D1" title="AFINSQ_1:def.1">AFINSQ_1:def 1</a></i>
<br/>.= 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c6">D</font>
<b>by </b><i><a class="ref" href="afinsq_1.html#D1" title="AFINSQ_1:def.1">AFINSQ_1:def 1</a></i>
;<br/><a NAME="E5:9_1_2_1_8_1_3"/><b>then </b><i><font color="Green" title="E49">A160</font></i>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c16">do</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">o</font>
 <b>by </b><i><a class="txt" href="numeral1.html#E6:9_1_2_1_8"><i><font color="Green" title="E44">A75</font></i></a>, <a class="ref" href="afinsq_1.html#T14" title="AFINSQ_1:th.14">AFINSQ_1:14</a></i>;<br/><a NAME="E6:9_1_2_1_8_1_3"/><b>then </b><i><font color="Green" title="E50">A161</font></i>: 
 <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c6">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c16">do</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p2"><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">&lt;%</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">%&gt;</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="numeral1.html#E6:9_1_2_1"><i><font color="Green" title="E13">A38</font></i></a>, <a class="txt" href="numeral1.html#E3:9_1_2_1_8_1_3"><i><font color="Green" title="E47">A158</font></i></a>, <a class="ref" href="afinsq_1.html#D1" title="AFINSQ_1:def.1">AFINSQ_1:def 1</a></i>;<br/><a NAME="E7:9_1_2_1_8_1_3"/><i><font color="Green" title="E51">A162</font></i>: 
for <font color="Olive" title="b1">x</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">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c16">do</font> holds <br/><font color="Maroon" title="c6">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">do</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 <b>by </b><i><a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>;<br/><a NAME="E8:9_1_2_1_8_1_3"/><i><font color="Green" title="E52">A163</font></i>: 
for <font color="Olive" title="b1">x</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">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <span class="p1"><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">%&gt;</a></span> holds <br/><font color="Maroon" title="c6">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c16">do</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">%&gt;</a></span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_2_1_8_1_3_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c17">x</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="E1:9_1_2_1_8_1_3_2"/>
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <span class="p1"><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">%&gt;</a></span>
 ;<br/>

<a NAME="E2:9_1_2_1_8_1_3_2"/><b>then </b>
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 1
 
<b>by </b><i><a class="txt" href="numeral1.html#E1:9_1_2_1_8_1_3"><i><font color="Green" title="E46">A157</font></i></a>, <a class="txt" href="numeral1.html#E3:9_1_2_1_8_1_3"><i><font color="Green" title="E47">A158</font></i></a>, <a class="ref" href="afinsq_1.html#D1" title="AFINSQ_1:def.1">AFINSQ_1:def 1</a></i>;<br/>
<a NAME="E3:9_1_2_1_8_1_3_2"/><b>then </b>
<font color="Maroon" title="c17">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1
 
<b>by </b><i><a class="ref" href="prgcor_2.html#T1" title="PRGCOR_2:th.1">PRGCOR_2:1</a></i>;<br/>
<a NAME="E4:9_1_2_1_8_1_3_2"/><b>then </b>
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></i>;<br/>
<b>hence </b><a NAME="E5:9_1_2_1_8_1_3_2"/>
<font color="Maroon" title="c6">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c16">do</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c17">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">%&gt;</a></span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c17">x</font>
 <b>by </b><i><a class="txt" href="numeral1.html#E5:9_1_2_1_8_1_3"><i><font color="Green" title="E49">A160</font></i></a>, <a class="ref" href="afinsq_1.html#T38" title="AFINSQ_1:th.38">AFINSQ_1:38</a></i>;<br/>


</div><b>end;</b></div><a NAME="E9:9_1_2_1_8_1_3"/><b>then </b>
<font color="Maroon" title="c6">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">do</font> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <span class="p1"><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">%&gt;</a></span>
 <b>by </b><i><a class="txt" href="numeral1.html#E6:9_1_2_1_8_1_3"><i><font color="Green" title="E50">A161</font></i></a>, <a class="txt" href="numeral1.html#E7:9_1_2_1_8_1_3"><i><font color="Green" title="E51">A162</font></i></a>, <a class="ref" href="afinsq_1.html#D4" title="AFINSQ_1:def.4">AFINSQ_1:def 4</a></i>;<br/><a NAME="E10:9_1_2_1_8_1_3"/><b>then </b><i><font color="Green" title="E53">A164</font></i>: 
<font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <font color="Maroon" title="c16">do</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <span class="p2"><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">&lt;%</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">%&gt;</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="numeral1.html#E6:9_1_2_1"><i><font color="Green" title="E13">A38</font></i></a>, <a class="ref" href="numeral1.html#T1" target="_self" title="NUMERAL1:th.1">Th1</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c17">bo</font> = <font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font> as    <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>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><a NAME="E12:9_1_2_1_8_1_3"/><i><font color="Green" title="E54">A165</font></i>: 
<font color="Maroon" title="c17">bo</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="numeral1.html#E1:9"><i><font color="Green" title="E9">A1</font></i></a>, <a class="ref" href="prepower.html#T12" title="PREPOWER:th.12">PREPOWER:12</a></i>;<br/><b>consider </b><font color="Maroon" title="c18">ok</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E14:9_1_2_1_8_1_3"/><i><font color="Green" title="E55">A166</font></i>: 
<font color="Maroon" title="c18">ok</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">o</font>
 <b>by </b><i><a class="txt" href="numeral1.html#E1:9_1_2_1_8_1_3"><i><font color="Green" title="E46">A157</font></i></a>, <a class="ref" href="nat_1.html#T6" title="NAT_1:th.6">NAT_1:6</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c19">ok</font> = <font color="Maroon" title="c18">ok</font> as    <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>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c20">g</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">ok</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> as   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="numeral1.html#T3" target="_self" title="NUMERAL1:th.3">Th3</a></i>;<br/><a NAME="E17:9_1_2_1_8_1_3"/>
 <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="c2">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><a NAME="E18:9_1_2_1_8_1_3"/><b>then </b>
<font color="Maroon" title="c13">o</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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="c2">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="ordinal1.html#D2" title="ORDINAL1:def.2">ORDINAL1:def 2</a></i>;<br/><a NAME="E19:9_1_2_1_8_1_3"/><b>then </b><i><font color="Green" title="E56">A167</font></i>: 
 <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c20">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">o</font>
 <b>by </b><i><a class="txt" href="numeral1.html#E14:9_1_2_1_8_1_3"><i><font color="Green" title="E55">A166</font></i></a>, <a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a></i>;<br/><a NAME="E20:9_1_2_1_8_1_3"/><i><font color="Green" title="E57">A168</font></i>: 
1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="numeral1.html#E1:9"><i><font color="Green" title="E9">A1</font></i></a></i>;<br/><i><font color="Green" title="E58">A169</font></i>:  <a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <font color="Maroon" title="c20">g</font> = 
<span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">ok</font>
<b>by </b><i><a class="ref" href="numeral1.html#T2" target="_self" title="NUMERAL1:th.2">Th2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">ok</font>
<b>by </b><i><a class="ref" href="series_1.html#T12" title="SERIES_1:th.12">SERIES_1:12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">ok</font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T13" title="SEQ_1:th.13">SEQ_1:13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p4">(<span class="default"><font color="Maroon" title="c19">ok</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="numeral1.html#E1:9"><i><font color="Green" title="E9">A1</font></i></a>, <a class="ref" href="series_1.html#T26" title="SERIES_1:th.26">SERIES_1:26</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p4">(<span class="default"><font color="Maroon" title="c19">ok</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="numeral1.html#E1:9"><i><font color="Green" title="E9">A1</font></i></a>, <a class="ref" href="power.html#T46" title="POWER:th.46">POWER:46</a></i>
<br/>.= 
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="numeral1.html#E14:9_1_2_1_8_1_3"><i><font color="Green" title="E55">A166</font></i></a></i>
<br/>.= 
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="numeral1.html#E20:9_1_2_1_8_1_3"><i><font color="Green" title="E57">A168</font></i></a>, <a class="ref" href="xcmplx_1.html#T88" title="XCMPLX_1:th.88">XCMPLX_1:88</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1

;<br/><a NAME="E22:9_1_2_1_8_1_3"/><i><font color="Green" title="E59">A170</font></i>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c16">do</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="numeral1.html#E1:9_1_2_1_8_1_3"><i><font color="Green" title="E46">A157</font></i></a>, <a class="txt" href="numeral1.html#E5:9_1_2_1_8_1_3"><i><font color="Green" title="E49">A160</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E23:9_1_2_1_8_1_3"/><i><font color="Green" title="E60">A171</font></i>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c16">do</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c20">g</font>
 <b>by </b><i><a class="txt" href="numeral1.html#E5:9_1_2_1_8_1_3"><i><font color="Green" title="E49">A160</font></i></a>, <a class="txt" href="numeral1.html#E19:9_1_2_1_8_1_3"><i><font color="Green" title="E56">A167</font></i></a>, <a class="ref" href="afinsq_1.html#D1" title="AFINSQ_1:def.1">AFINSQ_1:def 1</a></i>;<br/><a NAME="E24:9_1_2_1_8_1_3"/>
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c16">do</font> holds <br/><font color="Maroon" title="c16">do</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">g</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">i</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_2_1_8_1_3_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c21">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>reconsider </b><font color="Maroon" title="c22">I</font> = <font color="Maroon" title="c21">i</font> as    <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>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<b>assume </b><a NAME="E2:9_1_2_1_8_1_3_4"/><i><font color="Green" title="E61">A172</font></i>: 
<font color="Maroon" title="c21">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c16">do</font>
 ;<br/>

<a NAME="E3:9_1_2_1_8_1_3_4"/><b>then </b>
<font color="Maroon" title="c21">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">o</font>
 
<b>by </b><i><a class="txt" href="numeral1.html#E6:9_1_2_1_8"><i><font color="Green" title="E44">A75</font></i></a>, <a class="txt" href="numeral1.html#E4:9_1_2_1_8_1_3"><i><font color="Green" title="E48">A159</font></i></a>, <a class="ref" href="afinsq_1.html#T14" title="AFINSQ_1:th.14">AFINSQ_1:14</a></i>;<br/>
<b>then </b><i><font color="Green" title="E62">A173</font></i>: <font color="Maroon" title="c20">g</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c22">I</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">I</font>
<b>by </b><i><a class="txt" href="numeral1.html#E19:9_1_2_1_8_1_3"><i><font color="Green" title="E56">A167</font></i></a>, <a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">I</font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T13" title="SEQ_1:th.13">SEQ_1:13</a></i>
<br/>.= 
<font color="Maroon" title="c3">b1</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c22">I</font></span>)</span>
<b>by </b><i><a class="ref" href="prepower.html#D1" title="PREPOWER:def.1">PREPOWER:def 1</a></i>
;<br/>
<a NAME="E5:9_1_2_1_8_1_3_4"/><i><font color="Green" title="E63">A174</font></i>: 
 <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c16">do</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c6">D</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T89" title="RELAT_1:th.89">RELAT_1:89</a></i>;<br/>
<a NAME="E6:9_1_2_1_8_1_3_4"/><b>then </b>
<font color="Maroon" title="c4">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c21">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">b1</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 
<b>by </b><i><a class="txt" href="numeral1.html#E3:9_1_2_1"><i><font color="Green" title="E11">A36</font></i></a>, <a class="txt" href="numeral1.html#E6:9_1_2_1"><i><font color="Green" title="E13">A38</font></i></a>, <a class="txt" href="numeral1.html#E2:9_1_2_1_8_1_3_4"><i><font color="Green" title="E61">A172</font></i></a></i>;<br/>
<a NAME="E7:9_1_2_1_8_1_3_4"/><b>then </b><i><font color="Green" title="E64">A175</font></i>: 
<font color="Maroon" title="c4">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c21">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">b1</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<font color="Maroon" title="c16">do</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c21">i</font> = 
<font color="Maroon" title="c6">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c21">i</font>
<b>by </b><i><a class="txt" href="numeral1.html#E2:9_1_2_1_8_1_3_4"><i><font color="Green" title="E61">A172</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c21">i</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c21">i</font></span>)</span>
<b>by </b><i><a class="txt" href="numeral1.html#E6:9_1_2_1"><i><font color="Green" title="E13">A38</font></i></a>, <a class="txt" href="numeral1.html#E2:9_1_2_1_8_1_3_4"><i><font color="Green" title="E61">A172</font></i></a>, <a class="txt" href="numeral1.html#E5:9_1_2_1_8_1_3_4"><i><font color="Green" title="E63">A174</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E9:9_1_2_1_8_1_3_4"/>
<font color="Maroon" title="c16">do</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c21">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">g</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c21">i</font>
 <b>by </b><i><a class="txt" href="numeral1.html#E4:9_1_2_1_8_1_3_4"><i><font color="Green" title="E62">A173</font></i></a>, <a class="txt" href="numeral1.html#E7:9_1_2_1_8_1_3_4"><i><font color="Green" title="E64">A175</font></i></a>, <a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a></i>;<br/>


</div><b>end;</b></div><a NAME="E25:9_1_2_1_8_1_3"/><b>then </b>
 <a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <font color="Maroon" title="c16">do</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <b>by </b><i><a class="txt" href="numeral1.html#E21:9_1_2_1_8_1_3"><i><font color="Green" title="E58">A169</font></i></a>, <a class="txt" href="numeral1.html#E22:9_1_2_1_8_1_3"><i><font color="Green" title="E59">A170</font></i></a>, <a class="txt" href="numeral1.html#E23:9_1_2_1_8_1_3"><i><font color="Green" title="E60">A171</font></i></a>, <a class="ref" href="numeral1.html#T4" target="_self" title="NUMERAL1:th.4">Th4</a></i>;<br/><a NAME="E26:9_1_2_1_8_1_3"/><b>then </b>
 <a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <font color="Maroon" title="c16">do</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T147" title="XREAL_1:th.147">XREAL_1:147</a></i>;<br/><a NAME="E27:9_1_2_1_8_1_3"/><b>then </b><i><font color="Green" title="E61">A176</font></i>: 
<span class="p1">(<span class="default"><a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <font color="Maroon" title="c16">do</font></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="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="nat_d.html#T27" title="NAT_D:th.27">NAT_D:27</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_2_1_8_1_3_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c21">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>reconsider </b><font color="Maroon" title="c22">K</font> = <font color="Maroon" title="c21">k</font> as    <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>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><b>assume </b><a NAME="E2:9_1_2_1_8_1_3_5"/><i><font color="Green" title="E62">A177</font></i>: 
<font color="Maroon" title="c21">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <span class="p1"><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">%&gt;</a></span>
 ;<br/><a NAME="E3:9_1_2_1_8_1_3_5"/><b>then </b>
<font color="Maroon" title="c21">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1"><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">%&gt;</a></span>
 <b>by </b><i><a class="ref" href="afinsq_1.html#D1" title="AFINSQ_1:def.1">AFINSQ_1:def 1</a></i>;<br/><a NAME="E4:9_1_2_1_8_1_3_5"/><b>then </b>
<font color="Maroon" title="c21">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1"><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">%&gt;</a></span>
 <b>by </b><i><a class="ref" href="prgcor_2.html#T1" title="PRGCOR_2:th.1">PRGCOR_2:1</a></i>;<br/><a NAME="E5:9_1_2_1_8_1_3_5"/><b>then </b>
<font color="Maroon" title="c13">o</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c21">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c6">D</font>
 <b>by </b><i><a class="txt" href="numeral1.html#E5:9_1_2_1_8_1_3"><i><font color="Green" title="E49">A160</font></i></a>, <a class="txt" href="numeral1.html#E6:9_1_2_1_8_1_3"><i><font color="Green" title="E50">A161</font></i></a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/><a NAME="E6:9_1_2_1_8_1_3_5"/><b>then </b>
<font color="Maroon" title="c13">o</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c21">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c6">D</font>
 <b>by </b><i><a class="ref" href="prgcor_2.html#T1" title="PRGCOR_2:th.1">PRGCOR_2:1</a></i>;<br/><a NAME="E7:9_1_2_1_8_1_3_5"/><b>then </b>
<font color="Maroon" title="c6">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">o</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c21">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">o</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c21">k</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">o</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c21">k</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="numeral1.html#E6:9_1_2_1"><i><font color="Green" title="E13">A38</font></i></a></i>;<br/><b>then </b><span class="p1"><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">%&gt;</a></span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c22">K</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">o</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c21">k</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">o</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c21">k</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="numeral1.html#E5:9_1_2_1_8_1_3"><i><font color="Green" title="E49">A160</font></i></a>, <a class="txt" href="numeral1.html#E8:9_1_2_1_8_1_3"><i><font color="Green" title="E52">A163</font></i></a>, <a class="txt" href="numeral1.html#E2:9_1_2_1_8_1_3_5"><i><font color="Green" title="E62">A177</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">o</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c21">k</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c21">k</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T13" title="NEWTON:th.13">NEWTON:13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">o</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c21">k</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c21">k</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span>

;<br/><b>hence </b><a NAME="E9:9_1_2_1_8_1_3_5"/>
<font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <span class="p1"><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">%&gt;</a></span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c21">k</font>
 <b>by </b><i><a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">NAT_D:def 3</a></i>;<br/></div><b>end;</b></div><a NAME="E29:9_1_2_1_8_1_3"/><b>then </b><i><font color="Green" title="E62">A178</font></i>: 
<font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a>  <a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <span class="p1"><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">%&gt;</a></span>
 <b>by </b><i><a class="ref" href="numeral1.html#T5" target="_self" title="NUMERAL1:th.5">Th5</a></i>;<br/><a NAME="E30:9_1_2_1_8_1_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <span class="p2"><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">&lt;%</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">%&gt;</a></span></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="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="numeral1.html#E12:9_1_2_1_8_1_3"><i><font color="Green" title="E54">A165</font></i></a>, <a class="ref" href="pepin.html#T6" title="PEPIN:th.6">PEPIN:6</a></i>;<br/><a NAME="E31:9_1_2_1_8_1_3"/><b>then </b>
<font color="Maroon" title="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <span class="p3"><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">&lt;%</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">%&gt;</a></span></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="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></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"><a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <font color="Maroon" title="c16">do</font></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="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="numeral1.html#E10:9_1_2_1_8_1_3"><i><font color="Green" title="E53">A164</font></i></a>, <a class="ref" href="nat_d.html#T19" title="NAT_D:th.19">NAT_D:19</a></i>;<br/><a NAME="E32:9_1_2_1_8_1_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <span class="p1"><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">%&gt;</a></span>
 <b>by </b><i><a class="txt" href="numeral1.html#E27:9_1_2_1_8_1_3"><i><font color="Green" title="E61">A176</font></i></a>, <a class="txt" href="numeral1.html#E29:9_1_2_1_8_1_3"><i><font color="Green" title="E62">A178</font></i></a>, <a class="ref" href="nat_d.html#T3" title="NAT_D:th.3">NAT_D:3</a></i>;<br/><a NAME="E33:9_1_2_1_8_1_3"/><b>then </b>
<font color="Maroon" title="c6">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span>
 <b>by </b><i><a class="ref" href="stirl2_1.html#T44" title="STIRL2_1:th.44">STIRL2_1:44</a></i>;<br/><a NAME="E34:9_1_2_1_8_1_3"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <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="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span>
 <b>by </b><i><a class="txt" href="numeral1.html#E6:9_1_2_1"><i><font color="Green" title="E13">A38</font></i></a>, <a class="txt" href="numeral1.html#E1:9_1_2_1_8"><i><font color="Green" title="E41">A72</font></i></a></i>;<br/><a NAME="E35:9_1_2_1_8_1_3"/><b>then </b><i><font color="Green" title="E63">A179</font></i>: 
<font color="Maroon" title="c4">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</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="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span>
 <b>by </b><i><a class="txt" href="numeral1.html#E12:9_1_2_1_8_1_3"><i><font color="Green" title="E54">A165</font></i></a>, <a class="ref" href="xcmplx_1.html#T90" title="XCMPLX_1:th.90">XCMPLX_1:90</a></i>;<br/><b>set </b><font color="Maroon" title="c21">d'</font> = <span class="p1"><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">%&gt;</a></span>;<br/><b>reconsider </b><font color="Maroon" title="c22">do</font> = <font color="Maroon" title="c8">E</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c13">o</font> as   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="afinsq_1.html#T16" title="AFINSQ_1:th.16">AFINSQ_1:16</a></i>;<br/><a NAME="E37:9_1_2_1_8_1_3"/><i><font color="Green" title="E64">A180</font></i>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1"><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">%&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">oo</font>
 <b>by </b><i><a class="txt" href="numeral1.html#E1:9_1_2_1_8_1_3"><i><font color="Green" title="E46">A157</font></i></a>, <a class="ref" href="afinsq_1.html#T38" title="AFINSQ_1:th.38">AFINSQ_1:38</a></i>;<br/><i><font color="Green" title="E65">A181</font></i>:  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c5">e</font> = 
 <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c8">E</font>
<b>by </b><i><a class="txt" href="numeral1.html#E25:9_1_2_1"><i><font color="Green" title="E20">A45</font></i></a>, <a class="ref" href="afinsq_1.html#D1" title="AFINSQ_1:def.1">AFINSQ_1:def 1</a></i>
<br/>.= 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c8">E</font>
<b>by </b><i><a class="ref" href="afinsq_1.html#D1" title="AFINSQ_1:def.1">AFINSQ_1:def 1</a></i>
;<br/><a NAME="E39:9_1_2_1_8_1_3"/><b>then </b><i><font color="Green" title="E66">A182</font></i>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c22">do</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">o</font>
 <b>by </b><i><a class="txt" href="numeral1.html#E81:9_1_2_1"><i><font color="Green" title="E39">A70</font></i></a>, <a class="txt" href="numeral1.html#E6:9_1_2_1_8"><i><font color="Green" title="E44">A75</font></i></a>, <a class="ref" href="afinsq_1.html#T14" title="AFINSQ_1:th.14">AFINSQ_1:14</a></i>;<br/><a NAME="E40:9_1_2_1_8_1_3"/><b>then </b><i><font color="Green" title="E67">A183</font></i>: 
 <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c8">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c22">do</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p2"><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">&lt;%</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">%&gt;</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="numeral1.html#E25:9_1_2_1"><i><font color="Green" title="E20">A45</font></i></a>, <a class="txt" href="numeral1.html#E81:9_1_2_1"><i><font color="Green" title="E39">A70</font></i></a>, <a class="txt" href="numeral1.html#E37:9_1_2_1_8_1_3"><i><font color="Green" title="E64">A180</font></i></a>, <a class="ref" href="afinsq_1.html#D1" title="AFINSQ_1:def.1">AFINSQ_1:def 1</a></i>;<br/><a NAME="E41:9_1_2_1_8_1_3"/><i><font color="Green" title="E68">A184</font></i>: 
for <font color="Olive" title="b1">x</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">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c22">do</font> holds <br/><font color="Maroon" title="c8">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">do</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 <b>by </b><i><a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>;<br/><a NAME="E42:9_1_2_1_8_1_3"/><i><font color="Green" title="E69">A185</font></i>: 
for <font color="Olive" title="b1">x</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">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <span class="p1"><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">%&gt;</a></span> holds <br/><font color="Maroon" title="c8">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c22">do</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">%&gt;</a></span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_2_1_8_1_3_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c23">x</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="E1:9_1_2_1_8_1_3_7"/>
<font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <span class="p1"><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">%&gt;</a></span>
 ;<br/>

<a NAME="E2:9_1_2_1_8_1_3_7"/><b>then </b>
<font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 1
 
<b>by </b><i><a class="txt" href="numeral1.html#E1:9_1_2_1_8_1_3"><i><font color="Green" title="E46">A157</font></i></a>, <a class="txt" href="numeral1.html#E37:9_1_2_1_8_1_3"><i><font color="Green" title="E64">A180</font></i></a>, <a class="ref" href="afinsq_1.html#D1" title="AFINSQ_1:def.1">AFINSQ_1:def 1</a></i>;<br/>
<a NAME="E3:9_1_2_1_8_1_3_7"/><b>then </b>
<font color="Maroon" title="c23">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1
 
<b>by </b><i><a class="ref" href="prgcor_2.html#T1" title="PRGCOR_2:th.1">PRGCOR_2:1</a></i>;<br/>
<a NAME="E4:9_1_2_1_8_1_3_7"/><b>then </b>
<font color="Maroon" title="c23">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></i>;<br/>
<b>hence </b><a NAME="E5:9_1_2_1_8_1_3_7"/>
<font color="Maroon" title="c8">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c22">do</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c23">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">%&gt;</a></span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c23">x</font>
 <b>by </b><i><a class="txt" href="numeral1.html#E39:9_1_2_1_8_1_3"><i><font color="Green" title="E66">A182</font></i></a>, <a class="ref" href="afinsq_1.html#T38" title="AFINSQ_1:th.38">AFINSQ_1:38</a></i>;<br/>


</div><b>end;</b></div><a NAME="E43:9_1_2_1_8_1_3"/><b>then </b>
<font color="Maroon" title="c8">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">do</font> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <span class="p1"><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">%&gt;</a></span>
 <b>by </b><i><a class="txt" href="numeral1.html#E40:9_1_2_1_8_1_3"><i><font color="Green" title="E67">A183</font></i></a>, <a class="txt" href="numeral1.html#E41:9_1_2_1_8_1_3"><i><font color="Green" title="E68">A184</font></i></a>, <a class="ref" href="afinsq_1.html#D4" title="AFINSQ_1:def.4">AFINSQ_1:def 4</a></i>;<br/><a NAME="E44:9_1_2_1_8_1_3"/><b>then </b><i><font color="Green" title="E70">A186</font></i>: 
<font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <font color="Maroon" title="c22">do</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <span class="p2"><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">&lt;%</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">%&gt;</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="numeral1.html#E25:9_1_2_1"><i><font color="Green" title="E20">A45</font></i></a>, <a class="ref" href="numeral1.html#T1" target="_self" title="NUMERAL1:th.1">Th1</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c23">bo</font> = <font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font> as    <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>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><a NAME="E46:9_1_2_1_8_1_3"/><i><font color="Green" title="E71">A187</font></i>: 
<font color="Maroon" title="c23">bo</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="numeral1.html#E1:9"><i><font color="Green" title="E9">A1</font></i></a>, <a class="ref" href="prepower.html#T12" title="PREPOWER:th.12">PREPOWER:12</a></i>;<br/><b>consider </b><font color="Maroon" title="c24">ok</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E48:9_1_2_1_8_1_3"/><i><font color="Green" title="E72">A188</font></i>: 
<font color="Maroon" title="c24">ok</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">o</font>
 <b>by </b><i><a class="txt" href="numeral1.html#E1:9_1_2_1_8_1_3"><i><font color="Green" title="E46">A157</font></i></a>, <a class="ref" href="nat_1.html#T6" title="NAT_1:th.6">NAT_1:6</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c25">ok</font> = <font color="Maroon" title="c24">ok</font> as    <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>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c26">g</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">ok</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> as   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="numeral1.html#T3" target="_self" title="NUMERAL1:th.3">Th3</a></i>;<br/><a NAME="E51:9_1_2_1_8_1_3"/>
 <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="c2">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><a NAME="E52:9_1_2_1_8_1_3"/><b>then </b>
<font color="Maroon" title="c13">o</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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="c2">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="ordinal1.html#D2" title="ORDINAL1:def.2">ORDINAL1:def 2</a></i>;<br/><a NAME="E53:9_1_2_1_8_1_3"/><b>then </b><i><font color="Green" title="E73">A189</font></i>: 
 <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c26">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">o</font>
 <b>by </b><i><a class="txt" href="numeral1.html#E48:9_1_2_1_8_1_3"><i><font color="Green" title="E72">A188</font></i></a>, <a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a></i>;<br/><a NAME="E54:9_1_2_1_8_1_3"/><i><font color="Green" title="E74">A190</font></i>: 
1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="numeral1.html#E1:9"><i><font color="Green" title="E9">A1</font></i></a></i>;<br/><i><font color="Green" title="E75">A191</font></i>:  <a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <font color="Maroon" title="c26">g</font> = 
<span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c25">ok</font>
<b>by </b><i><a class="ref" href="numeral1.html#T2" target="_self" title="NUMERAL1:th.2">Th2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c25">ok</font>
<b>by </b><i><a class="ref" href="series_1.html#T12" title="SERIES_1:th.12">SERIES_1:12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c25">ok</font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T13" title="SEQ_1:th.13">SEQ_1:13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p4">(<span class="default"><font color="Maroon" title="c25">ok</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="numeral1.html#E1:9"><i><font color="Green" title="E9">A1</font></i></a>, <a class="ref" href="series_1.html#T26" title="SERIES_1:th.26">SERIES_1:26</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p4">(<span class="default"><font color="Maroon" title="c25">ok</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="numeral1.html#E1:9"><i><font color="Green" title="E9">A1</font></i></a>, <a class="ref" href="power.html#T46" title="POWER:th.46">POWER:46</a></i>
<br/>.= 
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="numeral1.html#E48:9_1_2_1_8_1_3"><i><font color="Green" title="E72">A188</font></i></a></i>
<br/>.= 
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="numeral1.html#E54:9_1_2_1_8_1_3"><i><font color="Green" title="E74">A190</font></i></a>, <a class="ref" href="xcmplx_1.html#T88" title="XCMPLX_1:th.88">XCMPLX_1:88</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1

;<br/><a NAME="E56:9_1_2_1_8_1_3"/><i><font color="Green" title="E76">A192</font></i>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c22">do</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="numeral1.html#E1:9_1_2_1_8_1_3"><i><font color="Green" title="E46">A157</font></i></a>, <a class="txt" href="numeral1.html#E39:9_1_2_1_8_1_3"><i><font color="Green" title="E66">A182</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E57:9_1_2_1_8_1_3"/><i><font color="Green" title="E77">A193</font></i>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c22">do</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c26">g</font>
 <b>by </b><i><a class="txt" href="numeral1.html#E39:9_1_2_1_8_1_3"><i><font color="Green" title="E66">A182</font></i></a>, <a class="txt" href="numeral1.html#E53:9_1_2_1_8_1_3"><i><font color="Green" title="E73">A189</font></i></a>, <a class="ref" href="afinsq_1.html#D1" title="AFINSQ_1:def.1">AFINSQ_1:def 1</a></i>;<br/><a NAME="E58:9_1_2_1_8_1_3"/>
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c22">do</font> holds <br/><font color="Maroon" title="c22">do</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">g</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">i</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_2_1_8_1_3_9"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c27">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>reconsider </b><font color="Maroon" title="c28">I</font> = <font color="Maroon" title="c27">i</font> as    <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>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<b>assume </b><a NAME="E2:9_1_2_1_8_1_3_9"/><i><font color="Green" title="E78">A194</font></i>: 
<font color="Maroon" title="c27">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c22">do</font>
 ;<br/>

<a NAME="E3:9_1_2_1_8_1_3_9"/><b>then </b>
<font color="Maroon" title="c27">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">o</font>
 
<b>by </b><i><a class="txt" href="numeral1.html#E81:9_1_2_1"><i><font color="Green" title="E39">A70</font></i></a>, <a class="txt" href="numeral1.html#E6:9_1_2_1_8"><i><font color="Green" title="E44">A75</font></i></a>, <a class="txt" href="numeral1.html#E38:9_1_2_1_8_1_3"><i><font color="Green" title="E65">A181</font></i></a>, <a class="ref" href="afinsq_1.html#T14" title="AFINSQ_1:th.14">AFINSQ_1:14</a></i>;<br/>
<b>then </b><i><font color="Green" title="E79">A195</font></i>: <font color="Maroon" title="c26">g</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c28">I</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c28">I</font>
<b>by </b><i><a class="txt" href="numeral1.html#E53:9_1_2_1_8_1_3"><i><font color="Green" title="E73">A189</font></i></a>, <a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c28">I</font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T13" title="SEQ_1:th.13">SEQ_1:13</a></i>
<br/>.= 
<font color="Maroon" title="c3">b1</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c28">I</font></span>)</span>
<b>by </b><i><a class="ref" href="prepower.html#D1" title="PREPOWER:def.1">PREPOWER:def 1</a></i>
;<br/>
<a NAME="E5:9_1_2_1_8_1_3_9"/><i><font color="Green" title="E80">A196</font></i>: 
 <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c22">do</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c8">E</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T89" title="RELAT_1:th.89">RELAT_1:89</a></i>;<br/>
<a NAME="E6:9_1_2_1_8_1_3_9"/><b>then </b>
<font color="Maroon" title="c5">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c27">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">b1</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 
<b>by </b><i><a class="txt" href="numeral1.html#E4:9_1_2_1"><i><font color="Green" title="E12">A37</font></i></a>, <a class="txt" href="numeral1.html#E25:9_1_2_1"><i><font color="Green" title="E20">A45</font></i></a>, <a class="txt" href="numeral1.html#E2:9_1_2_1_8_1_3_9"><i><font color="Green" title="E78">A194</font></i></a></i>;<br/>
<a NAME="E7:9_1_2_1_8_1_3_9"/><b>then </b><i><font color="Green" title="E81">A197</font></i>: 
<font color="Maroon" title="c5">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c27">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">b1</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<font color="Maroon" title="c22">do</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c27">i</font> = 
<font color="Maroon" title="c8">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c27">i</font>
<b>by </b><i><a class="txt" href="numeral1.html#E2:9_1_2_1_8_1_3_9"><i><font color="Green" title="E78">A194</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c27">i</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c27">i</font></span>)</span>
<b>by </b><i><a class="txt" href="numeral1.html#E25:9_1_2_1"><i><font color="Green" title="E20">A45</font></i></a>, <a class="txt" href="numeral1.html#E2:9_1_2_1_8_1_3_9"><i><font color="Green" title="E78">A194</font></i></a>, <a class="txt" href="numeral1.html#E5:9_1_2_1_8_1_3_9"><i><font color="Green" title="E80">A196</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E9:9_1_2_1_8_1_3_9"/>
<font color="Maroon" title="c22">do</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c27">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">g</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c27">i</font>
 <b>by </b><i><a class="txt" href="numeral1.html#E4:9_1_2_1_8_1_3_9"><i><font color="Green" title="E79">A195</font></i></a>, <a class="txt" href="numeral1.html#E7:9_1_2_1_8_1_3_9"><i><font color="Green" title="E81">A197</font></i></a>, <a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a></i>;<br/>


</div><b>end;</b></div><a NAME="E59:9_1_2_1_8_1_3"/><b>then </b>
 <a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <font color="Maroon" title="c22">do</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <b>by </b><i><a class="txt" href="numeral1.html#E55:9_1_2_1_8_1_3"><i><font color="Green" title="E75">A191</font></i></a>, <a class="txt" href="numeral1.html#E56:9_1_2_1_8_1_3"><i><font color="Green" title="E76">A192</font></i></a>, <a class="txt" href="numeral1.html#E57:9_1_2_1_8_1_3"><i><font color="Green" title="E77">A193</font></i></a>, <a class="ref" href="numeral1.html#T4" target="_self" title="NUMERAL1:th.4">Th4</a></i>;<br/><a NAME="E60:9_1_2_1_8_1_3"/><b>then </b>
 <a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <font color="Maroon" title="c22">do</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T147" title="XREAL_1:th.147">XREAL_1:147</a></i>;<br/><a NAME="E61:9_1_2_1_8_1_3"/><b>then </b><i><font color="Green" title="E78">A198</font></i>: 
<span class="p1">(<span class="default"><a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <font color="Maroon" title="c22">do</font></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="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="nat_d.html#T27" title="NAT_D:th.27">NAT_D:27</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_2_1_8_1_3_10"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c27">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>reconsider </b><font color="Maroon" title="c28">K</font> = <font color="Maroon" title="c27">k</font> as    <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>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><b>assume </b><a NAME="E2:9_1_2_1_8_1_3_10"/><i><font color="Green" title="E79">A199</font></i>: 
<font color="Maroon" title="c27">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <span class="p1"><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">%&gt;</a></span>
 ;<br/><a NAME="E3:9_1_2_1_8_1_3_10"/><b>then </b>
<font color="Maroon" title="c27">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1"><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">%&gt;</a></span>
 <b>by </b><i><a class="ref" href="afinsq_1.html#D1" title="AFINSQ_1:def.1">AFINSQ_1:def 1</a></i>;<br/><a NAME="E4:9_1_2_1_8_1_3_10"/><b>then </b>
<font color="Maroon" title="c27">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1"><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">%&gt;</a></span>
 <b>by </b><i><a class="ref" href="prgcor_2.html#T1" title="PRGCOR_2:th.1">PRGCOR_2:1</a></i>;<br/><a NAME="E5:9_1_2_1_8_1_3_10"/><b>then </b>
<font color="Maroon" title="c13">o</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c27">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c8">E</font>
 <b>by </b><i><a class="txt" href="numeral1.html#E39:9_1_2_1_8_1_3"><i><font color="Green" title="E66">A182</font></i></a>, <a class="txt" href="numeral1.html#E40:9_1_2_1_8_1_3"><i><font color="Green" title="E67">A183</font></i></a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/><a NAME="E6:9_1_2_1_8_1_3_10"/><b>then </b>
<font color="Maroon" title="c13">o</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c27">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c8">E</font>
 <b>by </b><i><a class="ref" href="prgcor_2.html#T1" title="PRGCOR_2:th.1">PRGCOR_2:1</a></i>;<br/><a NAME="E7:9_1_2_1_8_1_3_10"/><b>then </b>
<font color="Maroon" title="c8">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">o</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c27">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">o</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c27">k</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">o</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c27">k</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="numeral1.html#E25:9_1_2_1"><i><font color="Green" title="E20">A45</font></i></a></i>;<br/><b>then </b><span class="p1"><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">%&gt;</a></span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c28">K</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">o</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c27">k</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">o</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c27">k</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="numeral1.html#E39:9_1_2_1_8_1_3"><i><font color="Green" title="E66">A182</font></i></a>, <a class="txt" href="numeral1.html#E42:9_1_2_1_8_1_3"><i><font color="Green" title="E69">A185</font></i></a>, <a class="txt" href="numeral1.html#E2:9_1_2_1_8_1_3_10"><i><font color="Green" title="E79">A199</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">o</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c27">k</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c27">k</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T13" title="NEWTON:th.13">NEWTON:13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">o</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c27">k</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c27">k</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span>

;<br/><b>hence </b><a NAME="E9:9_1_2_1_8_1_3_10"/>
<font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <span class="p1"><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">%&gt;</a></span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c27">k</font>
 <b>by </b><i><a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">NAT_D:def 3</a></i>;<br/></div><b>end;</b></div><a NAME="E63:9_1_2_1_8_1_3"/><b>then </b><i><font color="Green" title="E79">A200</font></i>: 
<font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a>  <a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <span class="p1"><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">%&gt;</a></span>
 <b>by </b><i><a class="ref" href="numeral1.html#T5" target="_self" title="NUMERAL1:th.5">Th5</a></i>;<br/><a NAME="E64:9_1_2_1_8_1_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <span class="p2"><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">&lt;%</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">%&gt;</a></span></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="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="numeral1.html#E46:9_1_2_1_8_1_3"><i><font color="Green" title="E71">A187</font></i></a>, <a class="ref" href="pepin.html#T6" title="PEPIN:th.6">PEPIN:6</a></i>;<br/><a NAME="E65:9_1_2_1_8_1_3"/><b>then </b>
<font color="Maroon" title="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <span class="p3"><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">&lt;%</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">%&gt;</a></span></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="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></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"><a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <font color="Maroon" title="c22">do</font></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="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="numeral1.html#E44:9_1_2_1_8_1_3"><i><font color="Green" title="E70">A186</font></i></a>, <a class="ref" href="nat_d.html#T19" title="NAT_D:th.19">NAT_D:19</a></i>;<br/><a NAME="E66:9_1_2_1_8_1_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <span class="p1"><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">%&gt;</a></span>
 <b>by </b><i><a class="txt" href="numeral1.html#E61:9_1_2_1_8_1_3"><i><font color="Green" title="E78">A198</font></i></a>, <a class="txt" href="numeral1.html#E63:9_1_2_1_8_1_3"><i><font color="Green" title="E79">A200</font></i></a>, <a class="ref" href="nat_d.html#T3" title="NAT_D:th.3">NAT_D:3</a></i>;<br/><a NAME="E67:9_1_2_1_8_1_3"/><b>then </b>
<font color="Maroon" title="c8">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span>
 <b>by </b><i><a class="ref" href="stirl2_1.html#T44" title="STIRL2_1:th.44">STIRL2_1:44</a></i>;<br/><a NAME="E68:9_1_2_1_8_1_3"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <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="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span>
 <b>by </b><i><a class="txt" href="numeral1.html#E25:9_1_2_1"><i><font color="Green" title="E20">A45</font></i></a>, <a class="txt" href="numeral1.html#E4:9_1_2_1_8"><i><font color="Green" title="E42">A73</font></i></a></i>;<br/><b>hence </b><a NAME="E69:9_1_2_1_8_1_3"/>
<font color="Maroon" title="c4">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c12">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c12">a</font>
 <b>by </b><i><a class="txt" href="numeral1.html#E35:9_1_2_1_8_1_3"><i><font color="Green" title="E63">A179</font></i></a>, <a class="txt" href="numeral1.html#E46:9_1_2_1_8_1_3"><i><font color="Green" title="E71">A187</font></i></a>, <a class="ref" href="xcmplx_1.html#T90" title="XCMPLX_1:th.90">XCMPLX_1:90</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1_2_1_8_1_4"><b>suppose </b></a><a NAME="E1:9_1_2_1_8_1_4"/><i><font color="Green" title="E46">A201</font></i>: 
( <font color="Maroon" title="c13">o</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c13">o</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">k</font> )
 ;<br/><div class="add"><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> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>$2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default">$1 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c13">o</font></span>)</span>;<br/><a NAME="E2:9_1_2_1_8_1_4"/><i><font color="Green" title="E47">A202</font></i>: 
for <font color="Olive" title="b1">u</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">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">oo</font> holds <br/> ex <font color="Olive" title="b2">x</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">u</font>,<font color="Olive" title="b2">x</font>]
 ;<br/><b>consider </b><font color="Maroon" title="c15">d'</font> being   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E4:9_1_2_1_8_1_4"/><i><font color="Green" title="E48">A203</font></i>: 
(  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c15">d'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">oo</font> &amp; ( for <font color="Olive" title="b1">x</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">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">oo</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c15">d'</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">x</font>] ) )
 <b>from </b><i><a class="ref" href="stirl2_1.html#S6" title="STIRL2_1:sch.6">STIRL2_1:sch 6</a>(<a class="txt" href="numeral1.html#E2:9_1_2_1_8_1_4"><i><font color="Green" title="E47">A202</font></i></a>);<br/></i><b>reconsider </b><font color="Maroon" title="c16">do</font> = <font color="Maroon" title="c6">D</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c13">o</font> as   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="afinsq_1.html#T16" title="AFINSQ_1:th.16">AFINSQ_1:16</a></i>;<br/><a NAME="E6:9_1_2_1_8_1_4"/><i><font color="Green" title="E49">A204</font></i>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c15">d'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">oo</font>
 <b>by </b><i><a class="txt" href="numeral1.html#E4:9_1_2_1_8_1_4"><i><font color="Green" title="E48">A203</font></i></a>, <a class="ref" href="afinsq_1.html#D1" title="AFINSQ_1:def.1">AFINSQ_1:def 1</a></i>;<br/><i><font color="Green" title="E50">A205</font></i>:  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c4">d</font> = 
 <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c6">D</font>
<b>by </b><i><a class="txt" href="numeral1.html#E6:9_1_2_1"><i><font color="Green" title="E13">A38</font></i></a>, <a class="ref" href="afinsq_1.html#D1" title="AFINSQ_1:def.1">AFINSQ_1:def 1</a></i>
<br/>.= 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c6">D</font>
<b>by </b><i><a class="ref" href="afinsq_1.html#D1" title="AFINSQ_1:def.1">AFINSQ_1:def 1</a></i>
;<br/><a NAME="E8:9_1_2_1_8_1_4"/><b>then </b><i><font color="Green" title="E51">A206</font></i>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c16">do</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">o</font>
 <b>by </b><i><a class="txt" href="numeral1.html#E6:9_1_2_1_8"><i><font color="Green" title="E44">A75</font></i></a>, <a class="ref" href="afinsq_1.html#T14" title="AFINSQ_1:th.14">AFINSQ_1:14</a></i>;<br/><a NAME="E9:9_1_2_1_8_1_4"/><b>then </b><i><font color="Green" title="E52">A207</font></i>: 
 <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c6">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c16">do</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c15">d'</font></span>)</span>
 <b>by </b><i><a class="txt" href="numeral1.html#E6:9_1_2_1"><i><font color="Green" title="E13">A38</font></i></a>, <a class="txt" href="numeral1.html#E6:9_1_2_1_8_1_4"><i><font color="Green" title="E49">A204</font></i></a>, <a class="ref" href="afinsq_1.html#D1" title="AFINSQ_1:def.1">AFINSQ_1:def 1</a></i>;<br/><a NAME="E10:9_1_2_1_8_1_4"/><i><font color="Green" title="E53">A208</font></i>: 
for <font color="Olive" title="b1">x</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">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c16">do</font> holds <br/><font color="Maroon" title="c6">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">do</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 <b>by </b><i><a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>;<br/><a NAME="E11:9_1_2_1_8_1_4"/>
for <font color="Olive" title="b1">x</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">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c15">d'</font> holds <br/><font color="Maroon" title="c6">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c16">do</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">d'</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 <b>by </b><i><a class="txt" href="numeral1.html#E4:9_1_2_1_8_1_4"><i><font color="Green" title="E48">A203</font></i></a>, <a class="txt" href="numeral1.html#E8:9_1_2_1_8_1_4"><i><font color="Green" title="E51">A206</font></i></a></i>;<br/><a NAME="E12:9_1_2_1_8_1_4"/><b>then </b>
<font color="Maroon" title="c6">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">do</font> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <font color="Maroon" title="c15">d'</font>
 <b>by </b><i><a class="txt" href="numeral1.html#E9:9_1_2_1_8_1_4"><i><font color="Green" title="E52">A207</font></i></a>, <a class="txt" href="numeral1.html#E10:9_1_2_1_8_1_4"><i><font color="Green" title="E53">A208</font></i></a>, <a class="ref" href="afinsq_1.html#D4" title="AFINSQ_1:def.4">AFINSQ_1:def 4</a></i>;<br/><a NAME="E13:9_1_2_1_8_1_4"/><b>then </b><i><font color="Green" title="E54">A209</font></i>: 
<font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <font color="Maroon" title="c16">do</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <font color="Maroon" title="c15">d'</font></span>)</span>
 <b>by </b><i><a class="txt" href="numeral1.html#E6:9_1_2_1"><i><font color="Green" title="E13">A38</font></i></a>, <a class="ref" href="numeral1.html#T1" target="_self" title="NUMERAL1:th.1">Th1</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c17">bo</font> = <font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font> as    <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>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><a NAME="E15:9_1_2_1_8_1_4"/><i><font color="Green" title="E55">A210</font></i>: 
<font color="Maroon" title="c17">bo</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="numeral1.html#E1:9"><i><font color="Green" title="E9">A1</font></i></a>, <a class="ref" href="prepower.html#T12" title="PREPOWER:th.12">PREPOWER:12</a></i>;<br/><b>consider </b><font color="Maroon" title="c18">ok</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E17:9_1_2_1_8_1_4"/><i><font color="Green" title="E56">A211</font></i>: 
<font color="Maroon" title="c18">ok</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">o</font>
 <b>by </b><i><a class="txt" href="numeral1.html#E1:9_1_2_1_8_1_4"><i><font color="Green" title="E46">A201</font></i></a>, <a class="ref" href="nat_1.html#T6" title="NAT_1:th.6">NAT_1:6</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c19">ok</font> = <font color="Maroon" title="c18">ok</font> as    <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>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c20">g</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">ok</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> as   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="numeral1.html#T3" target="_self" title="NUMERAL1:th.3">Th3</a></i>;<br/><a NAME="E20:9_1_2_1_8_1_4"/>
 <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="c2">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><a NAME="E21:9_1_2_1_8_1_4"/><b>then </b>
<font color="Maroon" title="c13">o</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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="c2">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="ordinal1.html#D2" title="ORDINAL1:def.2">ORDINAL1:def 2</a></i>;<br/><a NAME="E22:9_1_2_1_8_1_4"/><b>then </b><i><font color="Green" title="E57">A212</font></i>: 
 <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c20">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">o</font>
 <b>by </b><i><a class="txt" href="numeral1.html#E17:9_1_2_1_8_1_4"><i><font color="Green" title="E56">A211</font></i></a>, <a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a></i>;<br/><a NAME="E23:9_1_2_1_8_1_4"/><i><font color="Green" title="E58">A213</font></i>: 
1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="numeral1.html#E1:9"><i><font color="Green" title="E9">A1</font></i></a></i>;<br/><i><font color="Green" title="E59">A214</font></i>:  <a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <font color="Maroon" title="c20">g</font> = 
<span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">ok</font>
<b>by </b><i><a class="ref" href="numeral1.html#T2" target="_self" title="NUMERAL1:th.2">Th2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">ok</font>
<b>by </b><i><a class="ref" href="series_1.html#T12" title="SERIES_1:th.12">SERIES_1:12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">ok</font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T13" title="SEQ_1:th.13">SEQ_1:13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p4">(<span class="default"><font color="Maroon" title="c19">ok</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="numeral1.html#E1:9"><i><font color="Green" title="E9">A1</font></i></a>, <a class="ref" href="series_1.html#T26" title="SERIES_1:th.26">SERIES_1:26</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p4">(<span class="default"><font color="Maroon" title="c19">ok</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="numeral1.html#E1:9"><i><font color="Green" title="E9">A1</font></i></a>, <a class="ref" href="power.html#T46" title="POWER:th.46">POWER:46</a></i>
<br/>.= 
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="numeral1.html#E17:9_1_2_1_8_1_4"><i><font color="Green" title="E56">A211</font></i></a></i>
<br/>.= 
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="numeral1.html#E23:9_1_2_1_8_1_4"><i><font color="Green" title="E58">A213</font></i></a>, <a class="ref" href="xcmplx_1.html#T88" title="XCMPLX_1:th.88">XCMPLX_1:88</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1

;<br/><a NAME="E25:9_1_2_1_8_1_4"/><i><font color="Green" title="E60">A215</font></i>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c16">do</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="numeral1.html#E1:9_1_2_1_8_1_4"><i><font color="Green" title="E46">A201</font></i></a>, <a class="txt" href="numeral1.html#E8:9_1_2_1_8_1_4"><i><font color="Green" title="E51">A206</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E26:9_1_2_1_8_1_4"/><i><font color="Green" title="E61">A216</font></i>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c16">do</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c20">g</font>
 <b>by </b><i><a class="txt" href="numeral1.html#E8:9_1_2_1_8_1_4"><i><font color="Green" title="E51">A206</font></i></a>, <a class="txt" href="numeral1.html#E22:9_1_2_1_8_1_4"><i><font color="Green" title="E57">A212</font></i></a>, <a class="ref" href="afinsq_1.html#D1" title="AFINSQ_1:def.1">AFINSQ_1:def 1</a></i>;<br/><a NAME="E27:9_1_2_1_8_1_4"/>
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c16">do</font> holds <br/><font color="Maroon" title="c16">do</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">g</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">i</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_2_1_8_1_4_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c21">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>reconsider </b><font color="Maroon" title="c22">I</font> = <font color="Maroon" title="c21">i</font> as    <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>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<b>assume </b><a NAME="E2:9_1_2_1_8_1_4_3"/><i><font color="Green" title="E62">A217</font></i>: 
<font color="Maroon" title="c21">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c16">do</font>
 ;<br/>

<a NAME="E3:9_1_2_1_8_1_4_3"/><b>then </b>
<font color="Maroon" title="c21">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">o</font>
 
<b>by </b><i><a class="txt" href="numeral1.html#E6:9_1_2_1_8"><i><font color="Green" title="E44">A75</font></i></a>, <a class="txt" href="numeral1.html#E7:9_1_2_1_8_1_4"><i><font color="Green" title="E50">A205</font></i></a>, <a class="ref" href="afinsq_1.html#T14" title="AFINSQ_1:th.14">AFINSQ_1:14</a></i>;<br/>
<b>then </b><i><font color="Green" title="E63">A218</font></i>: <font color="Maroon" title="c20">g</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c22">I</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">I</font>
<b>by </b><i><a class="txt" href="numeral1.html#E22:9_1_2_1_8_1_4"><i><font color="Green" title="E57">A212</font></i></a>, <a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">I</font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T13" title="SEQ_1:th.13">SEQ_1:13</a></i>
<br/>.= 
<font color="Maroon" title="c3">b1</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c22">I</font></span>)</span>
<b>by </b><i><a class="ref" href="prepower.html#D1" title="PREPOWER:def.1">PREPOWER:def 1</a></i>
;<br/>
<a NAME="E5:9_1_2_1_8_1_4_3"/><i><font color="Green" title="E64">A219</font></i>: 
 <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c16">do</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c6">D</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T89" title="RELAT_1:th.89">RELAT_1:89</a></i>;<br/>
<a NAME="E6:9_1_2_1_8_1_4_3"/><b>then </b>
<font color="Maroon" title="c4">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c21">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">b1</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 
<b>by </b><i><a class="txt" href="numeral1.html#E3:9_1_2_1"><i><font color="Green" title="E11">A36</font></i></a>, <a class="txt" href="numeral1.html#E6:9_1_2_1"><i><font color="Green" title="E13">A38</font></i></a>, <a class="txt" href="numeral1.html#E2:9_1_2_1_8_1_4_3"><i><font color="Green" title="E62">A217</font></i></a></i>;<br/>
<a NAME="E7:9_1_2_1_8_1_4_3"/><b>then </b><i><font color="Green" title="E65">A220</font></i>: 
<font color="Maroon" title="c4">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c21">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">b1</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<font color="Maroon" title="c16">do</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c21">i</font> = 
<font color="Maroon" title="c6">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c21">i</font>
<b>by </b><i><a class="txt" href="numeral1.html#E2:9_1_2_1_8_1_4_3"><i><font color="Green" title="E62">A217</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c21">i</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c21">i</font></span>)</span>
<b>by </b><i><a class="txt" href="numeral1.html#E6:9_1_2_1"><i><font color="Green" title="E13">A38</font></i></a>, <a class="txt" href="numeral1.html#E2:9_1_2_1_8_1_4_3"><i><font color="Green" title="E62">A217</font></i></a>, <a class="txt" href="numeral1.html#E5:9_1_2_1_8_1_4_3"><i><font color="Green" title="E64">A219</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E9:9_1_2_1_8_1_4_3"/>
<font color="Maroon" title="c16">do</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c21">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">g</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c21">i</font>
 <b>by </b><i><a class="txt" href="numeral1.html#E4:9_1_2_1_8_1_4_3"><i><font color="Green" title="E63">A218</font></i></a>, <a class="txt" href="numeral1.html#E7:9_1_2_1_8_1_4_3"><i><font color="Green" title="E65">A220</font></i></a>, <a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a></i>;<br/>


</div><b>end;</b></div><a NAME="E28:9_1_2_1_8_1_4"/><b>then </b>
 <a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <font color="Maroon" title="c16">do</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <b>by </b><i><a class="txt" href="numeral1.html#E24:9_1_2_1_8_1_4"><i><font color="Green" title="E59">A214</font></i></a>, <a class="txt" href="numeral1.html#E25:9_1_2_1_8_1_4"><i><font color="Green" title="E60">A215</font></i></a>, <a class="txt" href="numeral1.html#E26:9_1_2_1_8_1_4"><i><font color="Green" title="E61">A216</font></i></a>, <a class="ref" href="numeral1.html#T4" target="_self" title="NUMERAL1:th.4">Th4</a></i>;<br/><a NAME="E29:9_1_2_1_8_1_4"/><b>then </b>
 <a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <font color="Maroon" title="c16">do</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T147" title="XREAL_1:th.147">XREAL_1:147</a></i>;<br/><a NAME="E30:9_1_2_1_8_1_4"/><b>then </b><i><font color="Green" title="E62">A221</font></i>: 
<span class="p1">(<span class="default"><a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <font color="Maroon" title="c16">do</font></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="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="nat_d.html#T27" title="NAT_D:th.27">NAT_D:27</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_2_1_8_1_4_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c21">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>reconsider </b><font color="Maroon" title="c22">K</font> = <font color="Maroon" title="c21">k</font> as    <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>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><b>assume </b><a NAME="E2:9_1_2_1_8_1_4_4"/><i><font color="Green" title="E63">A222</font></i>: 
<font color="Maroon" title="c21">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c15">d'</font>
 ;<br/><a NAME="E3:9_1_2_1_8_1_4_4"/><b>then </b>
<font color="Maroon" title="c21">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c15">d'</font>
 <b>by </b><i><a class="ref" href="afinsq_1.html#D1" title="AFINSQ_1:def.1">AFINSQ_1:def 1</a></i>;<br/><a NAME="E4:9_1_2_1_8_1_4_4"/><b>then </b>
<font color="Maroon" title="c21">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c15">d'</font>
 <b>by </b><i><a class="ref" href="prgcor_2.html#T1" title="PRGCOR_2:th.1">PRGCOR_2:1</a></i>;<br/><a NAME="E5:9_1_2_1_8_1_4_4"/><b>then </b>
<font color="Maroon" title="c13">o</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c21">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c6">D</font>
 <b>by </b><i><a class="txt" href="numeral1.html#E8:9_1_2_1_8_1_4"><i><font color="Green" title="E51">A206</font></i></a>, <a class="txt" href="numeral1.html#E9:9_1_2_1_8_1_4"><i><font color="Green" title="E52">A207</font></i></a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/><a NAME="E6:9_1_2_1_8_1_4_4"/><b>then </b>
<font color="Maroon" title="c13">o</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c21">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c6">D</font>
 <b>by </b><i><a class="ref" href="prgcor_2.html#T1" title="PRGCOR_2:th.1">PRGCOR_2:1</a></i>;<br/><a NAME="E7:9_1_2_1_8_1_4_4"/><b>then </b>
<font color="Maroon" title="c6">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">o</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c21">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">o</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c21">k</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">o</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c21">k</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="numeral1.html#E6:9_1_2_1"><i><font color="Green" title="E13">A38</font></i></a></i>;<br/><b>then </b><font color="Maroon" title="c15">d'</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c22">K</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">o</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c21">k</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">o</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c21">k</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="numeral1.html#E4:9_1_2_1_8_1_4"><i><font color="Green" title="E48">A203</font></i></a>, <a class="txt" href="numeral1.html#E2:9_1_2_1_8_1_4_4"><i><font color="Green" title="E63">A222</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">o</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c21">k</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c21">k</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T13" title="NEWTON:th.13">NEWTON:13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">o</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c21">k</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c21">k</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span>

;<br/><b>hence </b><a NAME="E9:9_1_2_1_8_1_4_4"/>
<font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c15">d'</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c21">k</font>
 <b>by </b><i><a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">NAT_D:def 3</a></i>;<br/></div><b>end;</b></div><a NAME="E32:9_1_2_1_8_1_4"/><b>then </b><i><font color="Green" title="E63">A223</font></i>: 
<font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a>  <a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <font color="Maroon" title="c15">d'</font>
 <b>by </b><i><a class="ref" href="numeral1.html#T5" target="_self" title="NUMERAL1:th.5">Th5</a></i>;<br/><a NAME="E33:9_1_2_1_8_1_4"/><b>then </b>
<span class="p1">(<span class="default"><a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <font color="Maroon" title="c15">d'</font></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="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="numeral1.html#E15:9_1_2_1_8_1_4"><i><font color="Green" title="E55">A210</font></i></a>, <a class="ref" href="pepin.html#T6" title="PEPIN:th.6">PEPIN:6</a></i>;<br/><a NAME="E34:9_1_2_1_8_1_4"/><b>then </b>
<font color="Maroon" title="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <font color="Maroon" title="c15">d'</font></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="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></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"><a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <font color="Maroon" title="c16">do</font></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="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="numeral1.html#E13:9_1_2_1_8_1_4"><i><font color="Green" title="E54">A209</font></i></a>, <a class="ref" href="nat_d.html#T19" title="NAT_D:th.19">NAT_D:19</a></i>;<br/><a NAME="E35:9_1_2_1_8_1_4"/><b>then </b><i><font color="Green" title="E64">A224</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <font color="Maroon" title="c15">d'</font>
 <b>by </b><i><a class="txt" href="numeral1.html#E30:9_1_2_1_8_1_4"><i><font color="Green" title="E62">A221</font></i></a>, <a class="txt" href="numeral1.html#E32:9_1_2_1_8_1_4"><i><font color="Green" title="E63">A223</font></i></a>, <a class="ref" href="nat_d.html#T3" title="NAT_D:th.3">NAT_D:3</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c21">o1</font> = <font color="Maroon" title="c13">o</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 as    <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/><a NAME="E37:9_1_2_1_8_1_4"/><i><font color="Green" title="E65">A225</font></i>: 
( <font color="Maroon" title="c21">o1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">k</font> &amp; <font color="Maroon" title="c7">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c4">d</font> )
 <b>by </b><i><a class="txt" href="numeral1.html#E1:9_1_2_1_8_1_4"><i><font color="Green" title="E46">A201</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a>, <a class="ref" href="xreal_1.html#T149" title="XREAL_1:th.149">XREAL_1:149</a></i>;<br/><a NAME="E38:9_1_2_1_8_1_4"/><b>then </b><i><font color="Green" title="E66">A226</font></i>: 
<font color="Maroon" title="c21">o1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c4">d</font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c22">oo1</font> = <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c4">d</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c21">o1</font> as    <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>by </b><i><a class="txt" href="numeral1.html#E37:9_1_2_1_8_1_4"><i><font color="Green" title="E65">A225</font></i></a>, <a class="ref" href="nat_1.html#T21" title="NAT_1:th.21">NAT_1:21</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>2</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> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>$2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default">$1 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c21">o1</font></span>)</span>;<br/><a NAME="E40:9_1_2_1_8_1_4"/><i><font color="Green" title="E67">A227</font></i>: 
for <font color="Olive" title="b1">u</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">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c22">oo1</font> holds <br/> ex <font color="Olive" title="b2">x</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>2</sub></font>[<font color="Olive" title="b1">u</font>,<font color="Olive" title="b2">x</font>]
 ;<br/><b>consider </b><font color="Maroon" title="c23">d'1</font> being   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E42:9_1_2_1_8_1_4"/><i><font color="Green" title="E68">A228</font></i>: 
(  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c23">d'1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">oo1</font> &amp; ( for <font color="Olive" title="b1">x</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">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c22">oo1</font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c23">d'1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">x</font>] ) )
 <b>from </b><i><a class="ref" href="stirl2_1.html#S6" title="STIRL2_1:sch.6">STIRL2_1:sch 6</a>(<a class="txt" href="numeral1.html#E40:9_1_2_1_8_1_4"><i><font color="Green" title="E67">A227</font></i></a>);<br/></i><b>reconsider </b><font color="Maroon" title="c24">do1</font> = <font color="Maroon" title="c6">D</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c21">o1</font> as   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="afinsq_1.html#T16" title="AFINSQ_1:th.16">AFINSQ_1:16</a></i>;<br/><a NAME="E44:9_1_2_1_8_1_4"/><i><font color="Green" title="E69">A229</font></i>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c23">d'1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">oo1</font>
 <b>by </b><i><a class="txt" href="numeral1.html#E42:9_1_2_1_8_1_4"><i><font color="Green" title="E68">A228</font></i></a>, <a class="ref" href="afinsq_1.html#D1" title="AFINSQ_1:def.1">AFINSQ_1:def 1</a></i>;<br/><i><font color="Green" title="E70">A230</font></i>:  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c4">d</font> = 
 <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c6">D</font>
<b>by </b><i><a class="txt" href="numeral1.html#E6:9_1_2_1"><i><font color="Green" title="E13">A38</font></i></a>, <a class="ref" href="afinsq_1.html#D1" title="AFINSQ_1:def.1">AFINSQ_1:def 1</a></i>
<br/>.= 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c6">D</font>
<b>by </b><i><a class="ref" href="afinsq_1.html#D1" title="AFINSQ_1:def.1">AFINSQ_1:def 1</a></i>
;<br/><a NAME="E46:9_1_2_1_8_1_4"/><b>then </b><i><font color="Green" title="E71">A231</font></i>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c24">do1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">o1</font>
 <b>by </b><i><a class="txt" href="numeral1.html#E38:9_1_2_1_8_1_4"><i><font color="Green" title="E66">A226</font></i></a>, <a class="ref" href="afinsq_1.html#T14" title="AFINSQ_1:th.14">AFINSQ_1:14</a></i>;<br/><a NAME="E47:9_1_2_1_8_1_4"/><b>then </b><i><font color="Green" title="E72">A232</font></i>: 
 <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c6">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c24">do1</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c23">d'1</font></span>)</span>
 <b>by </b><i><a class="txt" href="numeral1.html#E6:9_1_2_1"><i><font color="Green" title="E13">A38</font></i></a>, <a class="txt" href="numeral1.html#E44:9_1_2_1_8_1_4"><i><font color="Green" title="E69">A229</font></i></a>, <a class="ref" href="afinsq_1.html#D1" title="AFINSQ_1:def.1">AFINSQ_1:def 1</a></i>;<br/><a NAME="E48:9_1_2_1_8_1_4"/><i><font color="Green" title="E73">A233</font></i>: 
for <font color="Olive" title="b1">x</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">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c24">do1</font> holds <br/><font color="Maroon" title="c6">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">do1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 <b>by </b><i><a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>;<br/><a NAME="E49:9_1_2_1_8_1_4"/>
for <font color="Olive" title="b1">x</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">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c23">d'1</font> holds <br/><font color="Maroon" title="c6">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c24">do1</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">d'1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 <b>by </b><i><a class="txt" href="numeral1.html#E42:9_1_2_1_8_1_4"><i><font color="Green" title="E68">A228</font></i></a>, <a class="txt" href="numeral1.html#E46:9_1_2_1_8_1_4"><i><font color="Green" title="E71">A231</font></i></a></i>;<br/><a NAME="E50:9_1_2_1_8_1_4"/><b>then </b>
<font color="Maroon" title="c6">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">do1</font> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <font color="Maroon" title="c23">d'1</font>
 <b>by </b><i><a class="txt" href="numeral1.html#E47:9_1_2_1_8_1_4"><i><font color="Green" title="E72">A232</font></i></a>, <a class="txt" href="numeral1.html#E48:9_1_2_1_8_1_4"><i><font color="Green" title="E73">A233</font></i></a>, <a class="ref" href="afinsq_1.html#D4" title="AFINSQ_1:def.4">AFINSQ_1:def 4</a></i>;<br/><a NAME="E51:9_1_2_1_8_1_4"/><b>then </b><i><font color="Green" title="E74">A234</font></i>: 
<font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <font color="Maroon" title="c24">do1</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <font color="Maroon" title="c23">d'1</font></span>)</span>
 <b>by </b><i><a class="txt" href="numeral1.html#E6:9_1_2_1"><i><font color="Green" title="E13">A38</font></i></a>, <a class="ref" href="numeral1.html#T1" target="_self" title="NUMERAL1:th.1">Th1</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c25">bo1</font> = <font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c21">o1</font> as    <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>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><a NAME="E53:9_1_2_1_8_1_4"/><i><font color="Green" title="E75">A235</font></i>: 
<font color="Maroon" title="c25">bo1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="numeral1.html#E1:9"><i><font color="Green" title="E9">A1</font></i></a>, <a class="ref" href="prepower.html#T12" title="PREPOWER:th.12">PREPOWER:12</a></i>;<br/><b>consider </b><font color="Maroon" title="c26">ok1</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E55:9_1_2_1_8_1_4"/><i><font color="Green" title="E76">A236</font></i>: 
<font color="Maroon" title="c26">ok1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">o1</font>
 ;<br/><b>reconsider </b><font color="Maroon" title="c27">ok1</font> = <font color="Maroon" title="c26">ok1</font> as    <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>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c28">g1</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c27">ok1</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> as   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="numeral1.html#T3" target="_self" title="NUMERAL1:th.3">Th3</a></i>;<br/><a NAME="E58:9_1_2_1_8_1_4"/>
 <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="c2">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><a NAME="E59:9_1_2_1_8_1_4"/><b>then </b>
<font color="Maroon" title="c21">o1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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="c2">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="ordinal1.html#D2" title="ORDINAL1:def.2">ORDINAL1:def 2</a></i>;<br/><a NAME="E60:9_1_2_1_8_1_4"/><b>then </b><i><font color="Green" title="E77">A237</font></i>: 
 <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c28">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">o1</font>
 <b>by </b><i><a class="txt" href="numeral1.html#E55:9_1_2_1_8_1_4"><i><font color="Green" title="E76">A236</font></i></a>, <a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a></i>;<br/><a NAME="E61:9_1_2_1_8_1_4"/><i><font color="Green" title="E78">A238</font></i>: 
1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="numeral1.html#E1:9"><i><font color="Green" title="E9">A1</font></i></a></i>;<br/><i><font color="Green" title="E79">A239</font></i>:  <a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <font color="Maroon" title="c28">g1</font> = 
<span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c27">ok1</font>
<b>by </b><i><a class="ref" href="numeral1.html#T2" target="_self" title="NUMERAL1:th.2">Th2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c27">ok1</font>
<b>by </b><i><a class="ref" href="series_1.html#T12" title="SERIES_1:th.12">SERIES_1:12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c27">ok1</font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T13" title="SEQ_1:th.13">SEQ_1:13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p4">(<span class="default"><font color="Maroon" title="c27">ok1</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="numeral1.html#E1:9"><i><font color="Green" title="E9">A1</font></i></a>, <a class="ref" href="series_1.html#T26" title="SERIES_1:th.26">SERIES_1:26</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p4">(<span class="default"><font color="Maroon" title="c27">ok1</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="numeral1.html#E1:9"><i><font color="Green" title="E9">A1</font></i></a>, <a class="ref" href="power.html#T46" title="POWER:th.46">POWER:46</a></i>
<br/>.= 
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c21">o1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="numeral1.html#E55:9_1_2_1_8_1_4"><i><font color="Green" title="E76">A236</font></i></a></i>
<br/>.= 
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c21">o1</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="numeral1.html#E61:9_1_2_1_8_1_4"><i><font color="Green" title="E78">A238</font></i></a>, <a class="ref" href="xcmplx_1.html#T88" title="XCMPLX_1:th.88">XCMPLX_1:88</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c21">o1</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1

;<br/><a NAME="E63:9_1_2_1_8_1_4"/><i><font color="Green" title="E80">A240</font></i>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c24">do1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="numeral1.html#E46:9_1_2_1_8_1_4"><i><font color="Green" title="E71">A231</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E64:9_1_2_1_8_1_4"/><i><font color="Green" title="E81">A241</font></i>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c24">do1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c28">g1</font>
 <b>by </b><i><a class="txt" href="numeral1.html#E46:9_1_2_1_8_1_4"><i><font color="Green" title="E71">A231</font></i></a>, <a class="txt" href="numeral1.html#E60:9_1_2_1_8_1_4"><i><font color="Green" title="E77">A237</font></i></a>, <a class="ref" href="afinsq_1.html#D1" title="AFINSQ_1:def.1">AFINSQ_1:def 1</a></i>;<br/><a NAME="E65:9_1_2_1_8_1_4"/>
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c24">do1</font> holds <br/><font color="Maroon" title="c24">do1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">g1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">i</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_2_1_8_1_4_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c29">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>reconsider </b><font color="Maroon" title="c30">I</font> = <font color="Maroon" title="c29">i</font> as    <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>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<b>assume </b><a NAME="E2:9_1_2_1_8_1_4_7"/><i><font color="Green" title="E82">A242</font></i>: 
<font color="Maroon" title="c29">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c24">do1</font>
 ;<br/>

<a NAME="E3:9_1_2_1_8_1_4_7"/><b>then </b>
<font color="Maroon" title="c29">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c21">o1</font>
 
<b>by </b><i><a class="txt" href="numeral1.html#E38:9_1_2_1_8_1_4"><i><font color="Green" title="E66">A226</font></i></a>, <a class="txt" href="numeral1.html#E45:9_1_2_1_8_1_4"><i><font color="Green" title="E70">A230</font></i></a>, <a class="ref" href="afinsq_1.html#T14" title="AFINSQ_1:th.14">AFINSQ_1:14</a></i>;<br/>
<b>then </b><i><font color="Green" title="E83">A243</font></i>: <font color="Maroon" title="c28">g1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c30">I</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">I</font>
<b>by </b><i><a class="txt" href="numeral1.html#E60:9_1_2_1_8_1_4"><i><font color="Green" title="E77">A237</font></i></a>, <a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">I</font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T13" title="SEQ_1:th.13">SEQ_1:13</a></i>
<br/>.= 
<font color="Maroon" title="c3">b1</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c30">I</font></span>)</span>
<b>by </b><i><a class="ref" href="prepower.html#D1" title="PREPOWER:def.1">PREPOWER:def 1</a></i>
;<br/>
<a NAME="E5:9_1_2_1_8_1_4_7"/><i><font color="Green" title="E84">A244</font></i>: 
 <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c24">do1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c6">D</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T89" title="RELAT_1:th.89">RELAT_1:89</a></i>;<br/>
<a NAME="E6:9_1_2_1_8_1_4_7"/><b>then </b>
<font color="Maroon" title="c4">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c29">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">b1</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 
<b>by </b><i><a class="txt" href="numeral1.html#E3:9_1_2_1"><i><font color="Green" title="E11">A36</font></i></a>, <a class="txt" href="numeral1.html#E6:9_1_2_1"><i><font color="Green" title="E13">A38</font></i></a>, <a class="txt" href="numeral1.html#E2:9_1_2_1_8_1_4_7"><i><font color="Green" title="E82">A242</font></i></a></i>;<br/>
<a NAME="E7:9_1_2_1_8_1_4_7"/><b>then </b><i><font color="Green" title="E85">A245</font></i>: 
<font color="Maroon" title="c4">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c29">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">b1</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<font color="Maroon" title="c24">do1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c29">i</font> = 
<font color="Maroon" title="c6">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c29">i</font>
<b>by </b><i><a class="txt" href="numeral1.html#E2:9_1_2_1_8_1_4_7"><i><font color="Green" title="E82">A242</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c29">i</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c29">i</font></span>)</span>
<b>by </b><i><a class="txt" href="numeral1.html#E6:9_1_2_1"><i><font color="Green" title="E13">A38</font></i></a>, <a class="txt" href="numeral1.html#E2:9_1_2_1_8_1_4_7"><i><font color="Green" title="E82">A242</font></i></a>, <a class="txt" href="numeral1.html#E5:9_1_2_1_8_1_4_7"><i><font color="Green" title="E84">A244</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E9:9_1_2_1_8_1_4_7"/>
<font color="Maroon" title="c24">do1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c29">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">g1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c29">i</font>
 <b>by </b><i><a class="txt" href="numeral1.html#E4:9_1_2_1_8_1_4_7"><i><font color="Green" title="E83">A243</font></i></a>, <a class="txt" href="numeral1.html#E7:9_1_2_1_8_1_4_7"><i><font color="Green" title="E85">A245</font></i></a>, <a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a></i>;<br/>


</div><b>end;</b></div><a NAME="E66:9_1_2_1_8_1_4"/><b>then </b>
 <a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <font color="Maroon" title="c24">do1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c21">o1</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <b>by </b><i><a class="txt" href="numeral1.html#E62:9_1_2_1_8_1_4"><i><font color="Green" title="E79">A239</font></i></a>, <a class="txt" href="numeral1.html#E63:9_1_2_1_8_1_4"><i><font color="Green" title="E80">A240</font></i></a>, <a class="txt" href="numeral1.html#E64:9_1_2_1_8_1_4"><i><font color="Green" title="E81">A241</font></i></a>, <a class="ref" href="numeral1.html#T4" target="_self" title="NUMERAL1:th.4">Th4</a></i>;<br/><a NAME="E67:9_1_2_1_8_1_4"/><b>then </b>
 <a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <font color="Maroon" title="c24">do1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c21">o1</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T147" title="XREAL_1:th.147">XREAL_1:147</a></i>;<br/><a NAME="E68:9_1_2_1_8_1_4"/><b>then </b><i><font color="Green" title="E82">A246</font></i>: 
<span class="p1">(<span class="default"><a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <font color="Maroon" title="c24">do1</font></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="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c21">o1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="nat_d.html#T27" title="NAT_D:th.27">NAT_D:27</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_2_1_8_1_4_8"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c29">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>reconsider </b><font color="Maroon" title="c30">K</font> = <font color="Maroon" title="c29">k</font> as    <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>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><b>assume </b><a NAME="E2:9_1_2_1_8_1_4_8"/><i><font color="Green" title="E83">A247</font></i>: 
<font color="Maroon" title="c29">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c23">d'1</font>
 ;<br/><a NAME="E3:9_1_2_1_8_1_4_8"/><b>then </b>
<font color="Maroon" title="c29">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c23">d'1</font>
 <b>by </b><i><a class="ref" href="afinsq_1.html#D1" title="AFINSQ_1:def.1">AFINSQ_1:def 1</a></i>;<br/><a NAME="E4:9_1_2_1_8_1_4_8"/><b>then </b>
<font color="Maroon" title="c29">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c23">d'1</font>
 <b>by </b><i><a class="ref" href="prgcor_2.html#T1" title="PRGCOR_2:th.1">PRGCOR_2:1</a></i>;<br/><a NAME="E5:9_1_2_1_8_1_4_8"/><b>then </b>
<font color="Maroon" title="c21">o1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c29">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c6">D</font>
 <b>by </b><i><a class="txt" href="numeral1.html#E46:9_1_2_1_8_1_4"><i><font color="Green" title="E71">A231</font></i></a>, <a class="txt" href="numeral1.html#E47:9_1_2_1_8_1_4"><i><font color="Green" title="E72">A232</font></i></a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/><a NAME="E6:9_1_2_1_8_1_4_8"/><b>then </b>
<font color="Maroon" title="c21">o1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c29">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c6">D</font>
 <b>by </b><i><a class="ref" href="prgcor_2.html#T1" title="PRGCOR_2:th.1">PRGCOR_2:1</a></i>;<br/><a NAME="E7:9_1_2_1_8_1_4_8"/><b>then </b>
<font color="Maroon" title="c6">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">o1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c29">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">o1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c29">k</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">o1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c29">k</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="numeral1.html#E6:9_1_2_1"><i><font color="Green" title="E13">A38</font></i></a></i>;<br/><b>then </b><font color="Maroon" title="c23">d'1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c30">K</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">o1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c29">k</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">o1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c29">k</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="numeral1.html#E42:9_1_2_1_8_1_4"><i><font color="Green" title="E68">A228</font></i></a>, <a class="txt" href="numeral1.html#E2:9_1_2_1_8_1_4_8"><i><font color="Green" title="E83">A247</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">o1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c29">k</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c21">o1</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c29">k</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T13" title="NEWTON:th.13">NEWTON:13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c21">o1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c29">k</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c29">k</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c21">o1</font></span>)</span>

;<br/><b>hence </b><a NAME="E9:9_1_2_1_8_1_4_8"/>
<font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c21">o1</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c23">d'1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c29">k</font>
 <b>by </b><i><a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">NAT_D:def 3</a></i>;<br/></div><b>end;</b></div><a NAME="E70:9_1_2_1_8_1_4"/><b>then </b><i><font color="Green" title="E83">A248</font></i>: 
<font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c21">o1</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a>  <a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <font color="Maroon" title="c23">d'1</font>
 <b>by </b><i><a class="ref" href="numeral1.html#T5" target="_self" title="NUMERAL1:th.5">Th5</a></i>;<br/><a NAME="E71:9_1_2_1_8_1_4"/><b>then </b>
<span class="p1">(<span class="default"><a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <font color="Maroon" title="c23">d'1</font></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="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c21">o1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="numeral1.html#E53:9_1_2_1_8_1_4"><i><font color="Green" title="E75">A235</font></i></a>, <a class="ref" href="pepin.html#T6" title="PEPIN:th.6">PEPIN:6</a></i>;<br/><a NAME="E72:9_1_2_1_8_1_4"/><b>then </b>
<font color="Maroon" title="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c21">o1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <font color="Maroon" title="c23">d'1</font></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="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c21">o1</font></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"><a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <font color="Maroon" title="c24">do1</font></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="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c21">o1</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="numeral1.html#E51:9_1_2_1_8_1_4"><i><font color="Green" title="E74">A234</font></i></a>, <a class="ref" href="nat_d.html#T19" title="NAT_D:th.19">NAT_D:19</a></i>;<br/><a NAME="E73:9_1_2_1_8_1_4"/><b>then </b><i><font color="Green" title="E84">A249</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c21">o1</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c21">o1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <font color="Maroon" title="c23">d'1</font>
 <b>by </b><i><a class="txt" href="numeral1.html#E68:9_1_2_1_8_1_4"><i><font color="Green" title="E82">A246</font></i></a>, <a class="txt" href="numeral1.html#E70:9_1_2_1_8_1_4"><i><font color="Green" title="E83">A248</font></i></a>, <a class="ref" href="nat_d.html#T3" title="NAT_D:th.3">NAT_D:3</a></i>;<br/><i><font color="Green" title="E85">A250</font></i>:  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c15">d'</font> = 
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c23">d'1</font></span>)</span>
<b>by </b><i><a class="txt" href="numeral1.html#E4:9_1_2_1_8_1_4"><i><font color="Green" title="E48">A203</font></i></a>, <a class="txt" href="numeral1.html#E42:9_1_2_1_8_1_4"><i><font color="Green" title="E68">A228</font></i></a></i>
<br/>.= 
1 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c23">d'1</font></span>)</span>
<b>by </b><i><a class="ref" href="afinsq_1.html#D1" title="AFINSQ_1:def.1">AFINSQ_1:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p2"><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">&lt;%</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">%&gt;</a></span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c23">d'1</font></span>)</span>
<b>by </b><i><a class="ref" href="afinsq_1.html#T38" title="AFINSQ_1:th.38">AFINSQ_1:38</a></i>
;<br/><div><i><font color="Green" title="E86">A251</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_2_1_8_1_4_10"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c29">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="E1:9_1_2_1_8_1_4_10"/>
<font color="Maroon" title="c29">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <span class="p1"><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">%&gt;</a></span>
 ;<br/><a NAME="E2:9_1_2_1_8_1_4_10"/><b>then </b>
<font color="Maroon" title="c29">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 1
 <b>by </b><i><a class="ref" href="afinsq_1.html#T36" title="AFINSQ_1:th.36">AFINSQ_1:36</a></i>;<br/><a NAME="E3:9_1_2_1_8_1_4_10"/><b>then </b>
<font color="Maroon" title="c29">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1
 <b>by </b><i><a class="ref" href="prgcor_2.html#T1" title="PRGCOR_2:th.1">PRGCOR_2:1</a></i>;<br/><a NAME="E4:9_1_2_1_8_1_4_10"/><b>then </b><i><font color="Green" title="E87">A252</font></i>: 
<font color="Maroon" title="c29">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></i>;<br/><a NAME="E5:9_1_2_1_8_1_4_10"/>
<font color="Maroon" title="c14">oo</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="numeral1.html#E6:9_1_2_1_8"><i><font color="Green" title="E44">A75</font></i></a>, <a class="ref" href="xreal_1.html#T52" title="XREAL_1:th.52">XREAL_1:52</a></i>;<br/><a NAME="E6:9_1_2_1_8_1_4_10"/><b>then </b>
<font color="Maroon" title="c29">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c15">d'</font>
 <b>by </b><i><a class="txt" href="numeral1.html#E4:9_1_2_1_8_1_4"><i><font color="Green" title="E48">A203</font></i></a>, <a class="txt" href="numeral1.html#E4:9_1_2_1_8_1_4_10"><i><font color="Green" title="E87">A252</font></i></a>, <a class="ref" href="prgcor_2.html#T1" title="PRGCOR_2:th.1">PRGCOR_2:1</a></i>;<br/><b>hence </b><font color="Maroon" title="c15">d'</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c29">k</font> = 
<font color="Maroon" title="c6">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c29">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c13">o</font></span>)</span>
<b>by </b><i><a class="txt" href="numeral1.html#E4:9_1_2_1_8_1_4"><i><font color="Green" title="E48">A203</font></i></a></i>
<br/>.= 
<span class="p1"><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">%&gt;</a></span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c29">k</font>
<b>by </b><i><a class="txt" href="numeral1.html#E4:9_1_2_1_8_1_4_10"><i><font color="Green" title="E87">A252</font></i></a>, <a class="ref" href="afinsq_1.html#D5" title="AFINSQ_1:def.5">AFINSQ_1:def 5</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_2_1_8_1_4_11"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c29">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="E1:9_1_2_1_8_1_4_11"/><i><font color="Green" title="E87">A253</font></i>: 
<font color="Maroon" title="c29">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c23">d'1</font>
 ;<br/><a NAME="E2:9_1_2_1_8_1_4_11"/><b>then </b>
<font color="Maroon" title="c29">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c22">oo1</font>
 <b>by </b><i><a class="txt" href="numeral1.html#E42:9_1_2_1_8_1_4"><i><font color="Green" title="E68">A228</font></i></a>, <a class="ref" href="prgcor_2.html#T1" title="PRGCOR_2:th.1">PRGCOR_2:1</a></i>;<br/><a NAME="E3:9_1_2_1_8_1_4_11"/><b>then </b>
<font color="Maroon" title="c29">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c22">oo1</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/><a NAME="E4:9_1_2_1_8_1_4_11"/><b>then </b><i><font color="Green" title="E88">A254</font></i>: 
<font color="Maroon" title="c29">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c15">d'</font>
 <b>by </b><i><a class="txt" href="numeral1.html#E4:9_1_2_1_8_1_4"><i><font color="Green" title="E48">A203</font></i></a>, <a class="ref" href="prgcor_2.html#T1" title="PRGCOR_2:th.1">PRGCOR_2:1</a></i>;<br/><b>thus </b><font color="Maroon" title="c15">d'</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p3"><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">&lt;%</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">%&gt;</a></span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c29">k</font></span>)</span> = 
<font color="Maroon" title="c15">d'</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c29">k</font></span>)</span>
<b>by </b><i><a class="ref" href="afinsq_1.html#T37" title="AFINSQ_1:th.37">AFINSQ_1:37</a></i>
<br/>.= 
<font color="Maroon" title="c6">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c16">do</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c29">k</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="numeral1.html#E4:9_1_2_1_8_1_4"><i><font color="Green" title="E48">A203</font></i></a>, <a class="txt" href="numeral1.html#E8:9_1_2_1_8_1_4"><i><font color="Green" title="E51">A206</font></i></a>, <a class="txt" href="numeral1.html#E4:9_1_2_1_8_1_4_11"><i><font color="Green" title="E88">A254</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c6">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c24">do1</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c29">k</font></span>)</span>
<b>by </b><i><a class="txt" href="numeral1.html#E8:9_1_2_1_8_1_4"><i><font color="Green" title="E51">A206</font></i></a>, <a class="txt" href="numeral1.html#E46:9_1_2_1_8_1_4"><i><font color="Green" title="E71">A231</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c23">d'1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c29">k</font>
<b>by </b><i><a class="txt" href="numeral1.html#E42:9_1_2_1_8_1_4"><i><font color="Green" title="E68">A228</font></i></a>, <a class="txt" href="numeral1.html#E46:9_1_2_1_8_1_4"><i><font color="Green" title="E71">A231</font></i></a>, <a class="txt" href="numeral1.html#E1:9_1_2_1_8_1_4_11"><i><font color="Green" title="E87">A253</font></i></a></i>
;<br/></div><b>end;</b></div><a NAME="E77:9_1_2_1_8_1_4"/><b>then </b>
<font color="Maroon" title="c15">d'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">%&gt;</a></span> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <font color="Maroon" title="c23">d'1</font>
 <b>by </b><i><a class="txt" href="numeral1.html#E74:9_1_2_1_8_1_4"><i><font color="Green" title="E85">A250</font></i></a>, <a class="txt" href="numeral1.html#E75:9_1_2_1_8_1_4"><i><font color="Green" title="E86">A251</font></i></a>, <a class="ref" href="afinsq_1.html#D4" title="AFINSQ_1:def.4">AFINSQ_1:def 4</a></i>;<br/><a NAME="E78:9_1_2_1_8_1_4"/><b>then </b>
 <a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <font color="Maroon" title="c15">d'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <span class="p2"><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">&lt;%</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">%&gt;</a></span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <font color="Maroon" title="c23">d'1</font></span>)</span>
 <b>by </b><i><a class="ref" href="numeral1.html#T1" target="_self" title="NUMERAL1:th.1">Th1</a></i>;<br/><a NAME="E79:9_1_2_1_8_1_4"/><b>then </b>
<font color="Maroon" title="c6">D</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</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="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c21">o1</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c21">o1</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="numeral1.html#E35:9_1_2_1_8_1_4"><i><font color="Green" title="E64">A224</font></i></a>, <a class="txt" href="numeral1.html#E73:9_1_2_1_8_1_4"><i><font color="Green" title="E84">A249</font></i></a>, <a class="ref" href="stirl2_1.html#T44" title="STIRL2_1:th.44">STIRL2_1:44</a></i>;<br/><a NAME="E80:9_1_2_1_8_1_4"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <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="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c21">o1</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c21">o1</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="numeral1.html#E6:9_1_2_1"><i><font color="Green" title="E13">A38</font></i></a>, <a class="txt" href="numeral1.html#E1:9_1_2_1_8"><i><font color="Green" title="E41">A72</font></i></a></i>;<br/><a NAME="E81:9_1_2_1_8_1_4"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <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="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c21">o1</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> 1</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="newton.html#T13" title="NEWTON:th.13">NEWTON:13</a></i>;<br/><a NAME="E82:9_1_2_1_8_1_4"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c21">o1</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> 1</span>)</span></span>)</span></span>)</span>
 ;<br/><a NAME="E83:9_1_2_1_8_1_4"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <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="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c21">o1</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span>
 <b>by </b><i><a class="ref" href="newton.html#T10" title="NEWTON:th.10">NEWTON:10</a></i>;<br/><a NAME="E84:9_1_2_1_8_1_4"/><b>then </b><i><font color="Green" title="E87">A255</font></i>: 
<font color="Maroon" title="c4">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</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="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c21">o1</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span>
 <b>by </b><i><a class="txt" href="numeral1.html#E15:9_1_2_1_8_1_4"><i><font color="Green" title="E55">A210</font></i></a>, <a class="ref" href="xcmplx_1.html#T90" title="XCMPLX_1:th.90">XCMPLX_1:90</a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>3</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> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>$2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default">$1 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c13">o</font></span>)</span>;<br/><a NAME="E85:9_1_2_1_8_1_4"/><i><font color="Green" title="E88">A256</font></i>: 
for <font color="Olive" title="b1">u</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">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">oo</font> holds <br/> ex <font color="Olive" title="b2">x</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>3</sub></font>[<font color="Olive" title="b1">u</font>,<font color="Olive" title="b2">x</font>]
 ;<br/><b>consider </b><font color="Maroon" title="c29">d'</font> being   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E87:9_1_2_1_8_1_4"/><i><font color="Green" title="E89">A257</font></i>: 
(  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c29">d'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">oo</font> &amp; ( for <font color="Olive" title="b1">x</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">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">oo</font> holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c29">d'</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">x</font>] ) )
 <b>from </b><i><a class="ref" href="stirl2_1.html#S6" title="STIRL2_1:sch.6">STIRL2_1:sch 6</a>(<a class="txt" href="numeral1.html#E85:9_1_2_1_8_1_4"><i><font color="Green" title="E88">A256</font></i></a>);<br/></i><b>reconsider </b><font color="Maroon" title="c30">do</font> = <font color="Maroon" title="c8">E</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c13">o</font> as   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="afinsq_1.html#T16" title="AFINSQ_1:th.16">AFINSQ_1:16</a></i>;<br/><a NAME="E89:9_1_2_1_8_1_4"/><i><font color="Green" title="E90">A258</font></i>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c29">d'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">oo</font>
 <b>by </b><i><a class="txt" href="numeral1.html#E87:9_1_2_1_8_1_4"><i><font color="Green" title="E89">A257</font></i></a>, <a class="ref" href="afinsq_1.html#D1" title="AFINSQ_1:def.1">AFINSQ_1:def 1</a></i>;<br/><i><font color="Green" title="E91">A259</font></i>:  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c5">e</font> = 
 <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c8">E</font>
<b>by </b><i><a class="txt" href="numeral1.html#E25:9_1_2_1"><i><font color="Green" title="E20">A45</font></i></a>, <a class="ref" href="afinsq_1.html#D1" title="AFINSQ_1:def.1">AFINSQ_1:def 1</a></i>
<br/>.= 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c8">E</font>
<b>by </b><i><a class="ref" href="afinsq_1.html#D1" title="AFINSQ_1:def.1">AFINSQ_1:def 1</a></i>
;<br/><a NAME="E91:9_1_2_1_8_1_4"/><b>then </b><i><font color="Green" title="E92">A260</font></i>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c30">do</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">o</font>
 <b>by </b><i><a class="txt" href="numeral1.html#E81:9_1_2_1"><i><font color="Green" title="E39">A70</font></i></a>, <a class="txt" href="numeral1.html#E6:9_1_2_1_8"><i><font color="Green" title="E44">A75</font></i></a>, <a class="ref" href="afinsq_1.html#T14" title="AFINSQ_1:th.14">AFINSQ_1:14</a></i>;<br/><a NAME="E92:9_1_2_1_8_1_4"/><b>then </b><i><font color="Green" title="E93">A261</font></i>: 
 <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c8">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c30">do</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c29">d'</font></span>)</span>
 <b>by </b><i><a class="txt" href="numeral1.html#E25:9_1_2_1"><i><font color="Green" title="E20">A45</font></i></a>, <a class="txt" href="numeral1.html#E81:9_1_2_1"><i><font color="Green" title="E39">A70</font></i></a>, <a class="txt" href="numeral1.html#E89:9_1_2_1_8_1_4"><i><font color="Green" title="E90">A258</font></i></a>, <a class="ref" href="afinsq_1.html#D1" title="AFINSQ_1:def.1">AFINSQ_1:def 1</a></i>;<br/><a NAME="E93:9_1_2_1_8_1_4"/><i><font color="Green" title="E94">A262</font></i>: 
for <font color="Olive" title="b1">x</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">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c30">do</font> holds <br/><font color="Maroon" title="c8">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">do</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 <b>by </b><i><a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>;<br/><a NAME="E94:9_1_2_1_8_1_4"/>
for <font color="Olive" title="b1">x</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">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c29">d'</font> holds <br/><font color="Maroon" title="c8">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c30">do</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">d'</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 <b>by </b><i><a class="txt" href="numeral1.html#E87:9_1_2_1_8_1_4"><i><font color="Green" title="E89">A257</font></i></a>, <a class="txt" href="numeral1.html#E91:9_1_2_1_8_1_4"><i><font color="Green" title="E92">A260</font></i></a></i>;<br/><a NAME="E95:9_1_2_1_8_1_4"/><b>then </b>
<font color="Maroon" title="c8">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">do</font> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <font color="Maroon" title="c29">d'</font>
 <b>by </b><i><a class="txt" href="numeral1.html#E92:9_1_2_1_8_1_4"><i><font color="Green" title="E93">A261</font></i></a>, <a class="txt" href="numeral1.html#E93:9_1_2_1_8_1_4"><i><font color="Green" title="E94">A262</font></i></a>, <a class="ref" href="afinsq_1.html#D4" title="AFINSQ_1:def.4">AFINSQ_1:def 4</a></i>;<br/><a NAME="E96:9_1_2_1_8_1_4"/><b>then </b><i><font color="Green" title="E95">A263</font></i>: 
<font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <font color="Maroon" title="c30">do</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <font color="Maroon" title="c29">d'</font></span>)</span>
 <b>by </b><i><a class="txt" href="numeral1.html#E25:9_1_2_1"><i><font color="Green" title="E20">A45</font></i></a>, <a class="ref" href="numeral1.html#T1" target="_self" title="NUMERAL1:th.1">Th1</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c31">bo</font> = <font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font> as    <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>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><a NAME="E98:9_1_2_1_8_1_4"/><i><font color="Green" title="E96">A264</font></i>: 
<font color="Maroon" title="c31">bo</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="numeral1.html#E1:9"><i><font color="Green" title="E9">A1</font></i></a>, <a class="ref" href="prepower.html#T12" title="PREPOWER:th.12">PREPOWER:12</a></i>;<br/><b>consider </b><font color="Maroon" title="c32">ok</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E100:9_1_2_1_8_1_4"/><i><font color="Green" title="E97">A265</font></i>: 
<font color="Maroon" title="c32">ok</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">o</font>
 <b>by </b><i><a class="txt" href="numeral1.html#E1:9_1_2_1_8_1_4"><i><font color="Green" title="E46">A201</font></i></a>, <a class="ref" href="nat_1.html#T6" title="NAT_1:th.6">NAT_1:6</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c33">ok</font> = <font color="Maroon" title="c32">ok</font> as    <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>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c34">g</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c33">ok</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> as   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="numeral1.html#T3" target="_self" title="NUMERAL1:th.3">Th3</a></i>;<br/><a NAME="E103:9_1_2_1_8_1_4"/>
 <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="c2">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><a NAME="E104:9_1_2_1_8_1_4"/><b>then </b>
<font color="Maroon" title="c13">o</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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="c2">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="ordinal1.html#D2" title="ORDINAL1:def.2">ORDINAL1:def 2</a></i>;<br/><a NAME="E105:9_1_2_1_8_1_4"/><b>then </b><i><font color="Green" title="E98">A266</font></i>: 
 <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c34">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">o</font>
 <b>by </b><i><a class="txt" href="numeral1.html#E100:9_1_2_1_8_1_4"><i><font color="Green" title="E97">A265</font></i></a>, <a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a></i>;<br/><a NAME="E106:9_1_2_1_8_1_4"/><i><font color="Green" title="E99">A267</font></i>: 
1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="numeral1.html#E1:9"><i><font color="Green" title="E9">A1</font></i></a></i>;<br/><i><font color="Green" title="E100">A268</font></i>:  <a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <font color="Maroon" title="c34">g</font> = 
<span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c33">ok</font>
<b>by </b><i><a class="ref" href="numeral1.html#T2" target="_self" title="NUMERAL1:th.2">Th2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c33">ok</font>
<b>by </b><i><a class="ref" href="series_1.html#T12" title="SERIES_1:th.12">SERIES_1:12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c33">ok</font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T13" title="SEQ_1:th.13">SEQ_1:13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p4">(<span class="default"><font color="Maroon" title="c33">ok</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="numeral1.html#E1:9"><i><font color="Green" title="E9">A1</font></i></a>, <a class="ref" href="series_1.html#T26" title="SERIES_1:th.26">SERIES_1:26</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p4">(<span class="default"><font color="Maroon" title="c33">ok</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="numeral1.html#E1:9"><i><font color="Green" title="E9">A1</font></i></a>, <a class="ref" href="power.html#T46" title="POWER:th.46">POWER:46</a></i>
<br/>.= 
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="numeral1.html#E100:9_1_2_1_8_1_4"><i><font color="Green" title="E97">A265</font></i></a></i>
<br/>.= 
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="numeral1.html#E106:9_1_2_1_8_1_4"><i><font color="Green" title="E99">A267</font></i></a>, <a class="ref" href="xcmplx_1.html#T88" title="XCMPLX_1:th.88">XCMPLX_1:88</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1

;<br/><a NAME="E108:9_1_2_1_8_1_4"/><i><font color="Green" title="E101">A269</font></i>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c30">do</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="numeral1.html#E1:9_1_2_1_8_1_4"><i><font color="Green" title="E46">A201</font></i></a>, <a class="txt" href="numeral1.html#E91:9_1_2_1_8_1_4"><i><font color="Green" title="E92">A260</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E109:9_1_2_1_8_1_4"/><i><font color="Green" title="E102">A270</font></i>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c30">do</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c34">g</font>
 <b>by </b><i><a class="txt" href="numeral1.html#E91:9_1_2_1_8_1_4"><i><font color="Green" title="E92">A260</font></i></a>, <a class="txt" href="numeral1.html#E105:9_1_2_1_8_1_4"><i><font color="Green" title="E98">A266</font></i></a>, <a class="ref" href="afinsq_1.html#D1" title="AFINSQ_1:def.1">AFINSQ_1:def 1</a></i>;<br/><a NAME="E110:9_1_2_1_8_1_4"/>
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c30">do</font> holds <br/><font color="Maroon" title="c30">do</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c34">g</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">i</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_2_1_8_1_4_14"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c35">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>reconsider </b><font color="Maroon" title="c36">I</font> = <font color="Maroon" title="c35">i</font> as    <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>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<b>assume </b><a NAME="E2:9_1_2_1_8_1_4_14"/><i><font color="Green" title="E103">A271</font></i>: 
<font color="Maroon" title="c35">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c30">do</font>
 ;<br/>

<a NAME="E3:9_1_2_1_8_1_4_14"/><b>then </b>
<font color="Maroon" title="c35">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">o</font>
 
<b>by </b><i><a class="txt" href="numeral1.html#E81:9_1_2_1"><i><font color="Green" title="E39">A70</font></i></a>, <a class="txt" href="numeral1.html#E6:9_1_2_1_8"><i><font color="Green" title="E44">A75</font></i></a>, <a class="txt" href="numeral1.html#E90:9_1_2_1_8_1_4"><i><font color="Green" title="E91">A259</font></i></a>, <a class="ref" href="afinsq_1.html#T14" title="AFINSQ_1:th.14">AFINSQ_1:14</a></i>;<br/>
<b>then </b><i><font color="Green" title="E104">A272</font></i>: <font color="Maroon" title="c34">g</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c36">I</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c36">I</font>
<b>by </b><i><a class="txt" href="numeral1.html#E105:9_1_2_1_8_1_4"><i><font color="Green" title="E98">A266</font></i></a>, <a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c36">I</font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T13" title="SEQ_1:th.13">SEQ_1:13</a></i>
<br/>.= 
<font color="Maroon" title="c3">b1</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c36">I</font></span>)</span>
<b>by </b><i><a class="ref" href="prepower.html#D1" title="PREPOWER:def.1">PREPOWER:def 1</a></i>
;<br/>
<a NAME="E5:9_1_2_1_8_1_4_14"/><i><font color="Green" title="E105">A273</font></i>: 
 <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c30">do</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c8">E</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T89" title="RELAT_1:th.89">RELAT_1:89</a></i>;<br/>
<a NAME="E6:9_1_2_1_8_1_4_14"/><b>then </b>
<font color="Maroon" title="c5">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c35">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">b1</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 
<b>by </b><i><a class="txt" href="numeral1.html#E4:9_1_2_1"><i><font color="Green" title="E12">A37</font></i></a>, <a class="txt" href="numeral1.html#E25:9_1_2_1"><i><font color="Green" title="E20">A45</font></i></a>, <a class="txt" href="numeral1.html#E2:9_1_2_1_8_1_4_14"><i><font color="Green" title="E103">A271</font></i></a></i>;<br/>
<a NAME="E7:9_1_2_1_8_1_4_14"/><b>then </b><i><font color="Green" title="E106">A274</font></i>: 
<font color="Maroon" title="c5">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c35">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">b1</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<font color="Maroon" title="c30">do</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c35">i</font> = 
<font color="Maroon" title="c8">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c35">i</font>
<b>by </b><i><a class="txt" href="numeral1.html#E2:9_1_2_1_8_1_4_14"><i><font color="Green" title="E103">A271</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c35">i</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c35">i</font></span>)</span>
<b>by </b><i><a class="txt" href="numeral1.html#E25:9_1_2_1"><i><font color="Green" title="E20">A45</font></i></a>, <a class="txt" href="numeral1.html#E2:9_1_2_1_8_1_4_14"><i><font color="Green" title="E103">A271</font></i></a>, <a class="txt" href="numeral1.html#E5:9_1_2_1_8_1_4_14"><i><font color="Green" title="E105">A273</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E9:9_1_2_1_8_1_4_14"/>
<font color="Maroon" title="c30">do</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c35">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c34">g</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c35">i</font>
 <b>by </b><i><a class="txt" href="numeral1.html#E4:9_1_2_1_8_1_4_14"><i><font color="Green" title="E104">A272</font></i></a>, <a class="txt" href="numeral1.html#E7:9_1_2_1_8_1_4_14"><i><font color="Green" title="E106">A274</font></i></a>, <a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a></i>;<br/>


</div><b>end;</b></div><a NAME="E111:9_1_2_1_8_1_4"/><b>then </b>
 <a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <font color="Maroon" title="c30">do</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <b>by </b><i><a class="txt" href="numeral1.html#E107:9_1_2_1_8_1_4"><i><font color="Green" title="E100">A268</font></i></a>, <a class="txt" href="numeral1.html#E108:9_1_2_1_8_1_4"><i><font color="Green" title="E101">A269</font></i></a>, <a class="txt" href="numeral1.html#E109:9_1_2_1_8_1_4"><i><font color="Green" title="E102">A270</font></i></a>, <a class="ref" href="numeral1.html#T4" target="_self" title="NUMERAL1:th.4">Th4</a></i>;<br/><a NAME="E112:9_1_2_1_8_1_4"/><b>then </b>
 <a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <font color="Maroon" title="c30">do</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T147" title="XREAL_1:th.147">XREAL_1:147</a></i>;<br/><a NAME="E113:9_1_2_1_8_1_4"/><b>then </b><i><font color="Green" title="E103">A275</font></i>: 
<span class="p1">(<span class="default"><a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <font color="Maroon" title="c30">do</font></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="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="nat_d.html#T27" title="NAT_D:th.27">NAT_D:27</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_2_1_8_1_4_15"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c35">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>reconsider </b><font color="Maroon" title="c36">K</font> = <font color="Maroon" title="c35">k</font> as    <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>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><b>assume </b><a NAME="E2:9_1_2_1_8_1_4_15"/><i><font color="Green" title="E104">A276</font></i>: 
<font color="Maroon" title="c35">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c29">d'</font>
 ;<br/><a NAME="E3:9_1_2_1_8_1_4_15"/><b>then </b>
<font color="Maroon" title="c35">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c29">d'</font>
 <b>by </b><i><a class="ref" href="afinsq_1.html#D1" title="AFINSQ_1:def.1">AFINSQ_1:def 1</a></i>;<br/><a NAME="E4:9_1_2_1_8_1_4_15"/><b>then </b>
<font color="Maroon" title="c35">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c29">d'</font>
 <b>by </b><i><a class="ref" href="prgcor_2.html#T1" title="PRGCOR_2:th.1">PRGCOR_2:1</a></i>;<br/><a NAME="E5:9_1_2_1_8_1_4_15"/><b>then </b>
<font color="Maroon" title="c13">o</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c35">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c8">E</font>
 <b>by </b><i><a class="txt" href="numeral1.html#E91:9_1_2_1_8_1_4"><i><font color="Green" title="E92">A260</font></i></a>, <a class="txt" href="numeral1.html#E92:9_1_2_1_8_1_4"><i><font color="Green" title="E93">A261</font></i></a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/><a NAME="E6:9_1_2_1_8_1_4_15"/><b>then </b>
<font color="Maroon" title="c13">o</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c35">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c8">E</font>
 <b>by </b><i><a class="ref" href="prgcor_2.html#T1" title="PRGCOR_2:th.1">PRGCOR_2:1</a></i>;<br/><a NAME="E7:9_1_2_1_8_1_4_15"/><b>then </b>
<font color="Maroon" title="c8">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">o</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c35">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">o</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c35">k</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">o</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c35">k</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="numeral1.html#E25:9_1_2_1"><i><font color="Green" title="E20">A45</font></i></a></i>;<br/><b>then </b><font color="Maroon" title="c29">d'</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c36">K</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">o</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c35">k</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">o</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c35">k</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="numeral1.html#E87:9_1_2_1_8_1_4"><i><font color="Green" title="E89">A257</font></i></a>, <a class="txt" href="numeral1.html#E2:9_1_2_1_8_1_4_15"><i><font color="Green" title="E104">A276</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">o</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c35">k</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c35">k</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T13" title="NEWTON:th.13">NEWTON:13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">o</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c35">k</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c35">k</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span>

;<br/><b>hence </b><a NAME="E9:9_1_2_1_8_1_4_15"/>
<font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c29">d'</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c35">k</font>
 <b>by </b><i><a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">NAT_D:def 3</a></i>;<br/></div><b>end;</b></div><a NAME="E115:9_1_2_1_8_1_4"/><b>then </b><i><font color="Green" title="E104">A277</font></i>: 
<font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a>  <a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <font color="Maroon" title="c29">d'</font>
 <b>by </b><i><a class="ref" href="numeral1.html#T5" target="_self" title="NUMERAL1:th.5">Th5</a></i>;<br/><a NAME="E116:9_1_2_1_8_1_4"/><b>then </b>
<span class="p1">(<span class="default"><a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <font color="Maroon" title="c29">d'</font></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="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="numeral1.html#E98:9_1_2_1_8_1_4"><i><font color="Green" title="E96">A264</font></i></a>, <a class="ref" href="pepin.html#T6" title="PEPIN:th.6">PEPIN:6</a></i>;<br/><a NAME="E117:9_1_2_1_8_1_4"/><b>then </b>
<font color="Maroon" title="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <font color="Maroon" title="c29">d'</font></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="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></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"><a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <font color="Maroon" title="c30">do</font></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="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="numeral1.html#E96:9_1_2_1_8_1_4"><i><font color="Green" title="E95">A263</font></i></a>, <a class="ref" href="nat_d.html#T19" title="NAT_D:th.19">NAT_D:19</a></i>;<br/><a NAME="E118:9_1_2_1_8_1_4"/><b>then </b><i><font color="Green" title="E105">A278</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <font color="Maroon" title="c29">d'</font>
 <b>by </b><i><a class="txt" href="numeral1.html#E113:9_1_2_1_8_1_4"><i><font color="Green" title="E103">A275</font></i></a>, <a class="txt" href="numeral1.html#E115:9_1_2_1_8_1_4"><i><font color="Green" title="E104">A277</font></i></a>, <a class="ref" href="nat_d.html#T3" title="NAT_D:th.3">NAT_D:3</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c35">o1</font> = <font color="Maroon" title="c13">o</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 as    <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/><a NAME="E120:9_1_2_1_8_1_4"/><i><font color="Green" title="E106">A279</font></i>: 
( <font color="Maroon" title="c35">o1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">k</font> &amp; <font color="Maroon" title="c7">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c4">d</font> )
 <b>by </b><i><a class="txt" href="numeral1.html#E1:9_1_2_1_8_1_4"><i><font color="Green" title="E46">A201</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a>, <a class="ref" href="xreal_1.html#T149" title="XREAL_1:th.149">XREAL_1:149</a></i>;<br/><a NAME="E121:9_1_2_1_8_1_4"/><b>then </b><i><font color="Green" title="E107">A280</font></i>: 
<font color="Maroon" title="c35">o1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c4">d</font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c36">oo1</font> = <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c4">d</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c35">o1</font> as    <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>by </b><i><a class="txt" href="numeral1.html#E120:9_1_2_1_8_1_4"><i><font color="Green" title="E106">A279</font></i></a>, <a class="ref" href="nat_1.html#T21" title="NAT_1:th.21">NAT_1:21</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>4</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> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>$2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default">$1 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c35">o1</font></span>)</span>;<br/><a NAME="E123:9_1_2_1_8_1_4"/><i><font color="Green" title="E108">A281</font></i>: 
for <font color="Olive" title="b1">u</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">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c36">oo1</font> holds <br/> ex <font color="Olive" title="b2">x</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>4</sub></font>[<font color="Olive" title="b1">u</font>,<font color="Olive" title="b2">x</font>]
 ;<br/><b>consider </b><font color="Maroon" title="c37">d'1</font> being   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E125:9_1_2_1_8_1_4"/><i><font color="Green" title="E109">A282</font></i>: 
(  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c37">d'1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c36">oo1</font> &amp; ( for <font color="Olive" title="b1">x</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">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c36">oo1</font> holds <br/><font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c37">d'1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">x</font>] ) )
 <b>from </b><i><a class="ref" href="stirl2_1.html#S6" title="STIRL2_1:sch.6">STIRL2_1:sch 6</a>(<a class="txt" href="numeral1.html#E123:9_1_2_1_8_1_4"><i><font color="Green" title="E108">A281</font></i></a>);<br/></i><b>reconsider </b><font color="Maroon" title="c38">do1</font> = <font color="Maroon" title="c8">E</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c35">o1</font> as   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="afinsq_1.html#T16" title="AFINSQ_1:th.16">AFINSQ_1:16</a></i>;<br/><a NAME="E127:9_1_2_1_8_1_4"/><i><font color="Green" title="E110">A283</font></i>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c37">d'1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c36">oo1</font>
 <b>by </b><i><a class="txt" href="numeral1.html#E125:9_1_2_1_8_1_4"><i><font color="Green" title="E109">A282</font></i></a>, <a class="ref" href="afinsq_1.html#D1" title="AFINSQ_1:def.1">AFINSQ_1:def 1</a></i>;<br/><i><font color="Green" title="E111">A284</font></i>:  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c5">e</font> = 
 <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c8">E</font>
<b>by </b><i><a class="txt" href="numeral1.html#E25:9_1_2_1"><i><font color="Green" title="E20">A45</font></i></a>, <a class="ref" href="afinsq_1.html#D1" title="AFINSQ_1:def.1">AFINSQ_1:def 1</a></i>
<br/>.= 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c8">E</font>
<b>by </b><i><a class="ref" href="afinsq_1.html#D1" title="AFINSQ_1:def.1">AFINSQ_1:def 1</a></i>
;<br/><a NAME="E129:9_1_2_1_8_1_4"/><b>then </b><i><font color="Green" title="E112">A285</font></i>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c38">do1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c35">o1</font>
 <b>by </b><i><a class="txt" href="numeral1.html#E81:9_1_2_1"><i><font color="Green" title="E39">A70</font></i></a>, <a class="txt" href="numeral1.html#E121:9_1_2_1_8_1_4"><i><font color="Green" title="E107">A280</font></i></a>, <a class="ref" href="afinsq_1.html#T14" title="AFINSQ_1:th.14">AFINSQ_1:14</a></i>;<br/><a NAME="E130:9_1_2_1_8_1_4"/><b>then </b><i><font color="Green" title="E113">A286</font></i>: 
 <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c8">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c38">do1</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c37">d'1</font></span>)</span>
 <b>by </b><i><a class="txt" href="numeral1.html#E25:9_1_2_1"><i><font color="Green" title="E20">A45</font></i></a>, <a class="txt" href="numeral1.html#E81:9_1_2_1"><i><font color="Green" title="E39">A70</font></i></a>, <a class="txt" href="numeral1.html#E127:9_1_2_1_8_1_4"><i><font color="Green" title="E110">A283</font></i></a>, <a class="ref" href="afinsq_1.html#D1" title="AFINSQ_1:def.1">AFINSQ_1:def 1</a></i>;<br/><a NAME="E131:9_1_2_1_8_1_4"/><i><font color="Green" title="E114">A287</font></i>: 
for <font color="Olive" title="b1">x</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">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c38">do1</font> holds <br/><font color="Maroon" title="c8">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c38">do1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 <b>by </b><i><a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>;<br/><a NAME="E132:9_1_2_1_8_1_4"/>
for <font color="Olive" title="b1">x</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">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c37">d'1</font> holds <br/><font color="Maroon" title="c8">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c38">do1</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c37">d'1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 <b>by </b><i><a class="txt" href="numeral1.html#E125:9_1_2_1_8_1_4"><i><font color="Green" title="E109">A282</font></i></a>, <a class="txt" href="numeral1.html#E129:9_1_2_1_8_1_4"><i><font color="Green" title="E112">A285</font></i></a></i>;<br/><a NAME="E133:9_1_2_1_8_1_4"/><b>then </b>
<font color="Maroon" title="c8">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c38">do1</font> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <font color="Maroon" title="c37">d'1</font>
 <b>by </b><i><a class="txt" href="numeral1.html#E130:9_1_2_1_8_1_4"><i><font color="Green" title="E113">A286</font></i></a>, <a class="txt" href="numeral1.html#E131:9_1_2_1_8_1_4"><i><font color="Green" title="E114">A287</font></i></a>, <a class="ref" href="afinsq_1.html#D4" title="AFINSQ_1:def.4">AFINSQ_1:def 4</a></i>;<br/><a NAME="E134:9_1_2_1_8_1_4"/><b>then </b><i><font color="Green" title="E115">A288</font></i>: 
<font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <font color="Maroon" title="c38">do1</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <font color="Maroon" title="c37">d'1</font></span>)</span>
 <b>by </b><i><a class="txt" href="numeral1.html#E25:9_1_2_1"><i><font color="Green" title="E20">A45</font></i></a>, <a class="ref" href="numeral1.html#T1" target="_self" title="NUMERAL1:th.1">Th1</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c39">bo1</font> = <font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c35">o1</font> as    <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>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><a NAME="E136:9_1_2_1_8_1_4"/><i><font color="Green" title="E116">A289</font></i>: 
<font color="Maroon" title="c39">bo1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="numeral1.html#E1:9"><i><font color="Green" title="E9">A1</font></i></a>, <a class="ref" href="prepower.html#T12" title="PREPOWER:th.12">PREPOWER:12</a></i>;<br/><b>consider </b><font color="Maroon" title="c40">ok1</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E138:9_1_2_1_8_1_4"/><i><font color="Green" title="E117">A290</font></i>: 
<font color="Maroon" title="c40">ok1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c35">o1</font>
 ;<br/><b>reconsider </b><font color="Maroon" title="c41">ok1</font> = <font color="Maroon" title="c40">ok1</font> as    <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>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c42">g1</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c41">ok1</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> as   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="numeral1.html#T3" target="_self" title="NUMERAL1:th.3">Th3</a></i>;<br/><a NAME="E141:9_1_2_1_8_1_4"/>
 <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="c2">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><a NAME="E142:9_1_2_1_8_1_4"/><b>then </b>
<font color="Maroon" title="c35">o1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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="c2">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="ordinal1.html#D2" title="ORDINAL1:def.2">ORDINAL1:def 2</a></i>;<br/><a NAME="E143:9_1_2_1_8_1_4"/><b>then </b><i><font color="Green" title="E118">A291</font></i>: 
 <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c42">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c35">o1</font>
 <b>by </b><i><a class="txt" href="numeral1.html#E138:9_1_2_1_8_1_4"><i><font color="Green" title="E117">A290</font></i></a>, <a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a></i>;<br/><a NAME="E144:9_1_2_1_8_1_4"/><i><font color="Green" title="E119">A292</font></i>: 
1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="numeral1.html#E1:9"><i><font color="Green" title="E9">A1</font></i></a></i>;<br/><i><font color="Green" title="E120">A293</font></i>:  <a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <font color="Maroon" title="c42">g1</font> = 
<span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c41">ok1</font>
<b>by </b><i><a class="ref" href="numeral1.html#T2" target="_self" title="NUMERAL1:th.2">Th2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c41">ok1</font>
<b>by </b><i><a class="ref" href="series_1.html#T12" title="SERIES_1:th.12">SERIES_1:12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c41">ok1</font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T13" title="SEQ_1:th.13">SEQ_1:13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p4">(<span class="default"><font color="Maroon" title="c41">ok1</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="numeral1.html#E1:9"><i><font color="Green" title="E9">A1</font></i></a>, <a class="ref" href="series_1.html#T26" title="SERIES_1:th.26">SERIES_1:26</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p4">(<span class="default"><font color="Maroon" title="c41">ok1</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="numeral1.html#E1:9"><i><font color="Green" title="E9">A1</font></i></a>, <a class="ref" href="power.html#T46" title="POWER:th.46">POWER:46</a></i>
<br/>.= 
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c35">o1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="numeral1.html#E138:9_1_2_1_8_1_4"><i><font color="Green" title="E117">A290</font></i></a></i>
<br/>.= 
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c35">o1</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="numeral1.html#E144:9_1_2_1_8_1_4"><i><font color="Green" title="E119">A292</font></i></a>, <a class="ref" href="xcmplx_1.html#T88" title="XCMPLX_1:th.88">XCMPLX_1:88</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c35">o1</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1

;<br/><a NAME="E146:9_1_2_1_8_1_4"/><i><font color="Green" title="E121">A294</font></i>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c38">do1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="numeral1.html#E129:9_1_2_1_8_1_4"><i><font color="Green" title="E112">A285</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E147:9_1_2_1_8_1_4"/><i><font color="Green" title="E122">A295</font></i>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c38">do1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c42">g1</font>
 <b>by </b><i><a class="txt" href="numeral1.html#E129:9_1_2_1_8_1_4"><i><font color="Green" title="E112">A285</font></i></a>, <a class="txt" href="numeral1.html#E143:9_1_2_1_8_1_4"><i><font color="Green" title="E118">A291</font></i></a>, <a class="ref" href="afinsq_1.html#D1" title="AFINSQ_1:def.1">AFINSQ_1:def 1</a></i>;<br/><a NAME="E148:9_1_2_1_8_1_4"/>
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c38">do1</font> holds <br/><font color="Maroon" title="c38">do1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c42">g1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">i</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_2_1_8_1_4_18"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c43">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>reconsider </b><font color="Maroon" title="c44">I</font> = <font color="Maroon" title="c43">i</font> as    <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>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<b>assume </b><a NAME="E2:9_1_2_1_8_1_4_18"/><i><font color="Green" title="E123">A296</font></i>: 
<font color="Maroon" title="c43">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c38">do1</font>
 ;<br/>

<a NAME="E3:9_1_2_1_8_1_4_18"/><b>then </b>
<font color="Maroon" title="c43">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c35">o1</font>
 
<b>by </b><i><a class="txt" href="numeral1.html#E81:9_1_2_1"><i><font color="Green" title="E39">A70</font></i></a>, <a class="txt" href="numeral1.html#E121:9_1_2_1_8_1_4"><i><font color="Green" title="E107">A280</font></i></a>, <a class="txt" href="numeral1.html#E128:9_1_2_1_8_1_4"><i><font color="Green" title="E111">A284</font></i></a>, <a class="ref" href="afinsq_1.html#T14" title="AFINSQ_1:th.14">AFINSQ_1:14</a></i>;<br/>
<b>then </b><i><font color="Green" title="E124">A297</font></i>: <font color="Maroon" title="c42">g1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c44">I</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c44">I</font>
<b>by </b><i><a class="txt" href="numeral1.html#E143:9_1_2_1_8_1_4"><i><font color="Green" title="E118">A291</font></i></a>, <a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c44">I</font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T13" title="SEQ_1:th.13">SEQ_1:13</a></i>
<br/>.= 
<font color="Maroon" title="c3">b1</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c44">I</font></span>)</span>
<b>by </b><i><a class="ref" href="prepower.html#D1" title="PREPOWER:def.1">PREPOWER:def 1</a></i>
;<br/>
<a NAME="E5:9_1_2_1_8_1_4_18"/><i><font color="Green" title="E125">A298</font></i>: 
 <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c38">do1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c8">E</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T89" title="RELAT_1:th.89">RELAT_1:89</a></i>;<br/>
<a NAME="E6:9_1_2_1_8_1_4_18"/><b>then </b>
<font color="Maroon" title="c5">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c43">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">b1</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 
<b>by </b><i><a class="txt" href="numeral1.html#E4:9_1_2_1"><i><font color="Green" title="E12">A37</font></i></a>, <a class="txt" href="numeral1.html#E25:9_1_2_1"><i><font color="Green" title="E20">A45</font></i></a>, <a class="txt" href="numeral1.html#E2:9_1_2_1_8_1_4_18"><i><font color="Green" title="E123">A296</font></i></a></i>;<br/>
<a NAME="E7:9_1_2_1_8_1_4_18"/><b>then </b><i><font color="Green" title="E126">A299</font></i>: 
<font color="Maroon" title="c5">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c43">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">b1</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<font color="Maroon" title="c38">do1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c43">i</font> = 
<font color="Maroon" title="c8">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c43">i</font>
<b>by </b><i><a class="txt" href="numeral1.html#E2:9_1_2_1_8_1_4_18"><i><font color="Green" title="E123">A296</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c43">i</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c43">i</font></span>)</span>
<b>by </b><i><a class="txt" href="numeral1.html#E25:9_1_2_1"><i><font color="Green" title="E20">A45</font></i></a>, <a class="txt" href="numeral1.html#E2:9_1_2_1_8_1_4_18"><i><font color="Green" title="E123">A296</font></i></a>, <a class="txt" href="numeral1.html#E5:9_1_2_1_8_1_4_18"><i><font color="Green" title="E125">A298</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E9:9_1_2_1_8_1_4_18"/>
<font color="Maroon" title="c38">do1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c43">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c42">g1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c43">i</font>
 <b>by </b><i><a class="txt" href="numeral1.html#E4:9_1_2_1_8_1_4_18"><i><font color="Green" title="E124">A297</font></i></a>, <a class="txt" href="numeral1.html#E7:9_1_2_1_8_1_4_18"><i><font color="Green" title="E126">A299</font></i></a>, <a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a></i>;<br/>


</div><b>end;</b></div><a NAME="E149:9_1_2_1_8_1_4"/><b>then </b>
 <a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <font color="Maroon" title="c38">do1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c35">o1</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <b>by </b><i><a class="txt" href="numeral1.html#E145:9_1_2_1_8_1_4"><i><font color="Green" title="E120">A293</font></i></a>, <a class="txt" href="numeral1.html#E146:9_1_2_1_8_1_4"><i><font color="Green" title="E121">A294</font></i></a>, <a class="txt" href="numeral1.html#E147:9_1_2_1_8_1_4"><i><font color="Green" title="E122">A295</font></i></a>, <a class="ref" href="numeral1.html#T4" target="_self" title="NUMERAL1:th.4">Th4</a></i>;<br/><a NAME="E150:9_1_2_1_8_1_4"/><b>then </b>
 <a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <font color="Maroon" title="c38">do1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c35">o1</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T147" title="XREAL_1:th.147">XREAL_1:147</a></i>;<br/><a NAME="E151:9_1_2_1_8_1_4"/><b>then </b><i><font color="Green" title="E123">A300</font></i>: 
<span class="p1">(<span class="default"><a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <font color="Maroon" title="c38">do1</font></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="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c35">o1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="nat_d.html#T27" title="NAT_D:th.27">NAT_D:27</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_2_1_8_1_4_19"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c43">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>reconsider </b><font color="Maroon" title="c44">K</font> = <font color="Maroon" title="c43">k</font> as    <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>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><b>assume </b><a NAME="E2:9_1_2_1_8_1_4_19"/><i><font color="Green" title="E124">A301</font></i>: 
<font color="Maroon" title="c43">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c37">d'1</font>
 ;<br/><a NAME="E3:9_1_2_1_8_1_4_19"/><b>then </b>
<font color="Maroon" title="c43">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c37">d'1</font>
 <b>by </b><i><a class="ref" href="afinsq_1.html#D1" title="AFINSQ_1:def.1">AFINSQ_1:def 1</a></i>;<br/><a NAME="E4:9_1_2_1_8_1_4_19"/><b>then </b>
<font color="Maroon" title="c43">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c37">d'1</font>
 <b>by </b><i><a class="ref" href="prgcor_2.html#T1" title="PRGCOR_2:th.1">PRGCOR_2:1</a></i>;<br/><a NAME="E5:9_1_2_1_8_1_4_19"/><b>then </b>
<font color="Maroon" title="c35">o1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c43">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c8">E</font>
 <b>by </b><i><a class="txt" href="numeral1.html#E129:9_1_2_1_8_1_4"><i><font color="Green" title="E112">A285</font></i></a>, <a class="txt" href="numeral1.html#E130:9_1_2_1_8_1_4"><i><font color="Green" title="E113">A286</font></i></a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/><a NAME="E6:9_1_2_1_8_1_4_19"/><b>then </b>
<font color="Maroon" title="c35">o1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c43">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c8">E</font>
 <b>by </b><i><a class="ref" href="prgcor_2.html#T1" title="PRGCOR_2:th.1">PRGCOR_2:1</a></i>;<br/><a NAME="E7:9_1_2_1_8_1_4_19"/><b>then </b>
<font color="Maroon" title="c8">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c35">o1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c43">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c35">o1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c43">k</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c35">o1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c43">k</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="numeral1.html#E25:9_1_2_1"><i><font color="Green" title="E20">A45</font></i></a></i>;<br/><b>then </b><font color="Maroon" title="c37">d'1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c44">K</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c35">o1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c43">k</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c35">o1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c43">k</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="numeral1.html#E125:9_1_2_1_8_1_4"><i><font color="Green" title="E109">A282</font></i></a>, <a class="txt" href="numeral1.html#E2:9_1_2_1_8_1_4_19"><i><font color="Green" title="E124">A301</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c35">o1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c43">k</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c35">o1</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c43">k</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T13" title="NEWTON:th.13">NEWTON:13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c35">o1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c43">k</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c43">k</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c35">o1</font></span>)</span>

;<br/><b>hence </b><a NAME="E9:9_1_2_1_8_1_4_19"/>
<font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c35">o1</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c37">d'1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c43">k</font>
 <b>by </b><i><a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">NAT_D:def 3</a></i>;<br/></div><b>end;</b></div><a NAME="E153:9_1_2_1_8_1_4"/><b>then </b><i><font color="Green" title="E124">A302</font></i>: 
<font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c35">o1</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a>  <a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <font color="Maroon" title="c37">d'1</font>
 <b>by </b><i><a class="ref" href="numeral1.html#T5" target="_self" title="NUMERAL1:th.5">Th5</a></i>;<br/><a NAME="E154:9_1_2_1_8_1_4"/><b>then </b>
<span class="p1">(<span class="default"><a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <font color="Maroon" title="c37">d'1</font></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="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c35">o1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="numeral1.html#E136:9_1_2_1_8_1_4"><i><font color="Green" title="E116">A289</font></i></a>, <a class="ref" href="pepin.html#T6" title="PEPIN:th.6">PEPIN:6</a></i>;<br/><a NAME="E155:9_1_2_1_8_1_4"/><b>then </b>
<font color="Maroon" title="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c35">o1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <font color="Maroon" title="c37">d'1</font></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="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c35">o1</font></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"><a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <font color="Maroon" title="c38">do1</font></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="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c35">o1</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="numeral1.html#E134:9_1_2_1_8_1_4"><i><font color="Green" title="E115">A288</font></i></a>, <a class="ref" href="nat_d.html#T19" title="NAT_D:th.19">NAT_D:19</a></i>;<br/><a NAME="E156:9_1_2_1_8_1_4"/><b>then </b><i><font color="Green" title="E125">A303</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c35">o1</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c35">o1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <font color="Maroon" title="c37">d'1</font>
 <b>by </b><i><a class="txt" href="numeral1.html#E151:9_1_2_1_8_1_4"><i><font color="Green" title="E123">A300</font></i></a>, <a class="txt" href="numeral1.html#E153:9_1_2_1_8_1_4"><i><font color="Green" title="E124">A302</font></i></a>, <a class="ref" href="nat_d.html#T3" title="NAT_D:th.3">NAT_D:3</a></i>;<br/><a NAME="E157:9_1_2_1_8_1_4"/>
<font color="Maroon" title="c13">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c5">e</font>
 <b>by </b><i><a class="txt" href="numeral1.html#E81:9_1_2_1"><i><font color="Green" title="E39">A70</font></i></a>, <a class="txt" href="numeral1.html#E1:9_1_2_1_8"><i><font color="Green" title="E41">A72</font></i></a>, <a class="ref" href="afinsq_1.html#D1" title="AFINSQ_1:def.1">AFINSQ_1:def 1</a></i>;<br/><a NAME="E158:9_1_2_1_8_1_4"/><b>then </b><i><font color="Green" title="E126">A304</font></i>: 
<font color="Maroon" title="c13">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c8">E</font>
 <b>by </b><i><a class="txt" href="numeral1.html#E25:9_1_2_1"><i><font color="Green" title="E20">A45</font></i></a>, <a class="ref" href="afinsq_1.html#D1" title="AFINSQ_1:def.1">AFINSQ_1:def 1</a></i>;<br/><i><font color="Green" title="E127">A305</font></i>:  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c29">d'</font> = 
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c37">d'1</font></span>)</span>
<b>by </b><i><a class="txt" href="numeral1.html#E87:9_1_2_1_8_1_4"><i><font color="Green" title="E89">A257</font></i></a>, <a class="txt" href="numeral1.html#E125:9_1_2_1_8_1_4"><i><font color="Green" title="E109">A282</font></i></a></i>
<br/>.= 
1 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c37">d'1</font></span>)</span>
<b>by </b><i><a class="ref" href="afinsq_1.html#D1" title="AFINSQ_1:def.1">AFINSQ_1:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p2"><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">&lt;%</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">%&gt;</a></span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c37">d'1</font></span>)</span>
<b>by </b><i><a class="ref" href="afinsq_1.html#T38" title="AFINSQ_1:th.38">AFINSQ_1:38</a></i>
;<br/><div><i><font color="Green" title="E128">A306</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_2_1_8_1_4_21"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c43">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="E1:9_1_2_1_8_1_4_21"/>
<font color="Maroon" title="c43">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <span class="p1"><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">%&gt;</a></span>
 ;<br/><a NAME="E2:9_1_2_1_8_1_4_21"/><b>then </b>
<font color="Maroon" title="c43">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 1
 <b>by </b><i><a class="ref" href="afinsq_1.html#T36" title="AFINSQ_1:th.36">AFINSQ_1:36</a></i>;<br/><a NAME="E3:9_1_2_1_8_1_4_21"/><b>then </b>
<font color="Maroon" title="c43">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1
 <b>by </b><i><a class="ref" href="prgcor_2.html#T1" title="PRGCOR_2:th.1">PRGCOR_2:1</a></i>;<br/><a NAME="E4:9_1_2_1_8_1_4_21"/><b>then </b><i><font color="Green" title="E129">A307</font></i>: 
<font color="Maroon" title="c43">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></i>;<br/><a NAME="E5:9_1_2_1_8_1_4_21"/>
<font color="Maroon" title="c14">oo</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="numeral1.html#E6:9_1_2_1_8"><i><font color="Green" title="E44">A75</font></i></a>, <a class="ref" href="xreal_1.html#T52" title="XREAL_1:th.52">XREAL_1:52</a></i>;<br/><a NAME="E6:9_1_2_1_8_1_4_21"/><b>then </b>
<font color="Maroon" title="c43">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c29">d'</font>
 <b>by </b><i><a class="txt" href="numeral1.html#E87:9_1_2_1_8_1_4"><i><font color="Green" title="E89">A257</font></i></a>, <a class="txt" href="numeral1.html#E4:9_1_2_1_8_1_4_21"><i><font color="Green" title="E129">A307</font></i></a>, <a class="ref" href="prgcor_2.html#T1" title="PRGCOR_2:th.1">PRGCOR_2:1</a></i>;<br/><b>hence </b><font color="Maroon" title="c29">d'</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c43">k</font> = 
<font color="Maroon" title="c8">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c43">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c13">o</font></span>)</span>
<b>by </b><i><a class="txt" href="numeral1.html#E87:9_1_2_1_8_1_4"><i><font color="Green" title="E89">A257</font></i></a></i>
<br/>.= 
<span class="p1"><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">%&gt;</a></span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c43">k</font>
<b>by </b><i><a class="txt" href="numeral1.html#E4:9_1_2_1_8_1_4_21"><i><font color="Green" title="E129">A307</font></i></a>, <a class="ref" href="afinsq_1.html#D5" title="AFINSQ_1:def.5">AFINSQ_1:def 5</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_2_1_8_1_4_22"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c43">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="E1:9_1_2_1_8_1_4_22"/><i><font color="Green" title="E129">A308</font></i>: 
<font color="Maroon" title="c43">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c37">d'1</font>
 ;<br/><a NAME="E2:9_1_2_1_8_1_4_22"/><b>then </b>
<font color="Maroon" title="c43">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c36">oo1</font>
 <b>by </b><i><a class="txt" href="numeral1.html#E125:9_1_2_1_8_1_4"><i><font color="Green" title="E109">A282</font></i></a>, <a class="ref" href="prgcor_2.html#T1" title="PRGCOR_2:th.1">PRGCOR_2:1</a></i>;<br/><a NAME="E3:9_1_2_1_8_1_4_22"/><b>then </b>
<font color="Maroon" title="c43">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c36">oo1</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/><a NAME="E4:9_1_2_1_8_1_4_22"/><b>then </b><i><font color="Green" title="E130">A309</font></i>: 
<font color="Maroon" title="c43">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">oo</font>
 <b>by </b><i><a class="ref" href="prgcor_2.html#T1" title="PRGCOR_2:th.1">PRGCOR_2:1</a></i>;<br/><b>thus </b><font color="Maroon" title="c29">d'</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p3"><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">&lt;%</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">%&gt;</a></span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c43">k</font></span>)</span> = 
<font color="Maroon" title="c29">d'</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c43">k</font></span>)</span>
<b>by </b><i><a class="ref" href="afinsq_1.html#T37" title="AFINSQ_1:th.37">AFINSQ_1:37</a></i>
<br/>.= 
<font color="Maroon" title="c8">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c30">do</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c43">k</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="numeral1.html#E87:9_1_2_1_8_1_4"><i><font color="Green" title="E89">A257</font></i></a>, <a class="txt" href="numeral1.html#E91:9_1_2_1_8_1_4"><i><font color="Green" title="E92">A260</font></i></a>, <a class="txt" href="numeral1.html#E4:9_1_2_1_8_1_4_22"><i><font color="Green" title="E130">A309</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c8">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c38">do1</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c43">k</font></span>)</span>
<b>by </b><i><a class="txt" href="numeral1.html#E91:9_1_2_1_8_1_4"><i><font color="Green" title="E92">A260</font></i></a>, <a class="txt" href="numeral1.html#E129:9_1_2_1_8_1_4"><i><font color="Green" title="E112">A285</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c37">d'1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c43">k</font>
<b>by </b><i><a class="txt" href="numeral1.html#E125:9_1_2_1_8_1_4"><i><font color="Green" title="E109">A282</font></i></a>, <a class="txt" href="numeral1.html#E129:9_1_2_1_8_1_4"><i><font color="Green" title="E112">A285</font></i></a>, <a class="txt" href="numeral1.html#E1:9_1_2_1_8_1_4_22"><i><font color="Green" title="E129">A308</font></i></a></i>
;<br/></div><b>end;</b></div><a NAME="E162:9_1_2_1_8_1_4"/><b>then </b>
<font color="Maroon" title="c29">d'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">%&gt;</a></span> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <font color="Maroon" title="c37">d'1</font>
 <b>by </b><i><a class="txt" href="numeral1.html#E159:9_1_2_1_8_1_4"><i><font color="Green" title="E127">A305</font></i></a>, <a class="txt" href="numeral1.html#E160:9_1_2_1_8_1_4"><i><font color="Green" title="E128">A306</font></i></a>, <a class="ref" href="afinsq_1.html#D4" title="AFINSQ_1:def.4">AFINSQ_1:def 4</a></i>;<br/><a NAME="E163:9_1_2_1_8_1_4"/><b>then </b>
 <a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <font color="Maroon" title="c29">d'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <span class="p2"><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">&lt;%</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">%&gt;</a></span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <font color="Maroon" title="c37">d'1</font></span>)</span>
 <b>by </b><i><a class="ref" href="numeral1.html#T1" target="_self" title="NUMERAL1:th.1">Th1</a></i>;<br/><a NAME="E164:9_1_2_1_8_1_4"/><b>then </b>
<font color="Maroon" title="c8">E</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</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="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c35">o1</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c35">o1</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="numeral1.html#E118:9_1_2_1_8_1_4"><i><font color="Green" title="E105">A278</font></i></a>, <a class="txt" href="numeral1.html#E156:9_1_2_1_8_1_4"><i><font color="Green" title="E125">A303</font></i></a>, <a class="ref" href="stirl2_1.html#T44" title="STIRL2_1:th.44">STIRL2_1:44</a></i>;<br/><a NAME="E165:9_1_2_1_8_1_4"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <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="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c35">o1</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c35">o1</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="numeral1.html#E25:9_1_2_1"><i><font color="Green" title="E20">A45</font></i></a>, <a class="txt" href="numeral1.html#E158:9_1_2_1_8_1_4"><i><font color="Green" title="E126">A304</font></i></a></i>;<br/><a NAME="E166:9_1_2_1_8_1_4"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <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="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c35">o1</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> 1</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="newton.html#T13" title="NEWTON:th.13">NEWTON:13</a></i>;<br/><a NAME="E167:9_1_2_1_8_1_4"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c35">o1</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> 1</span>)</span></span>)</span></span>)</span>
 ;<br/><a NAME="E168:9_1_2_1_8_1_4"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <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="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c35">o1</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c13">o</font></span>)</span>
 <b>by </b><i><a class="ref" href="newton.html#T10" title="NEWTON:th.10">NEWTON:10</a></i>;<br/><b>hence </b><a NAME="E169:9_1_2_1_8_1_4"/>
<font color="Maroon" title="c4">d</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c12">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">e</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c12">a</font>
 <b>by </b><i><a class="txt" href="numeral1.html#E84:9_1_2_1_8_1_4"><i><font color="Green" title="E87">A255</font></i></a>, <a class="txt" href="numeral1.html#E98:9_1_2_1_8_1_4"><i><font color="Green" title="E96">A264</font></i></a>, <a class="ref" href="xcmplx_1.html#T90" title="XCMPLX_1:th.90">XCMPLX_1:90</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E84:9_1_2_1"/>
<font color="Maroon" title="c4">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">e</font>
 <b>by </b><i><a class="txt" href="numeral1.html#E82:9_1_2_1"><i><font color="Green" title="E40">A71</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>;<br/>


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

<b>assume </b><a NAME="E3:9_1_2"/>
( <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c4">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">&lt;%</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">%&gt;</a></span> &amp; <font color="Maroon" title="c5">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">&lt;%</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">%&gt;</a></span> )
 ;<br/>

<b>hence </b><a NAME="E4:9_1_2"/>
<font color="Maroon" title="c4">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">e</font>
 ;<br/>


</div>
