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

<b>let </b><font color="Maroon" title="c1">n</font>, <font color="Maroon" title="c2">f</font>, <font color="Maroon" title="c3">d</font>, <font color="Maroon" title="c4">n1</font>, <font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c6">q</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="E1:26"/><i><font color="Green" title="E27">A1</font></i>: 
( <font color="Maroon" title="c1">n</font> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">q</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c3">d</font> &amp; <font color="Maroon" title="c6">q</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c4">n1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c3">d</font> &amp; <font color="Maroon" title="c3">d</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c6">q</font> <a href="nat_d.html#K6" title="NAT_D:func.6">hcf</a> <font color="Maroon" title="c3">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c6">q</font> is <a href="int_2.html#V1" title="INT_2:attr.1">prime</a> )
 ;<br/>

<b>assume </b><a NAME="E2:26"/><i><font color="Green" title="E28">A2</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/>

<b>assume </b><a NAME="E3:26"/><i><font color="Green" title="E29">A3</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c6">q</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_d.html#K6" title="NAT_D:func.6">hcf</a> <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/>

<b>set </b><font color="Maroon" title="c7">f</font> = <font color="Maroon" title="c6">q</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c4">n1</font>;<br/>
<a NAME="E4:26"/>
<font color="Maroon" title="c6">q</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 
<b>by </b><i><a class="txt" href="nat_4.html#E1:26"><i><font color="Green" title="E27">A1</font></i></a>, <a class="ref" href="int_2.html#D5" title="INT_2:def.5">INT_2:def 5</a></i>;<br/>
<a NAME="E5:26"/><b>then </b><i><font color="Green" title="E30">A4</font></i>: 
1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">q</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c4">n1</font></span>)</span> <a href="nat_d.html#K6" title="NAT_D:func.6">hcf</a> <font color="Maroon" title="c3">d</font>
 
<b>by </b><i><a class="txt" href="nat_4.html#E1:26"><i><font color="Green" title="E27">A1</font></i></a>, <a class="ref" href="nat_4.html#T10" target="_self" title="NAT_4:th.10">Th10</a></i>;<br/>
<a NAME="E6:26"/>
for <font color="Olive" title="b1">q1</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">q1</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c6">q</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c4">n1</font> &amp; <font color="Olive" title="b1">q1</font> is <a href="int_2.html#V1" title="INT_2:attr.1">prime</a> holds <br/> ex <font color="Olive" title="b2">a</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/>( <span class="p1">(<span class="default"><font color="Olive" title="b2">a</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b2">a</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Olive" title="b1">q1</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_d.html#K6" title="NAT_D:func.6">hcf</a> <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">q1</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="E1:26_1"/><i><font color="Green" title="E31">A5</font></i>: 
<font color="Maroon" title="c8">q1</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c6">q</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c4">n1</font>
 ;<br/>

<b>assume </b><a NAME="E2:26_1"/><i><font color="Green" title="E32">A6</font></i>: 
<font color="Maroon" title="c8">q1</font> is <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>
 ;<br/>

<a NAME="E3:26_1"/><i><font color="Green" title="E33">A7</font></i>: 
<font color="Maroon" title="c4">n1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:26_1_1"/>
<font color="Maroon" title="c4">n1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/>

<a NAME="E2:26_1_1"/><b>then </b>
<font color="Maroon" title="c4">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
;<br/>
<a NAME="E3:26_1_1"/><b>then </b>
<font color="Maroon" title="c3">d</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1
 
<b>by </b><i><a class="txt" href="nat_4.html#E1:26"><i><font color="Green" title="E27">A1</font></i></a>, <a class="ref" href="newton.html#T9" title="NEWTON:th.9">NEWTON:9</a></i>;<br/>
<b>hence </b><a NAME="E4:26_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="nat_4.html#E1:26"><i><font color="Green" title="E27">A1</font></i></a>, <a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c9">a</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="E5:26_1"/><i><font color="Green" title="E34">A8</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c6">q</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_d.html#K6" title="NAT_D:func.6">hcf</a> <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <b>by </b><i><a class="txt" href="nat_4.html#E2:26"><i><font color="Green" title="E28">A2</font></i></a>, <a class="txt" href="nat_4.html#E3:26"><i><font color="Green" title="E29">A3</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c9">a</font>
;<br/>

<b>thus </b><a NAME="E6:26_1"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c8">q1</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_d.html#K6" title="NAT_D:func.6">hcf</a> <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <b>by </b><i><a class="txt" href="nat_4.html#E1:26"><i><font color="Green" title="E27">A1</font></i></a>, <a class="txt" href="nat_4.html#E1:26_1"><i><font color="Green" title="E31">A5</font></i></a>, <a class="txt" href="nat_4.html#E2:26_1"><i><font color="Green" title="E32">A6</font></i></a>, <a class="txt" href="nat_4.html#E3:26_1"><i><font color="Green" title="E33">A7</font></i></a>, <a class="txt" href="nat_4.html#E5:26_1"><i><font color="Green" title="E34">A8</font></i></a>, <a class="ref" href="nat_4.html#T18" target="_self" title="NAT_4:th.18">Th18</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:26"/>
<font color="Maroon" title="c1">n</font> is <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>
 <b>by </b><i><a class="txt" href="nat_4.html#E1:26"><i><font color="Green" title="E27">A1</font></i></a>, <a class="txt" href="nat_4.html#E5:26"><i><font color="Green" title="E30">A4</font></i></a>, <a class="ref" href="nat_4.html#T24" target="_self" title="NAT_4:th.24">Th24</a></i>;<br/>


</div>
