<?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><font color="Maroon">a<sub>1</sub></font> <a href="stirl2_1.html#K3" target="_self">block</a> 3 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K12">/</a> 6</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">3 <a href="newton.html#K3">|^</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p3">(<span class="default">3 <a href="binop_2.html#K11">*</a> <span class="p4">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">a<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> 3</span>)</span>;<br/>
<a NAME="E1:76"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[2]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1"><b>proof </b></a><div class="add">

<span class="p1">(<span class="default">1 <a href="binop_2.html#K12">/</a> 6</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">3 <a href="newton.html#K3">|^</a> 2</span>)</span> <a href="binop_2.html#K10">-</a> <span class="p3">(<span class="default">3 <a href="binop_2.html#K11">*</a> <span class="p4">(<span class="default">2 <a href="newton.html#K3">|^</a> 2</span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> 3</span>)</span> = 
<span class="p1">(<span class="default">1 <a href="binop_2.html#K12">/</a> 6</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">3 <a href="binop_2.html#K24">*</a> 3</span>)</span> <a href="binop_2.html#K10">-</a> <span class="p3">(<span class="default">3 <a href="binop_2.html#K11">*</a> <span class="p4">(<span class="default">2 <a href="newton.html#K3">|^</a> 2</span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> 3</span>)</span>
<b>by </b><i><a class="ref" href="wsierp_1.html#T2">WSIERP_1:2</a></i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="binop_2.html#K12">/</a> 6</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">3 <a href="binop_2.html#K24">*</a> 3</span>)</span> <a href="binop_2.html#K10">-</a> <span class="p3">(<span class="default">3 <a href="binop_2.html#K24">*</a> <span class="p4">(<span class="default">2 <a href="binop_2.html#K24">*</a> 2</span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> 3</span>)</span>
<b>by </b><i><a class="ref" href="wsierp_1.html#T2">WSIERP_1:2</a></i>
<br/>.= 
2 <a href="stirl2_1.html#K3" target="_self">block</a> 3
<b>by </b><i/>
;<br/>
<b>hence </b><a NAME="E2:76_1"/>
<font color="Maroon">S<sub>1</sub></font>[2]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E2:76"/><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="Olive">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 &amp; <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="76_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="E1:76_2"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>] )
 ;<br/>

<a NAME="E2:76_2"/>
<font color="Maroon">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E3:76_2"/><b>then </b>
<font color="Maroon">k</font> <a href="stirl2_1.html#K3" target="_self">block</a> 2 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K12">/</a> 2</span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K10">-</a> 2</span>)</span>
 
<b>by </b><i/>;<br/>
<b>hence </b><span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="stirl2_1.html#K3" target="_self">block</a> 3 = 
<span class="p1">(<span class="default">3 <a href="binop_2.html#K24">*</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="stirl2_1.html#K3" target="_self">block</a> <span class="p3">(<span class="default">2 <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K12">/</a> 2</span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K10">-</a> 2</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="stirl2_1.html#T12" target="_self">Th12</a></i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="binop_2.html#K12">/</a> 6</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">3 <a href="binop_2.html#K11">*</a> <span class="p4">(<span class="default">3 <a href="newton.html#K3">|^</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">3 <a href="binop_2.html#K24">*</a> 2</span>)</span> <a href="binop_2.html#K11">*</a> <span class="p4">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> 3</span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="binop_2.html#K12">/</a> 6</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">3 <a href="newton.html#K3">|^</a> <span class="p4">(<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#K10">-</a> <span class="p3">(<span class="default">3 <a href="binop_2.html#K11">*</a> <span class="p4">(<span class="default">2 <a href="binop_2.html#K11">*</a> <span class="p5">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> 3</span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T11">NEWTON:11</a></i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="binop_2.html#K12">/</a> 6</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">3 <a href="newton.html#K3">|^</a> <span class="p4">(<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#K10">-</a> <span class="p3">(<span class="default">3 <a href="binop_2.html#K11">*</a> <span class="p4">(<span class="default">2 <a href="newton.html#K3">|^</a> <span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> 3</span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T11">NEWTON:11</a></i>
;<br/><br/>


</div><b>end;</b></div>
<a NAME="E3:76"/>
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="Olive">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S9">NAT_1:sch 9</a>(<a class="ref" href="stirl2_1.html#T14" target="_self">Th14</a>, );<br/></i>
<b>hence </b><a NAME="E4:76"/>
( <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 implies <font color="Maroon">n</font> <a href="stirl2_1.html#K3" target="_self">block</a> 3 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K12">/</a> 6</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">3 <a href="newton.html#K3">|^</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p3">(<span class="default">3 <a href="binop_2.html#K11">*</a> <span class="p4">(<span class="default">2 <a href="newton.html#K3">|^</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> 3</span>)</span> )
 ;<br/>


</div>
