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

<b>let </b><font color="Maroon" title="c1">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b>$1 <a href="stirl2_1.html#K3" title="STIRL2_1:func.3">block</a> 4 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 24</span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">4 <a href="newton.html#K3" title="NEWTON:func.3">|^</a> $1</span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p4">(<span class="default">4 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p5">(<span class="default">3 <a href="newton.html#K3" title="NEWTON:func.3">|^</a> $1</span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p3">(<span class="default">6 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p4">(<span class="default">2 <a href="newton.html#K3" title="NEWTON:func.3">|^</a> $1</span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 4</span>)</span>;<br/>
<a NAME="E1:78"/><i><font color="Green" title="E50">A1</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[3]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="78_1"><b>proof </b></a><div class="add">

<span class="p1">(<span class="default">1 <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 24</span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">4 <a href="newton.html#K3" title="NEWTON:func.3">|^</a> 3</span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p4">(<span class="default">4 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p5">(<span class="default">3 <a href="newton.html#K3" title="NEWTON:func.3">|^</a> 3</span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p3">(<span class="default">6 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p4">(<span class="default">2 <a href="newton.html#K3" title="NEWTON:func.3">|^</a> 3</span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 4</span>)</span> = 
<span class="p1">(<span class="default">1 <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 24</span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">4 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> 4</span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> 4</span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p4">(<span class="default">4 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p5">(<span class="default">3 <a href="newton.html#K3" title="NEWTON:func.3">|^</a> 3</span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p3">(<span class="default">6 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p4">(<span class="default">2 <a href="newton.html#K3" title="NEWTON:func.3">|^</a> 3</span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 4</span>)</span>
<b>by </b><i><a class="txt" href="stirl2_1.html#E66"><i><font color="Green" title="E49">Lm4</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 24</span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">64 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p4">(<span class="default">4 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">3 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> 3</span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> 3</span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p3">(<span class="default">6 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p4">(<span class="default">2 <a href="newton.html#K3" title="NEWTON:func.3">|^</a> 3</span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 4</span>)</span>
<b>by </b><i><a class="txt" href="stirl2_1.html#E66"><i><font color="Green" title="E49">Lm4</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 24</span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">64 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p4">(<span class="default">4 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> 27</span>)</span></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p3">(<span class="default">6 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> 2</span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> 2</span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 4</span>)</span>
<b>by </b><i><a class="txt" href="stirl2_1.html#E66"><i><font color="Green" title="E49">Lm4</font></i></a></i>
<br/>.= 
3 <a href="stirl2_1.html#K3" title="STIRL2_1:func.3">block</a> 4
<b>by </b><i><a class="ref" href="stirl2_1.html#T29" target="_self" title="STIRL2_1:th.29">Th29</a></i>
;<br/>
<b>hence </b><a NAME="E2:78_1"/>
<font color="Maroon">S<sub>1</sub></font>[3]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E2:78"/><i><font color="Green" title="E51">A2</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 3 &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="78_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:78_2"/><i><font color="Green" title="E52">A3</font></i>: 
( <font color="Maroon" title="c2">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 3 &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">k</font>] )
 ;<br/>

<a NAME="E2:78_2"/>
<font color="Maroon" title="c2">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2
 
<b>by </b><i><a class="txt" href="stirl2_1.html#E1:78_2"><i><font color="Green" title="E52">A3</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E3:78_2"/><b>then </b>
<font color="Maroon" title="c2">k</font> <a href="stirl2_1.html#K3" title="STIRL2_1:func.3">block</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 6</span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">3 <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p3">(<span class="default">3 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p4">(<span class="default">2 <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c2">k</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> 3</span>)</span>
 
<b>by </b><i><a class="ref" href="stirl2_1.html#T58" target="_self" title="STIRL2_1:th.58">Th58</a></i>;<br/>
<b>hence </b><span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="stirl2_1.html#K3" title="STIRL2_1:func.3">block</a> 4 = 
<span class="p1">(<span class="default">4 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="stirl2_1.html#K3" title="STIRL2_1:func.3">block</a> <span class="p3">(<span class="default">3 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 6</span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">3 <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p4">(<span class="default">3 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p5">(<span class="default">2 <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c2">k</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> 3</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="stirl2_1.html#T56" target="_self" title="STIRL2_1:th.56">Th56</a></i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 24</span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">4 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p5">(<span class="default">4 <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c2">k</font></span>)</span></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p4">(<span class="default">4 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p5">(<span class="default">3 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p0">(<span class="default">3 <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c2">k</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p3">(<span class="default">6 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p4">(<span class="default">2 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p5">(<span class="default">2 <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c2">k</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 4</span>)</span>
<b>by </b><i><a class="txt" href="stirl2_1.html#E1:78_2"><i><font color="Green" title="E52">A3</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 24</span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">4 <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p4">(<span class="default">4 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p5">(<span class="default">3 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p0">(<span class="default">3 <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c2">k</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p3">(<span class="default">6 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p4">(<span class="default">2 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p5">(<span class="default">2 <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c2">k</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 4</span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T11" title="NEWTON:th.11">NEWTON:11</a></i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 24</span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">4 <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p4">(<span class="default">4 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p5">(<span class="default">3 <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <span class="p0">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p3">(<span class="default">6 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p4">(<span class="default">2 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p5">(<span class="default">2 <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c2">k</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 4</span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T11" title="NEWTON:th.11">NEWTON:11</a></i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 24</span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">4 <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p4">(<span class="default">4 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p5">(<span class="default">3 <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <span class="p0">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p3">(<span class="default">6 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p4">(<span class="default">2 <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 4</span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T11" title="NEWTON:th.11">NEWTON:11</a></i>
;<br/><br/>


</div><b>end;</b></div>
<a NAME="E3:78"/>
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 3 holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</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="stirl2_1.html#E1:78"><i><font color="Green" title="E50">A1</font></i></a>, <a class="txt" href="stirl2_1.html#E2:78"><i><font color="Green" title="E51">A2</font></i></a>);<br/></i>
<b>hence </b><a NAME="E4:78"/>
( <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 3 implies <font color="Maroon" title="c1">n</font> <a href="stirl2_1.html#K3" title="STIRL2_1:func.3">block</a> 4 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 24</span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">4 <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p4">(<span class="default">4 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p5">(<span class="default">3 <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p3">(<span class="default">6 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p4">(<span class="default">2 <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 4</span>)</span> )
 ;<br/>


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