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

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b><span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="binop_2.html#K23">+</a> 2</span>)</span> <a href="stirl2_1.html#K3" target="_self">block</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">3 <a href="binop_2.html#K24">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="binop_2.html#K23">+</a> 2</span>)</span> <a href="newton.html#K9">choose</a> 4</span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="binop_2.html#K23">+</a> 2</span>)</span> <a href="newton.html#K9">choose</a> 3</span>)</span>;<br/>
<a NAME="E1:82"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="82_1"><b>proof </b></a><div class="add">

<a NAME="E1:82_1"/>
( 2 <a href="newton.html#K9">choose</a> 4 <a href="hidden.html#R1">=</a> 0 &amp; 2 <a href="newton.html#K9">choose</a> 3 <a href="hidden.html#R1">=</a> 0 &amp; 2 <a href="stirl2_1.html#K3" target="_self">block</a> 0 <a href="hidden.html#R1">=</a> 0 )
 
<b>by </b><i><a class="ref" href="newton.html#D3">NEWTON:def 3</a>, </i>;<br/>
<b>hence </b><a NAME="E2:82_1"/>
<font color="Maroon">S<sub>1</sub></font>[0]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E2:82"/><i><font color="Green">E39</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font> <a href="binop_2.html#K23">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="82_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:82_2"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>]
 ;<br/>

<b>set </b><font color="Maroon">k2</font> = <font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 2;<br/>
<b>set </b><font color="Maroon">k1</font> = <font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1;<br/>
<b>set </b><font color="Maroon">k3</font> = <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 2</span>)</span> <a href="binop_2.html#K23">+</a> 1;<br/>
<i><font color="Green">E47</font></i>: <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 2</span>)</span> <a href="stirl2_1.html#K3" target="_self">block</a> <font color="Maroon">k</font> = 
<span class="p1">(<span class="default">3 <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 2</span>)</span> <a href="binop_2.html#K11">*</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 2</span>)</span> <a href="binop_2.html#K10">-</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 2</span>)</span> <a href="binop_2.html#K10">-</a> 2</span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 2</span>)</span> <a href="binop_2.html#K10">-</a> 3</span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> 24</span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 2</span>)</span> <a href="newton.html#K9">choose</a> 3</span>)</span>
<b>by </b><i><a class="ref" href="stirl2_1.html#T17" target="_self">Th17</a>, </i>
<br/>.= 
<span class="p1">(<span class="default">3 <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 2</span>)</span> <a href="binop_2.html#K11">*</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 2</span>)</span> <a href="binop_2.html#K10">-</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 2</span>)</span> <a href="binop_2.html#K10">-</a> 2</span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 2</span>)</span> <a href="binop_2.html#K10">-</a> 3</span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> 24</span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 2</span>)</span> <a href="binop_2.html#K11">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 2</span>)</span> <a href="binop_2.html#K10">-</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 2</span>)</span> <a href="binop_2.html#K10">-</a> 2</span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> 6</span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 2</span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 2</span>)</span> <a href="binop_2.html#K10">-</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K11">*</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">3 <a href="binop_2.html#K11">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 2</span>)</span> <a href="binop_2.html#K10">-</a> 2</span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 2</span>)</span> <a href="binop_2.html#K10">-</a> 3</span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> 24</span>)</span> <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">4 <a href="binop_2.html#K11">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 2</span>)</span> <a href="binop_2.html#K10">-</a> 2</span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> 24</span>)</span></span>)</span>

;<br/>
<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="stirl2_1.html#K3" target="_self">block</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 2</span>)</span> <a href="newton.html#K9">choose</a> 2</span>)</span>
<b>by </b><i><a class="ref" href="stirl2_1.html#D1" target="_self">Def1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 2</span>)</span> <a href="binop_2.html#K11">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 2</span>)</span> <a href="binop_2.html#K10">-</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> 2</span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 2</span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 2</span>)</span> <a href="binop_2.html#K10">-</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K24">*</a> 12</span>)</span> <a href="binop_2.html#K12">/</a> 24</span>)</span>

;<br/>
<b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 2</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="stirl2_1.html#K3" target="_self">block</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 2</span>)</span> <a href="binop_2.html#K24">*</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K24">*</a> 12</span>)</span> <a href="binop_2.html#K12">/</a> 24</span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 2</span>)</span> <a href="binop_2.html#K24">*</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">3 <a href="binop_2.html#K24">*</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 2</span>)</span> <a href="binop_2.html#K10">-</a> 3</span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> 24</span>)</span> <a href="binop_2.html#K9">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">4 <a href="binop_2.html#K24">*</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K12">/</a> 24</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="stirl2_1.html#T18" target="_self">Th18</a>, <a class="ref" href="stirl2_1.html#T12" target="_self">Th12</a></i>
<br/>.= 
<span class="p1">(<span class="default">3 <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 2</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K11">*</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 2</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K10">-</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 2</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K10">-</a> 2</span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 2</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K10">-</a> 3</span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> 24</span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 2</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K24">*</a> <span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 2</span>)</span></span>)</span> <a href="binop_2.html#K24">*</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> 6</span>)</span>

<br/>.= 
<span class="p1">(<span class="default">3 <a href="binop_2.html#K24">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 2</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="newton.html#K9">choose</a> 4</span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 2</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K11">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 2</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K10">-</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 2</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K10">-</a> 2</span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> 6</span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default">3 <a href="binop_2.html#K24">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 2</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="newton.html#K9">choose</a> 4</span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 2</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="newton.html#K9">choose</a> 3</span>)</span>
<b>by </b><i/>
;<br/>
<b>hence </b><a NAME="E5:82_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:82"/>
for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="ref" href="stirl2_1.html#T15" target="_self">Th15</a>, <a class="ref" href="stirl2_1.html#T16" target="_self">Th16</a>);<br/></i>
<b>hence </b><a NAME="E4:82"/>
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 2</span>)</span> <a href="stirl2_1.html#K3" target="_self">block</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">3 <a href="binop_2.html#K24">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 2</span>)</span> <a href="newton.html#K9">choose</a> 4</span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 2</span>)</span> <a href="newton.html#K9">choose</a> 3</span>)</span>
 ;<br/>


</div>
