<?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> <a href="pepin.html#K5" target="_self">Fermat</a> 2 <a href="nat_1.html#R1">divides</a> <span class="p1">(<span class="default">3 <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">16 <a href="nat_1.html#K2">*</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 8</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1;<br/>
<a NAME="E1:88"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="88_1"><b>proof </b></a><div class="add">

<a NAME="E1:88_1"/>
<span class="p1">(<span class="default">3 <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">16 <a href="nat_1.html#K2">*</a> 0</span>)</span> <a href="nat_1.html#K1">+</a> 8</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> 17 <a href="nat_1.html#K2">*</a> 386
 
<b>by </b><i><a class="ref" href="pepin.html#T10" target="_self">Th10</a></i>;<br/>
<b>hence </b><a NAME="E2:88_1"/>
<font color="Maroon">S<sub>1</sub></font>[0]
 <b>by </b><i><a class="txt" href="pepin.html#E4"><i><font color="Green">Lemma54</font></i></a>, <a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:88"/><i><font color="Green">E45</font></i>: 
for <font color="Olive">n</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">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="88_2"><b>proof </b></a><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>assume </b><a NAME="E1:88_2"/>
 <a href="pepin.html#K5" target="_self">Fermat</a> 2 <a href="nat_1.html#R1">divides</a> <span class="p1">(<span class="default">3 <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">16 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 8</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
 ;<br/>

<b>then consider </b><font color="Maroon">m</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:88_2"/><i><font color="Green">E49</font></i>: 
<span class="p1">(<span class="default">3 <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">16 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 8</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pepin.html#K5" target="_self">Fermat</a> 2</span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/>
<a NAME="E4:88_2"/><i><font color="Green">E50</font></i>: 
3 <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">16 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 8</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pepin.html#K5" target="_self">Fermat</a> 2</span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 
<b>by </b><i><a class="txt" href="pepin.html#E22"><i><font color="Green">Lemma75</font></i></a></i>;<br/>
<a NAME="E5:88_2"/><i><font color="Green">E51</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> 6561</span>)</span> <a href="xcmplx_0.html#K3">*</a> 6561</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default">386 <a href="nat_1.html#K2">*</a> 6560</span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="88_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:88_2_1"/>
3 <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default">16 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 3 <a href="wsierp_1.html#K2">|^</a> 0
 
<b>by </b><i><a class="ref" href="prepower.html#T107">PREPOWER:107</a></i>;<br/>
<a NAME="E2:88_2_1"/><b>then </b>
3 <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default">16 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<b>by </b><i><a class="ref" href="newton.html#T9">NEWTON:9</a></i>;<br/>
<a NAME="E3:88_2_1"/><b>then </b>
<span class="p1">(<span class="default">3 <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default">16 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E4:88_2_1"/><b>then </b>
3 <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default">16 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E5:88_2_1"/><b>then </b>
<span class="p1">(<span class="default">3 <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default">16 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default">3 <a href="wsierp_1.html#K2">|^</a> 8</span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default">3 <a href="wsierp_1.html#K2">|^</a> 8</span>)</span>
 
<b>by </b><i><a class="ref" href="pepin.html#T10" target="_self">Th10</a>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/>
<a NAME="E6:88_2_1"/><b>then </b>
3 <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">16 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 8</span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="newton.html#T13">NEWTON:13</a></i>;<br/>
<a NAME="E7:88_2_1"/><b>then </b>
<span class="p1">(<span class="default">3 <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">16 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 8</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E8:88_2_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pepin.html#K5" target="_self">Fermat</a> 2</span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> 6561 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 <a href="nat_1.html#K2">*</a> 6561
 
<b>by </b><i><a class="txt" href="pepin.html#E22"><i><font color="Green">Lemma75</font></i></a>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/>
<a NAME="E9:88_2_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pepin.html#K5" target="_self">Fermat</a> 2</span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> 6561</span>)</span> <a href="xcmplx_0.html#K3">*</a> 6561 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 6561 <a href="nat_1.html#K2">*</a> 6561
 
<b>by </b><i><a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/>
<a NAME="E10:88_2_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pepin.html#K5" target="_self">Fermat</a> 2</span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> 6561</span>)</span> <a href="xcmplx_0.html#K3">*</a> 6561 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 6562 <a href="nat_1.html#K2">*</a> 6560
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E11:88_2_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pepin.html#K5" target="_self">Fermat</a> 2</span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> 6561</span>)</span> <a href="xcmplx_0.html#K3">*</a> 6561</span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><a href="pepin.html#K5" target="_self">Fermat</a> 2</span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default">6560 <a href="nat_1.html#K2">*</a> 6562</span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><a href="pepin.html#K5" target="_self">Fermat</a> 2</span>)</span>
 
<b>by </b><i><a class="ref" href="real_1.html#T73">REAL_1:73</a></i>;<br/>
<a NAME="E12:88_2_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> 6561</span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><a href="pepin.html#K5" target="_self">Fermat</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> 6561</span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pepin.html#K5" target="_self">Fermat</a> 2</span>)</span> <a href="xcmplx_0.html#K5">"</a> </span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default">6560 <a href="nat_1.html#K2">*</a> 6562</span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><a href="pepin.html#K5" target="_self">Fermat</a> 2</span>)</span>
 
<b>by </b><i><a class="ref" href="xcmplx_0.html#D9">XCMPLX_0:def 9</a></i>;<br/>
<a NAME="E13:88_2_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> 6561</span>)</span> <a href="xcmplx_0.html#K3">*</a> 6561</span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pepin.html#K5" target="_self">Fermat</a> 2</span>)</span> <a href="xcmplx_0.html#K5">"</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="pepin.html#K5" target="_self">Fermat</a> 2</span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default">6560 <a href="nat_1.html#K2">*</a> 6562</span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><a href="pepin.html#K5" target="_self">Fermat</a> 2</span>)</span>
 
;<br/>
<a NAME="E14:88_2_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> 6561</span>)</span> <a href="xcmplx_0.html#K3">*</a> 6561</span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><a href="pepin.html#K5" target="_self">Fermat</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="pepin.html#K5" target="_self">Fermat</a> 2</span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default">6560 <a href="nat_1.html#K2">*</a> 6562</span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><a href="pepin.html#K5" target="_self">Fermat</a> 2</span>)</span>
 
<b>by </b><i><a class="ref" href="xcmplx_0.html#D9">XCMPLX_0:def 9</a></i>;<br/>
<a NAME="E15:88_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> 6561</span>)</span> <a href="xcmplx_0.html#K3">*</a> 6561 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default">6560 <a href="nat_1.html#K2">*</a> 6562</span>)</span> <a href="xcmplx_0.html#K7">/</a> 17
 
<b>by </b><i><a class="txt" href="pepin.html#E4"><i><font color="Green">Lemma54</font></i></a>, <a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a></i>;<br/>
<a NAME="E16:88_2_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> 6561</span>)</span> <a href="xcmplx_0.html#K3">*</a> 6561</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default">386 <a href="nat_1.html#K2">*</a> 6560</span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default">386 <a href="nat_1.html#K2">*</a> 6560</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default">386 <a href="nat_1.html#K2">*</a> 6560</span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<b>hence </b><a NAME="E17:88_2_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> 6561</span>)</span> <a href="xcmplx_0.html#K3">*</a> 6561</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default">386 <a href="nat_1.html#K2">*</a> 6560</span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/>


</div><b>end;</b></div>
<span class="p1">(<span class="default">3 <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">16 <a href="nat_1.html#K2">*</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 8</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 = 
<span class="p1">(<span class="default">3 <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">16 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 8</span>)</span> <a href="nat_1.html#K1">+</a> 16</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pepin.html#K5" target="_self">Fermat</a> 2</span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default">3 <a href="wsierp_1.html#K2">|^</a> 16</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1
<b>by </b><i><a class="txt" href="pepin.html#E23"><i><font color="Green">Lemma76</font></i></a>, <a class="ref" href="newton.html#T13">NEWTON:13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="pepin.html#K5" target="_self">Fermat</a> 2</span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> 6561</span>)</span> <a href="xcmplx_0.html#K3">*</a> 6561</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default">386 <a href="nat_1.html#K2">*</a> 6560</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="pepin.html#T11" target="_self">Th11</a>, <a class="txt" href="pepin.html#E4"><i><font color="Green">Lemma54</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="pepin.html#K5" target="_self">Fermat</a> 2</span>)</span> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> 6561</span>)</span> <a href="xcmplx_0.html#K3">*</a> 6561</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default">386 <a href="nat_1.html#K2">*</a> 6560</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="pepin.html#E24"><i><font color="Green">Lemma77</font></i></a>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>
;<br/>
<b>hence </b><a NAME="E7:88_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1]
 <b>by </b><i><a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:88"/>
for <font color="Olive">n</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">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="pepin.html#E20"><i><font color="Green">Lemma73</font></i></a>, <a class="txt" href="pepin.html#E21"><i><font color="Green">Lemma74</font></i></a>);<br/></i>
<b>hence </b><a NAME="E4:88"/>
 <a href="pepin.html#K5" target="_self">Fermat</a> 2 <a href="nat_1.html#R1">divides</a> <span class="p1">(<span class="default">3 <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">16 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 8</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
 ;<br/>


</div>
