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

<b>reconsider </b><font color="Maroon" title="c3">N</font> = <font color="Maroon" title="c1">n</font>, <font color="Maroon" title="c4">B</font> = <font color="Maroon" title="c2">b</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>thus </b><a NAME="E2:9_1_1"/>
( <font color="Maroon" title="c1">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 implies  ex <font color="Olive" title="b1">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>  st <br/>(  <a href="numeral1.html#K1" title="NUMERAL1:func.1">value</a> <font color="Olive" title="b1">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="Olive" title="b1">d</font> <a href="stirl2_1.html#K8" title="STIRL2_1:func.8">.</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="Olive" title="b1">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> 0 &amp; ( for <font color="Olive" title="b2">i</font> being <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b2">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="Olive" title="b1">d</font> holds <br/>( 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">d</font> <a href="stirl2_1.html#K8" title="STIRL2_1:func.8">.</a> <font color="Olive" title="b2">i</font> &amp; <font color="Olive" title="b1">d</font> <a href="stirl2_1.html#K8" title="STIRL2_1:func.8">.</a> <font color="Olive" title="b2">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">b</font> ) ) ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:9_1_1_1"/><i><font color="Green" title="E10">an</font></i>: 
<font color="Maroon" title="c1">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 ;<br/>

<b>deffunc </b><font color="Maroon">H<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="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ) <b>-&gt; </b>   <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>  = $2 <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c4">B</font>;<br/>
<b>consider </b><font color="Maroon" title="c5">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> , <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:9_1_1_1"/><i><font color="Green" title="E11">f</font></i>: 
( <font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">N</font> &amp; ( for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">i</font>,<font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font>) ) )
 <b>from </b><i><a class="ref" href="recdef_1.html#S4" title="RECDEF_1:sch.4">RECDEF_1:sch 4</a>();<br/></i>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[ <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ] <b>means </b><font color="Maroon" title="c5">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[ <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ] <b>means </b> ex <font color="Olive" title="b1">i</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" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1;<br/>
<a NAME="E4:9_1_1_1"/><i><font color="Green" title="E12">ak</font></i>: 
 ex <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">k</font>]
 
<b>by </b><i><a class="txt" href="numeral1.html#E3:9_1_1_1"><i><font color="Green" title="E11">f</font></i></a></i>;<br/>
<a NAME="E5:9_1_1_1"/><i><font color="Green" title="E13">fk</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/> ex <font color="Olive" title="b2">l</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b2">l</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">k</font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b2">l</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">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_1_1_1"/><i><font color="Green" title="E14">k</font></i>: 
( <font color="Maroon" title="c6">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c6">k</font>] )
 ;<br/>

<b>then consider </b><font color="Maroon" title="c7">i</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:9_1_1_1_1"/><i><font color="Green" title="E15">fk</font></i>: 
<font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">k</font>
 ;<br/>
<b>take </b><font color="Maroon" title="c8">l</font> = <font color="Maroon" title="c6">k</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c2">b</font>;<br/>

<a NAME="E4:9_1_1_1_1"/>
( <font color="Maroon" title="c6">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 &amp; <font color="Maroon" title="c2">b</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 )
 
<b>by </b><i><a class="txt" href="numeral1.html#E1:9_1_1_1_1"><i><font color="Green" title="E14">k</font></i></a>, <a class="txt" href="numeral1.html#E1:9"><i><font color="Green" title="E9">b</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:9_1_1_1_1"/>
<font color="Maroon" title="c8">l</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">k</font>
 <b>by </b><i><a class="ref" href="int_1.html#T83" title="INT_1:th.83">INT_1:83</a></i>;<br/>

<b>take </b>
<font color="Maroon" title="c7">i</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
;<br/>

<b>thus </b><a NAME="E6:9_1_1_1_1"/>
<font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">l</font>
 <b>by </b><i><a class="txt" href="numeral1.html#E3:9_1_1_1_1"><i><font color="Green" title="E15">fk</font></i></a>, <a class="txt" href="numeral1.html#E3:9_1_1_1"><i><font color="Green" title="E11">f</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:9_1_1_1"/>
<font color="Maroon">S<sub>2</sub></font>[0]
 
<b>from </b><i><a class="ref" href="nat_1.html#S7" title="NAT_1:sch.7">NAT_1:sch 7</a>(<a class="txt" href="numeral1.html#E4:9_1_1_1"><i><font color="Green" title="E12">ak</font></i></a>, <a class="txt" href="numeral1.html#E5:9_1_1_1"><i><font color="Green" title="E13">fk</font></i></a>);<br/></i>
<a NAME="E7:9_1_1_1"/><b>then </b><i><font color="Green" title="E14">q</font></i>: 
 ex <font color="Olive" title="b1">l</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">l</font>]
 
;<br/>
<b>consider </b><font color="Maroon" title="c6">l</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E9:9_1_1_1"/><i><font color="Green" title="E15">l</font></i>: 
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">l</font>] &amp; ( for <font color="Olive" title="b1">i</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">i</font>] holds <br/><font color="Maroon" title="c6">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> ) )
 <b>from </b><i><a class="ref" href="nat_1.html#S5" title="NAT_1:sch.5">NAT_1:sch 5</a>(<a class="txt" href="numeral1.html#E7:9_1_1_1"><i><font color="Green" title="E14">q</font></i></a>);<br/></i>
<b>consider </b><font color="Maroon" title="c7">m</font> being  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <b> such that </b><br/><a NAME="E11:9_1_1_1"/><i><font color="Green" title="E16">m1</font></i>: 
<font color="Maroon" title="c7">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">l</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T6" title="NAT_1:th.6">NAT_1:6</a>, <a class="txt" href="numeral1.html#E9:9_1_1_1"><i><font color="Green" title="E15">l</font></i></a>, <a class="txt" href="numeral1.html#E3:9_1_1_1"><i><font color="Green" title="E11">f</font></i></a>, <a class="txt" href="numeral1.html#E1:9_1_1_1"><i><font color="Green" title="E10">an</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">m</font> = <font color="Maroon" title="c7">m</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_1_1"/>
<font color="Maroon" title="c8">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">l</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a>, <a class="txt" href="numeral1.html#E11:9_1_1_1"><i><font color="Green" title="E16">m1</font></i></a></i>;<br/>
<a NAME="E14:9_1_1_1"/><b>then </b><i><font color="Green" title="E17">m</font></i>: 
<font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 
<b>by </b><i><a class="txt" href="numeral1.html#E9:9_1_1_1"><i><font color="Green" title="E15">l</font></i></a></i>;<br/>
<a NAME="E15:9_1_1_1"/><i><font color="Green" title="E18">r1</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">l</font>]
 
<b>by </b><i><a class="txt" href="numeral1.html#E9:9_1_1_1"><i><font color="Green" title="E15">l</font></i></a></i>;<br/>
<div><i><font color="Green" title="E19">r2</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c9">i</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_1_1_2"/>
<font color="Maroon" title="c6">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">i</font>
 ;<br/><b>assume </b><a NAME="E2:9_1_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">i</font>]
 ;<br/><a NAME="E3:9_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c2">b</font>
 <b>by </b><i><a class="txt" href="numeral1.html#E3:9_1_1_1"><i><font color="Green" title="E11">f</font></i></a></i>;<br/><b>hence </b><a NAME="E4:9_1_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">i</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1]
 <b>by </b><i><a class="ref" href="nat_2.html#T4" title="NAT_2:th.4">NAT_2:4</a></i>;<br/></div><b>end;</b></div>
<a NAME="E17:9_1_1_1"/><i><font color="Green" title="E20">fim</font></i>: 
for <font color="Olive" title="b1">i</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" title="c6">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S8" title="NAT_1:sch.8">NAT_1:sch 8</a>(<a class="txt" href="numeral1.html#E15:9_1_1_1"><i><font color="Green" title="E18">r1</font></i></a>, <a class="txt" href="numeral1.html#E16:9_1_1_1"><i><font color="Green" title="E19">r2</font></i></a>);<br/></i>
<div><i><font color="Green" title="E21">m'</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c9">i</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_1_1_3"/>
<font color="Maroon" title="c8">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">i</font>
 ;<br/><a NAME="E2:9_1_1_1_3"/><b>then </b>
<font color="Maroon" title="c6">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">i</font>
 <b>by </b><i><a class="txt" href="numeral1.html#E11:9_1_1_1"><i><font color="Green" title="E16">m1</font></i></a>, <a class="ref" href="nat_1.html#T25" title="NAT_1:th.25">NAT_1:25</a></i>;<br/><b>hence </b><a NAME="E3:9_1_1_1_3"/>
<font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="txt" href="numeral1.html#E17:9_1_1_1"><i><font color="Green" title="E20">fim</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E19:9_1_1_1"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c5">f</font> <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="E20:9_1_1_1"/><b>then </b>
<font color="Maroon" title="c8">m</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c5">f</font>
 
<b>by </b><i><a class="ref" href="ordinal1.html#D2" title="ORDINAL1:def.2">ORDINAL1:def 2</a></i>;<br/>
<a NAME="E21:9_1_1_1"/><b>then </b>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">m</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">m</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 
<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="E22:9_1_1_1"/><b>then </b><i><font color="Green" title="E22">fi</font></i>: 
<font color="Maroon" title="c5">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">m</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> is   <a href="afinsq_1.html#NM1" title="AFINSQ_1:NM.1">XFinSequence</a>
 
<b>by </b><i><a class="ref" href="ordinal1.html#D7" title="ORDINAL1:def.7">ORDINAL1:def 7</a></i>;<br/>
<a NAME="E23:9_1_1_1"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">m</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c9">g</font> = <font color="Maroon" title="c5">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">m</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="txt" href="numeral1.html#E22:9_1_1_1"><i><font color="Green" title="E22">fi</font></i></a>, <a class="ref" href="ordinal1.html#D8" title="ORDINAL1:def.8">ORDINAL1:def 8</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="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b>$2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">g</font> <a href="stirl2_1.html#K8" title="STIRL2_1:func.8">.</a> $1</span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">b</font>;<br/>
<a NAME="E25:9_1_1_1"/><i><font color="Green" title="E23">fe</font></i>: 
for <font color="Olive" title="b1">i</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">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">m</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 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">i</font>,<font color="Olive" title="b2">x</font>]
 
;<br/>
<b>consider </b><font color="Maroon" title="c10">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="E27:9_1_1_1"/><i><font color="Green" title="E24">d</font></i>: 
(  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c10">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">m</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 &amp; ( for <font color="Olive" title="b1">i</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">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">m</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">i</font>,<font color="Maroon" title="c10">d</font> <a href="stirl2_1.html#K8" title="STIRL2_1:func.8">.</a> <font color="Olive" title="b1">i</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_1_1"><i><font color="Green" title="E23">fe</font></i></a>);<br/></i>
<b>take </b>
<font color="Maroon" title="c10">d</font>
;<br/>

<b>deffunc </b><font color="Maroon">H<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> ) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = <span class="p1">(<span class="default"><font color="Maroon" title="c10">d</font> <a href="stirl2_1.html#K8" title="STIRL2_1:func.8">.</a> $1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> $1</span>)</span>;<br/>
<b>consider </b><font color="Maroon" title="c11">d'</font> being   <a href="afinsq_1.html#NM1" title="AFINSQ_1:NM.1">XFinSequence</a><b> such that </b><br/><a NAME="E29:9_1_1_1"/><i><font color="Green" title="E25">dd</font></i>: 
(  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c11">d'</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="c10">d</font> &amp; ( for <font color="Olive" title="b1">i</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">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="c10">d</font> holds <br/><font color="Maroon" title="c11">d'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">i</font>) ) )
 <b>from </b><i><a class="ref" href="afinsq_1.html#S2" title="AFINSQ_1:sch.2">AFINSQ_1:sch 2</a>();<br/></i>
<a NAME="E30:9_1_1_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c11">d'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1_1_4"><b>proof </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> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:9_1_1_1_4"/>
<font color="Maroon" title="c12">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c11">d'</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c13">i</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:9_1_1_1_4"/><i><font color="Green" title="E26">i</font></i>: 
( <font color="Maroon" title="c13">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="c11">d'</font> &amp; <font color="Maroon" title="c11">d'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">a</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c14">i</font> = <font color="Maroon" title="c13">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="txt" href="numeral1.html#E3:9_1_1_1_4"><i><font color="Green" title="E26">i</font></i></a></i>;<br/>
<a NAME="E5:9_1_1_1_4"/>
<font color="Maroon" title="c14">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="c11">d'</font>
 
<b>by </b><i><a class="txt" href="numeral1.html#E3:9_1_1_1_4"><i><font color="Green" title="E26">i</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_1_1_4"/><b>then </b>
<font color="Maroon" title="c12">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c14">i</font>)
 
<b>by </b><i><a class="txt" href="numeral1.html#E29:9_1_1_1"><i><font color="Green" title="E25">dd</font></i></a>, <a class="txt" href="numeral1.html#E3:9_1_1_1_4"><i><font color="Green" title="E26">i</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:9_1_1_1_4"/>
<font color="Maroon" title="c12">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c12">d'</font> = <font color="Maroon" title="c11">d'</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="ordinal1.html#D8" title="ORDINAL1:def.8">ORDINAL1:def 8</a></i>;<br/>
<i><font color="Green" title="E26">d'</font></i>:  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c12">d'</font> = 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c12">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="c10">d</font>
<b>by </b><i><a class="txt" href="numeral1.html#E29:9_1_1_1"><i><font color="Green" title="E25">dd</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_1_1"/><i><font color="Green" title="E27">e'</font></i>: 
for <font color="Olive" title="b1">i</font> being <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</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="c12">d'</font> holds <br/><font color="Maroon" title="c12">d'</font> <a href="stirl2_1.html#K8" title="STIRL2_1:func.8">.</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="c10">d</font> <a href="stirl2_1.html#K8" title="STIRL2_1:func.8">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1_1_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">i</font> be  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/>

<b>reconsider </b><font color="Maroon" title="c14">j</font> = <font color="Maroon" title="c13">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_1_1_6"/>
<font color="Maroon" title="c13">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="c12">d'</font>
 ;<br/>

<a NAME="E3:9_1_1_1_6"/><b>then </b>
<font color="Maroon" title="c14">j</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="c10">d</font>
 
<b>by </b><i><a class="txt" href="numeral1.html#E29:9_1_1_1"><i><font color="Green" title="E25">dd</font></i></a>, <a class="ref" href="afinsq_1.html#D1" title="AFINSQ_1:def.1">AFINSQ_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E4:9_1_1_1_6"/>
<font color="Maroon" title="c12">d'</font> <a href="stirl2_1.html#K8" title="STIRL2_1:func.8">.</a> <font color="Maroon" title="c13">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">d</font> <a href="stirl2_1.html#K8" title="STIRL2_1:func.8">.</a> <font color="Maroon" title="c13">i</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</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="txt" href="numeral1.html#E29:9_1_1_1"><i><font color="Green" title="E25">dd</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E34:9_1_1_1"/><i><font color="Green" title="E28">ma</font></i>: 
<font color="Maroon" title="c8">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">m</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E35:9_1_1_1"/><i><font color="Green" title="E29">mm</font></i>: 
<font color="Maroon" title="c8">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">m</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 
<b>by </b><i><a class="ref" href="afinsq_1.html#T1" title="AFINSQ_1:th.1">AFINSQ_1:1</a></i>;<br/>
<a NAME="E36:9_1_1_1"/><b>then </b>
<font color="Maroon" title="c9">g</font> <a href="stirl2_1.html#K8" title="STIRL2_1:func.8">.</a> <font color="Maroon" title="c8">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">m</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>;<br/>
<a NAME="E37:9_1_1_1"/><b>then </b>
<font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">m</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">g</font> <a href="stirl2_1.html#K8" title="STIRL2_1:func.8">.</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c2">b</font>
 
<b>by </b><i><a class="txt" href="numeral1.html#E3:9_1_1_1"><i><font color="Green" title="E11">f</font></i></a></i>;<br/>
<a NAME="E38:9_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c9">g</font> <a href="stirl2_1.html#K8" title="STIRL2_1:func.8">.</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 
<b>by </b><i><a class="txt" href="numeral1.html#E18:9_1_1_1"><i><font color="Green" title="E21">m'</font></i></a>, <a class="txt" href="numeral1.html#E34:9_1_1_1"><i><font color="Green" title="E28">ma</font></i></a></i>;<br/>
<a NAME="E39:9_1_1_1"/><b>then </b><i><font color="Green" title="E30">gm</font></i>: 
<font color="Maroon" title="c9">g</font> <a href="stirl2_1.html#K8" title="STIRL2_1:func.8">.</a> <font color="Maroon" title="c8">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">b</font>
 
<b>by </b><i><a class="txt" href="numeral1.html#E1:9"><i><font color="Green" title="E9">b</font></i></a>, <a class="ref" href="nat_2.html#T14" title="NAT_2:th.14">NAT_2:14</a></i>;<br/>
<a NAME="E40:9_1_1_1"/><i><font color="Green" title="E31">el</font></i>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c10">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">m</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 
<b>by </b><i><a class="txt" href="numeral1.html#E27:9_1_1_1"><i><font color="Green" title="E24">d</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 consider </b><font color="Maroon" title="c13">s</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> , <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E42:9_1_1_1"/><i><font color="Green" title="E32">s</font></i>: 
( <font color="Maroon" title="c13">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">d'</font> <a href="stirl2_1.html#K8" title="STIRL2_1:func.8">.</a> 0 &amp; ( for <font color="Olive" title="b1">i</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">i</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>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c12">d'</font> holds <br/><font color="Maroon" title="c13">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="binop_2.html#K47" title="BINOP_2:func.47">addnat</a>  <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">d'</font> <a href="stirl2_1.html#K8" title="STIRL2_1:func.8">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">i</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> ) &amp; <a href="binop_2.html#K47" title="BINOP_2:func.47">addnat</a>  <a href="stirl2_1.html#K4" title="STIRL2_1:func.4">"**"</a> <font color="Maroon" title="c12">d'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">s</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</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="c12">d'</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> )
 <b>by </b><i><a class="ref" href="stirl2_1.html#D3" title="STIRL2_1:def.3">STIRL2_1:def 3</a>, <a class="txt" href="numeral1.html#E29:9_1_1_1"><i><font color="Green" title="E25">dd</font></i></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> ] <b>means </b>( $1 <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="c12">d'</font> implies <font color="Maroon" title="c13">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default">$1 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></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> <span class="p3">(<span class="default">$1 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span>)</span> );<br/>
<a NAME="E43:9_1_1_1"/><i><font color="Green" title="E33">zm</font></i>: 
0 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">m</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</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/>
<b>then </b><font color="Maroon" title="c12">d'</font> <a href="stirl2_1.html#K8" title="STIRL2_1:func.8">.</a> 0 = 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">d</font> <a href="stirl2_1.html#K8" title="STIRL2_1:func.8">.</a> 0</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> 0</span>)</span>
<b>by </b><i><a class="txt" href="numeral1.html#E27:9_1_1_1"><i><font color="Green" title="E24">d</font></i></a>, <a class="txt" href="numeral1.html#E32:9_1_1_1"><i><font color="Green" title="E26">d'</font></i></a>, <a class="txt" href="numeral1.html#E33:9_1_1_1"><i><font color="Green" title="E27">e'</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">g</font> <a href="stirl2_1.html#K8" title="STIRL2_1:func.8">.</a> 0</span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> 0</span>)</span>
<b>by </b><i><a class="txt" href="numeral1.html#E27:9_1_1_1"><i><font color="Green" title="E24">d</font></i></a>, <a class="txt" href="numeral1.html#E43:9_1_1_1"><i><font color="Green" title="E33">zm</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 0</span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> 0</span>)</span>
<b>by </b><i><a class="txt" href="numeral1.html#E43:9_1_1_1"><i><font color="Green" title="E33">zm</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> 1
<b>by </b><i><a class="txt" href="numeral1.html#E3:9_1_1_1"><i><font color="Green" title="E11">f</font></i></a>, <a class="ref" href="newton.html#T9" title="NEWTON:th.9">NEWTON:9</a></i>
<br/>.= 
<font color="Maroon" title="c3">N</font> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">B</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">N</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c4">B</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T77" title="NEWTON:th.77">NEWTON:77</a>, <a class="txt" href="numeral1.html#E1:9"><i><font color="Green" title="E9">b</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</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> <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"><font color="Maroon" title="c2">b</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T10" title="NEWTON:th.10">NEWTON:10</a></i>
<br/>.= 
<font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default">0 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></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> <span class="p3">(<span class="default">0 <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#E3:9_1_1_1"><i><font color="Green" title="E11">f</font></i></a></i>
;<br/>
<a NAME="E45:9_1_1_1"/><b>then </b><i><font color="Green" title="E34">iz</font></i>: 
<font color="Maroon">S<sub>4</sub></font>[0]
 
<b>by </b><i><a class="txt" href="numeral1.html#E42:9_1_1_1"><i><font color="Green" title="E32">s</font></i></a></i>;<br/>
<div><i><font color="Green" title="E35">ik</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1_1_8"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c14">i</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_1_1_8"/><i><font color="Green" title="E36">i</font></i>: 
<font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c14">i</font>]
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1_1_8_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:9_1_1_1_8_1"/><i><font color="Green" title="E37">ii</font></i>: 
<font color="Maroon" title="c14">i</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>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c12">d'</font>
 ;<br/><a NAME="E2:9_1_1_1_8_1"/><b>then </b>
<font color="Maroon" title="c14">i</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#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c12">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="E3:9_1_1_1_8_1"/><b>then </b><i><font color="Green" title="E38">ie</font></i>: 
<font color="Maroon" title="c14">i</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="c12">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/><b>thus </b><font color="Maroon" title="c13">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">i</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> = 
<a href="binop_2.html#K47" title="BINOP_2:func.47">addnat</a>  <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">d'</font> <a href="stirl2_1.html#K8" title="STIRL2_1:func.8">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">i</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#E42:9_1_1_1"><i><font color="Green" title="E32">s</font></i></a>, <a class="txt" href="numeral1.html#E1:9_1_1_1_8_1"><i><font color="Green" title="E37">ii</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <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="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">i</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></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> <span class="p4">(<span class="default"><font color="Maroon" title="c14">i</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">d'</font> <a href="stirl2_1.html#K8" title="STIRL2_1:func.8">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">i</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_1_1_1_8"><i><font color="Green" title="E36">i</font></i></a>, <a class="txt" href="numeral1.html#E1:9_1_1_1_8_1"><i><font color="Green" title="E37">ii</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a>, <a class="ref" href="binop_2.html#D23" title="BINOP_2:def.23">BINOP_2:def 23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <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="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">i</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></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> <span class="p4">(<span class="default"><font color="Maroon" title="c14">i</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">d</font> <a href="stirl2_1.html#K8" title="STIRL2_1:func.8">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">i</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></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> <span class="p3">(<span class="default"><font color="Maroon" title="c14">i</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#E33:9_1_1_1"><i><font color="Green" title="E27">e'</font></i></a>, <a class="txt" href="numeral1.html#E3:9_1_1_1_8_1"><i><font color="Green" title="E38">ie</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <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="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">i</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></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> <span class="p4">(<span class="default"><font color="Maroon" title="c14">i</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">g</font> <a href="stirl2_1.html#K8" title="STIRL2_1:func.8">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">i</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</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"><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="c14">i</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#E27:9_1_1_1"><i><font color="Green" title="E24">d</font></i></a>, <a class="txt" href="numeral1.html#E32:9_1_1_1"><i><font color="Green" title="E26">d'</font></i></a>, <a class="txt" href="numeral1.html#E3:9_1_1_1_8_1"><i><font color="Green" title="E38">ie</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <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="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">i</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></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> <span class="p4">(<span class="default"><font color="Maroon" title="c14">i</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">i</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</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"><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="c14">i</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#E27:9_1_1_1"><i><font color="Green" title="E24">d</font></i></a>, <a class="txt" href="numeral1.html#E32:9_1_1_1"><i><font color="Green" title="E26">d'</font></i></a>, <a class="txt" href="numeral1.html#E3:9_1_1_1_8_1"><i><font color="Green" title="E38">ie</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <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="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">i</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></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> <span class="p4">(<span class="default"><font color="Maroon" title="c14">i</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">i</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> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p0">(<span class="default"><font color="Maroon" title="c14">i</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c4">B</font></span>)</span></span>)</span></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> <span class="p3">(<span class="default"><font color="Maroon" title="c14">i</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="ref" href="newton.html#T77" title="NEWTON:th.77">NEWTON:77</a>, <a class="txt" href="numeral1.html#E1:9"><i><font color="Green" title="E9">b</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <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="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">i</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></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> <span class="p4">(<span class="default"><font color="Maroon" title="c14">i</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">i</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> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c14">i</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span>)</span></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> <span class="p3">(<span class="default"><font color="Maroon" title="c14">i</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#E3:9_1_1_1"><i><font color="Green" title="E11">f</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <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="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">i</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></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> <span class="p4">(<span class="default"><font color="Maroon" title="c14">i</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">i</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></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> <span class="p4">(<span class="default"><font color="Maroon" title="c14">i</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#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c14">i</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></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="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</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="c14">i</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <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="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">i</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></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> <span class="p4">(<span class="default"><font color="Maroon" title="c14">i</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">i</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></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> <span class="p4">(<span class="default"><font color="Maroon" title="c14">i</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#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c14">i</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></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> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c14">i</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T11" title="NEWTON:th.11">NEWTON:11</a></i>
<br/>.= 
<font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c14">i</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c14">i</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span>)</span>

;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:9_1_1_1_8"/>
<font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c14">i</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1]
 ;<br/></div><b>end;</b></div>
<a NAME="E47:9_1_1_1"/><i><font color="Green" title="E36">ind</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">i</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="numeral1.html#E45:9_1_1_1"><i><font color="Green" title="E34">iz</font></i></a>, <a class="txt" href="numeral1.html#E46:9_1_1_1"><i><font color="Green" title="E35">ik</font></i></a>);<br/></i>
<b>reconsider </b><font color="Maroon" title="c14">ld</font> = <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c12">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>  <b>by </b><i><a class="txt" href="numeral1.html#E29:9_1_1_1"><i><font color="Green" title="E25">dd</font></i></a>, <a class="txt" href="numeral1.html#E40:9_1_1_1"><i><font color="Green" title="E31">el</font></i></a></i>;<br/>
<a NAME="E49:9_1_1_1"/>
<font color="Maroon" title="c14">ld</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="c12">d'</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T46" title="XREAL_1:th.46">XREAL_1:46</a></i>;<br/>
<b>then </b><font color="Maroon" title="c13">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">ld</font> = 
<font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">m</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></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> <span class="p3">(<span class="default"><font color="Maroon" title="c8">m</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#E40:9_1_1_1"><i><font color="Green" title="E31">el</font></i></a>, <a class="txt" href="numeral1.html#E29:9_1_1_1"><i><font color="Green" title="E25">dd</font></i></a>, <a class="txt" href="numeral1.html#E47:9_1_1_1"><i><font color="Green" title="E36">ind</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c1">n</font>
<b>by </b><i><a class="txt" href="numeral1.html#E9:9_1_1_1"><i><font color="Green" title="E15">l</font></i></a>, <a class="txt" href="numeral1.html#E11:9_1_1_1"><i><font color="Green" title="E16">m1</font></i></a></i>
;<br/>
<a NAME="E51:9_1_1_1"/><b>then </b>
<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="c12">d'</font>
 
<b>by </b><i><a class="txt" href="numeral1.html#E42:9_1_1_1"><i><font color="Green" title="E32">s</font></i></a></i>;<br/>
<b>hence </b><a NAME="E52:9_1_1_1"/>
 <a href="numeral1.html#K1" title="NUMERAL1:func.1">value</a> <font color="Maroon" title="c10">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>
 <b>by </b><i><a class="txt" href="numeral1.html#E32:9_1_1_1"><i><font color="Green" title="E26">d'</font></i></a>, <a class="txt" href="numeral1.html#E33:9_1_1_1"><i><font color="Green" title="E27">e'</font></i></a>, <a class="ref" href="numeral1.html#D1" target="_self" title="NUMERAL1:def.1">val</a></i>;<br/>

<font color="Maroon" title="c10">d</font> <a href="stirl2_1.html#K8" title="STIRL2_1:func.8">.</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="c10">d</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">g</font> <a href="stirl2_1.html#K8" title="STIRL2_1:func.8">.</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">b</font>
<b>by </b><i><a class="txt" href="numeral1.html#E35:9_1_1_1"><i><font color="Green" title="E29">mm</font></i></a>, <a class="txt" href="numeral1.html#E40:9_1_1_1"><i><font color="Green" title="E31">el</font></i></a>, <a class="txt" href="numeral1.html#E27:9_1_1_1"><i><font color="Green" title="E24">d</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c9">g</font> <a href="stirl2_1.html#K8" title="STIRL2_1:func.8">.</a> <font color="Maroon" title="c8">m</font>
<b>by </b><i><a class="ref" href="nat_d.html#T24" title="NAT_D:th.24">NAT_D:24</a>, <a class="txt" href="numeral1.html#E39:9_1_1_1"><i><font color="Green" title="E30">gm</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E54:9_1_1_1"/>
<font color="Maroon" title="c10">d</font> <a href="stirl2_1.html#K8" title="STIRL2_1:func.8">.</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="c10">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> 0
 <b>by </b><i><a class="txt" href="numeral1.html#E35:9_1_1_1"><i><font color="Green" title="E29">mm</font></i></a>, <a class="txt" href="numeral1.html#E14:9_1_1_1"><i><font color="Green" title="E17">m</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>;<br/>

<b>let </b><font color="Maroon" title="c15">i</font> be  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/>

<b>assume </b><a NAME="E55:9_1_1_1"/>
<font color="Maroon" title="c15">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="c10">d</font>
 ;<br/>

<a NAME="E56:9_1_1_1"/><b>then </b>
<font color="Maroon" title="c10">d</font> <a href="stirl2_1.html#K8" title="STIRL2_1:func.8">.</a> <font color="Maroon" title="c15">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">g</font> <a href="stirl2_1.html#K8" title="STIRL2_1:func.8">.</a> <font color="Maroon" title="c15">i</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">b</font>
 
<b>by </b><i><a class="txt" href="numeral1.html#E27:9_1_1_1"><i><font color="Green" title="E24">d</font></i></a></i>;<br/>
<b>hence </b><a NAME="E57:9_1_1_1"/>
( 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">d</font> <a href="stirl2_1.html#K8" title="STIRL2_1:func.8">.</a> <font color="Maroon" title="c15">i</font> &amp; <font color="Maroon" title="c10">d</font> <a href="stirl2_1.html#K8" title="STIRL2_1:func.8">.</a> <font color="Maroon" title="c15">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">b</font> )
 <b>by </b><i><a class="ref" href="nat_d.html#T1" title="NAT_D:th.1">NAT_D:1</a>, <a class="txt" href="numeral1.html#E1:9"><i><font color="Green" title="E9">b</font></i></a></i>;<br/>


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

<b>assume </b><a NAME="E3:9_1_1"/>
<font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c5">d</font> = <span class="p1"><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">&lt;%</a><span class="default">0</span><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">%&gt;</a></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>  ;<br/>
<b>take </b>
<font color="Maroon" title="c5">d</font>
;<br/>

<b>thus </b><a NAME="E5:9_1_1"/>
<font color="Maroon" title="c5">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">0</span><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">%&gt;</a></span>
 ;<br/>


</div>
