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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="group_1.html#V1">strict</a> <a href="group_1.html#NM1">Group</a>;<br/>

<b>assume </b><a NAME="E1:14"/><i><font color="Green">E11</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>2</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:14"/><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="group_8.html#E1:14"><i><font color="Green">E11</font></i></a></i>;<br/>
<a NAME="E4:14"/><i><font color="Green">E13</font></i>: 
( ( for <font color="Olive">b<sub>1</sub></font> being <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> or <font color="Olive">b<sub>1</sub></font> <a href="group_2.html#R1">=</a>  <a href="group_2.html#K5">(1).</a> <font color="Maroon">c<sub>1</sub></font> ) ) implies ( <font color="Maroon">c<sub>1</sub></font> is  <a href="gr_cy_1.html#V1">cyclic</a> <a href="group_1.html#NM1">Group</a> &amp; <font color="Maroon">c<sub>1</sub></font> is <a href="group_1.html#V6">finite</a> &amp; ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>(  <a href="group_1.html#K9">ord</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="int_2.html#V1">prime</a> ) ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:14_1"/><i><font color="Green">E14</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> or <font color="Olive">b<sub>1</sub></font> <a href="group_2.html#R1">=</a>  <a href="group_2.html#K5">(1).</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/>

<a NAME="E2:14_1"/><i><font color="Green">E15</font></i>: 
 <a href="group_4.html#K5">gr</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K6">}</a></span> is  <a href="gr_cy_1.html#V1">cyclic</a> <a href="group_1.html#NM1">Group</a>
 
<b>by </b><i><a class="ref" href="gr_cy_2.html#T10">GR_CY_2:10</a></i>;<br/>
<a NAME="E3:14_1"/>
 <a href="group_4.html#K5">gr</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="group_2.html#K5">(1).</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="group_8.html#E3:14"><i><font color="Green">E12</font></i></a>, <a class="ref" href="group_8.html#T7" target="_self">Th7</a></i>;<br/>
<a NAME="E4:14_1"/><b>then </b><i><font color="Green">E16</font></i>: 
 <a href="group_4.html#K5">gr</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="group_8.html#E1:14_1"><i><font color="Green">E14</font></i></a></i>;<br/>
<a NAME="E5:14_1"/><b>then </b><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is  <a href="gr_cy_1.html#V1">cyclic</a> <a href="group_1.html#NM1">Group</a>
 
<b>by </b><i><a class="txt" href="group_8.html#E2:14_1"><i><font color="Green">E15</font></i></a></i>;<br/>
<a NAME="E6:14_1"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="group_1.html#NR1" target="_self">is_not_of_order_0</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:14_1_1"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="group_1.html#NR2" target="_self">is_of_order_0</a> 
 ;<br/>

<a NAME="E2:14_1_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="group_1.html#K6">|^</a> 2 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="group_1.html#D11">GROUP_1:def 11</a></i>;<br/>
<a NAME="E3:14_1_1"/><b>then </b><i><font color="Green">E20</font></i>: 
 <a href="group_4.html#K5">gr</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_1.html#K6">|^</a> 2</span>)</span></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="group_2.html#K5">(1).</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="group_8.html#T7" target="_self">Th7</a></i>;<br/>
<a NAME="E4:14_1_1"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="group_8.html#E4:14_1"><i><font color="Green">E16</font></i></a>, <a class="ref" href="gr_cy_2.html#T8">GR_CY_2:8</a></i>;<br/>
<a NAME="E5:14_1_1"/><i><font color="Green">E22</font></i>: 
 <a href="group_4.html#K5">gr</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_1.html#K6">|^</a> 2</span>)</span></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:14_1_1_1"/>
not <font color="Maroon">c<sub>2</sub></font> <a href="rlvect_1.html#R1">in</a>  <a href="group_4.html#K5">gr</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_1.html#K6">|^</a> 2</span>)</span></span><a href="domain_1.html#K6">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:14_1_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="rlvect_1.html#R1">in</a>  <a href="group_4.html#K5">gr</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_1.html#K6">|^</a> 2</span>)</span></span><a href="domain_1.html#K6">}</a></span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>3</sub></font> being   <a href="int_1.html#NM1">Integer</a><b> such that </b><br/><a NAME="E3:14_1_1_1_1"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_1.html#K6">|^</a> 2</span>)</span> <a href="group_1.html#K6">|^</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="gr_cy_1.html#T25">GR_CY_1:25</a></i>;<br/>
<a NAME="E4:14_1_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="group_1.html#K6">|^</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="group_8.html#E3:14_1_1_1_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="group_1.html#T67">GROUP_1:67</a></i>;<br/>
<a NAME="E5:14_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_1.html#K6">|^</a> <span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="group_1.html#D6">GROUP_1:def 6</a></i>;<br/>
<a NAME="E6:14_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_1.html#K6">|^</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 1</span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_1.html#K6">|^</a> <span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="group_1.html#T62">GROUP_1:62</a></i>;<br/>
<a NAME="E7:14_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="group_1.html#K6">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="group_1.html#T63">GROUP_1:63</a></i>;<br/>
<a NAME="E8:14_1_1_1_1"/><b>then </b><i><font color="Green">E24</font></i>: 
<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="txt" href="group_8.html#E1:14_1_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="group_8.html#T13" target="_self">Th13</a></i>;<br/>
<a NAME="E9:14_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="ref" href="int_1.html#T22">INT_1:22</a></i>;<br/>
<b>hence </b><a NAME="E10:14_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="group_8.html#E8:14_1_1_1_1"><i><font color="Green">E24</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:14_1_1_1"/>
 <a href="group_4.html#K5">gr</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_1.html#K6">|^</a> 2</span>)</span></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="group_8.html#E4:14_1_1"><i><font color="Green">E21</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E6:14_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="group_8.html#E1:14_1"><i><font color="Green">E14</font></i></a>, <a class="txt" href="group_8.html#E3:14_1_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="group_8.html#E5:14_1_1"><i><font color="Green">E22</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon">c<sub>3</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E8:14_1"/><i><font color="Green">E19</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K7">ord</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="group_1.html#NR1" target="_self">is_not_of_order_0</a>  )
 ;<br/>
<a NAME="E9:14_1"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="group_1.html#V6">finite</a>
 
<b>by </b><i><a class="txt" href="group_8.html#E4:14_1"><i><font color="Green">E16</font></i></a>, <a class="txt" href="group_8.html#E6:14_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="group_8.html#T11" target="_self">Th11</a></i>;<br/>
<a NAME="E10:14_1"/><i><font color="Green">E21</font></i>: 
 <a href="group_1.html#K9">ord</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K9">ord</a> <span class="p1">(<span class="default"><a href="group_4.html#K5">gr</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span>
 
<b>by </b><i><a class="txt" href="group_8.html#E4:14_1"><i><font color="Green">E16</font></i></a></i>;<br/>
<a NAME="E11:14_1"/><i><font color="Green">E22</font></i>: 
 <a href="group_1.html#K7">ord</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K9">ord</a> <span class="p1">(<span class="default"><a href="group_4.html#K5">gr</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span>
 
<b>by </b><i><a class="txt" href="group_8.html#E9:14_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="gr_cy_1.html#T27">GR_CY_1:27</a></i>;<br/>
<a NAME="E12:14_1"/><i><font color="Green">E23</font></i>: 
 <a href="group_1.html#K9">ord</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="group_8.html#E8:14_1"><i><font color="Green">E19</font></i></a>, <a class="txt" href="group_8.html#E9:14_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="group_8.html#E10:14_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="gr_cy_1.html#T27">GR_CY_1:27</a></i>;<br/>
<a NAME="E13:14_1"/>
<font color="Maroon">c<sub>3</sub></font> is <a href="int_2.html#V1">prime</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:14_1_2"/><i><font color="Green">E24</font></i>: 
not <font color="Maroon">c<sub>3</sub></font> is <a href="int_2.html#V1">prime</a>
 ;<br/>

<a NAME="E2:14_1_2"/>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 &amp; <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K2">*</a> <font color="Olive">b<sub>2</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:14_1_2_1"/>
(  not <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 or ex <font color="Olive">b<sub>1</sub></font> being <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>3</sub></font> &amp; not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 1 &amp; not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> ) )
 
<b>by </b><i><a class="txt" href="group_8.html#E1:14_1_2"><i><font color="Green">E24</font></i></a>, <a class="ref" href="int_2.html#D5">INT_2:def 5</a></i>;<br/>
<a NAME="E2:14_1_2_1"/><b>then </b><i><font color="Green">E25</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 or ex <font color="Olive">b<sub>1</sub></font> being <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>3</sub></font> ) )
 
;<br/>
<a NAME="E3:14_1_2_1"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_2_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:14_1_2_1_1"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K9">ord</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="group_8.html#E12:14_1"><i><font color="Green">E23</font></i></a></i>;<br/>
<a NAME="E2:14_1_2_1_1"/><b>then </b><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<b>by </b><i><a class="txt" href="group_8.html#E9:14_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="group_1.html#T90">GROUP_1:90</a></i>;<br/>
<a NAME="E3:14_1_2_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 
<b>by </b><i><a class="txt" href="group_8.html#E3:14"><i><font color="Green">E12</font></i></a>, <a class="txt" href="group_8.html#E10:14_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="group_8.html#E11:14_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="group_8.html#E1:14_1_2_1_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="group_1.html#T85">GROUP_1:85</a></i>;<br/>
<b>hence </b><a NAME="E4:14_1_2_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 <b>by </b><i><a class="txt" href="group_8.html#E2:14_1_2_1_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon">c<sub>4</sub></font> being  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E5:14_1_2_1"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="group_8.html#E2:14_1_2_1"><i><font color="Green">E25</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E8:14_1_2_1"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="group_8.html#E5:14_1_2_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/>
<b>take </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>5</sub></font>;<br/>

<b>take </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>6</sub></font>;<br/>

<a NAME="E9:14_1_2_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_2_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:14_1_2_1_2"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_2_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:14_1_2_1_2_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/>

<a NAME="E2:14_1_2_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 0 <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K3">div</a> 0</span>)</span>
 
<b>by </b><i><a class="txt" href="group_8.html#E5:14_1_2_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="nat_1.html#T49">NAT_1:49</a></i>;<br/>
<b>hence </b><a NAME="E3:14_1_2_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="group_8.html#E3:14_1_2_1"><i><font color="Green">E26</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:14_1_2_1_2"/><b>then </b>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>7</sub></font>
 
;<br/>
<a NAME="E3:14_1_2_1_2"/><b>then </b>
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="ref" href="int_1.html#T20">INT_1:20</a></i>;<br/>
<b>hence </b><a NAME="E4:14_1_2_1_2"/>
<font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 <b>by </b><i><a class="txt" href="group_8.html#E5:14_1_2_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:14_1_2_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_2_1_3"><b>proof </b></a><div class="add">

<a NAME="E1:14_1_2_1_3"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="txt" href="group_8.html#E3:14_1_2_1"><i><font color="Green">E26</font></i></a>, <a class="txt" href="group_8.html#E8:14_1_2_1"><i><font color="Green">E28</font></i></a></i>;<br/>
<a NAME="E2:14_1_2_1_3"/><b>then </b>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>8</sub></font>
 
;<br/>
<a NAME="E3:14_1_2_1_3"/><b>then </b>
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="ref" href="int_1.html#T20">INT_1:20</a></i>;<br/>
<a NAME="E4:14_1_2_1_3"/><b>then </b><i><font color="Green">E30</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font>
 
;<br/>
<a NAME="E5:14_1_2_1_3"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 
<b>by </b><i><a class="txt" href="group_8.html#E5:14_1_2_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="group_8.html#E8:14_1_2_1"><i><font color="Green">E28</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:14_1_2_1_3"/>
<font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 <b>by </b><i><a class="txt" href="group_8.html#E4:14_1_2_1_3"><i><font color="Green">E30</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E11:14_1_2_1"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 &amp; <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>8</sub></font> )
 <b>by </b><i><a class="txt" href="group_8.html#E8:14_1_2_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="group_8.html#E9:14_1_2_1"><i><font color="Green">E29</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:14_1_2"/><i><font color="Green">E25</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font> )
 ;<br/>
<a NAME="E5:14_1_2"/>
 <a href="group_1.html#K7">ord</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_1.html#K6">|^</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="group_8.html#E4:14_1"><i><font color="Green">E16</font></i></a>, <a class="txt" href="group_8.html#E9:14_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="group_8.html#E12:14_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="group_8.html#E4:14_1_2"><i><font color="Green">E25</font></i></a>, <a class="ref" href="gr_cy_2.html#T14">GR_CY_2:14</a></i>;<br/>
<a NAME="E6:14_1_2"/><b>then </b><i><font color="Green">E26</font></i>: 
 <a href="group_1.html#K9">ord</a> <span class="p1">(<span class="default"><a href="group_4.html#K5">gr</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_1.html#K6">|^</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="group_8.html#E9:14_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="gr_cy_1.html#T27">GR_CY_1:27</a></i>;<br/>
<a NAME="E7:14_1_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="group_8.html#E4:14_1_2"><i><font color="Green">E25</font></i></a>, <a class="ref" href="xcmplx_1.html#T7">XCMPLX_1:7</a></i>;<br/>
<a NAME="E8:14_1_2"/><b>then </b><i><font color="Green">E27</font></i>: 
 <a href="group_4.html#K5">gr</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_1.html#K6">|^</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="group_8.html#E12:14_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="group_8.html#E6:14_1_2"><i><font color="Green">E26</font></i></a></i>;<br/>
<a NAME="E9:14_1_2"/>
 <a href="group_4.html#K5">gr</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_1.html#K6">|^</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="group_2.html#K5">(1).</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="group_8.html#E4:14_1_2"><i><font color="Green">E25</font></i></a>, <a class="txt" href="group_8.html#E6:14_1_2"><i><font color="Green">E26</font></i></a>, <a class="ref" href="group_2.html#T81">GROUP_2:81</a></i>;<br/>
<b>hence </b><a NAME="E10:14_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="group_8.html#E1:14_1"><i><font color="Green">E14</font></i></a>, <a class="txt" href="group_8.html#E8:14_1_2"><i><font color="Green">E27</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E14:14_1"/>
( <font color="Maroon">c<sub>1</sub></font> is  <a href="gr_cy_1.html#V1">cyclic</a> <a href="group_1.html#NM1">Group</a> &amp; <font color="Maroon">c<sub>1</sub></font> is <a href="group_1.html#V6">finite</a> &amp; ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>(  <a href="group_1.html#K9">ord</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="int_2.html#V1">prime</a> ) )
 <b>by </b><i><a class="txt" href="group_8.html#E5:14_1"><i><font color="Green">E17</font></i></a>, <a class="txt" href="group_8.html#E9:14_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="group_8.html#E12:14_1"><i><font color="Green">E23</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:14"/><i><font color="Green">E14</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">c<sub>1</sub></font> is  <a href="gr_cy_1.html#V1">cyclic</a> <a href="group_1.html#NM1">Group</a> &amp; <font color="Maroon">c<sub>1</sub></font> is <a href="group_1.html#V6">finite</a> &amp;  <a href="group_1.html#K9">ord</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="int_2.html#V1">prime</a> holds <br/>for <font color="Olive">b<sub>2</sub></font> being <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> or <font color="Olive">b<sub>2</sub></font> <a href="group_2.html#R1">=</a>  <a href="group_2.html#K5">(1).</a> <font color="Maroon">c<sub>1</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:14_2"/><i><font color="Green">E15</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> is  <a href="gr_cy_1.html#V1">cyclic</a> <a href="group_1.html#NM1">Group</a> &amp; <font color="Maroon">c<sub>1</sub></font> is <a href="group_1.html#V6">finite</a> &amp;  <a href="group_1.html#K9">ord</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> is <a href="int_2.html#V1">prime</a> )
 ;<br/>

<b>let </b><font color="Maroon">c<sub>4</sub></font> be  <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<a NAME="E2:14_2"/><i><font color="Green">E16</font></i>: 
 <a href="group_1.html#K9">ord</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="group_1.html#K9">ord</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><a href="group_2.html#K17">index</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="group_8.html#E1:14_2"><i><font color="Green">E15</font></i></a>, <a class="ref" href="group_2.html#T177">GROUP_2:177</a></i>;<br/>
<a NAME="E3:14_2"/>
 <a href="group_1.html#K9">ord</a> <font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#R1">divides</a>  <a href="group_1.html#K9">ord</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="group_8.html#E2:14_2"><i><font color="Green">E16</font></i></a>, <a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/>
<a NAME="E4:14_2"/><b>then </b><i><font color="Green">E17</font></i>: 
(  <a href="group_1.html#K9">ord</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K9">ord</a> <font color="Maroon">c<sub>1</sub></font> or  <a href="group_1.html#K9">ord</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 1 )
 
<b>by </b><i><a class="txt" href="group_8.html#E1:14_2"><i><font color="Green">E15</font></i></a>, <a class="ref" href="int_2.html#D5">INT_2:def 5</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:14_2_1"/>
(  <a href="group_1.html#K9">ord</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K9">ord</a> <font color="Maroon">c<sub>1</sub></font> or  <a href="group_1.html#K9">ord</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 1 )
 </span><b>by </b><i><a class="txt" href="group_8.html#E4:14_2"><i><font color="Green">E17</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_2_1_1"><b>suppose </b></a><a NAME="E1:14_2_1_1"/>
 <a href="group_1.html#K9">ord</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K9">ord</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:14_2_1_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> or <font color="Maroon">c<sub>4</sub></font> <a href="group_2.html#R1">=</a>  <a href="group_2.html#K5">(1).</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="group_8.html#E1:14_2"><i><font color="Green">E15</font></i></a>, <a class="ref" href="group_2.html#T85">GROUP_2:85</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_2_1_2"><b>suppose </b></a><a NAME="E1:14_2_1_2"/><i><font color="Green">E18</font></i>: 
 <a href="group_1.html#K9">ord</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:14_2_1_2"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>4</sub></font> is <a href="group_1.html#V6">finite</a>
 <b>by </b><i><a class="txt" href="group_8.html#E1:14_2"><i><font color="Green">E15</font></i></a>, <a class="ref" href="group_2.html#T48">GROUP_2:48</a></i>;<br/><a NAME="E3:14_2_1_2"/>
( <font color="Maroon">c<sub>4</sub></font> is <a href="group_1.html#V6">finite</a> &amp;  <a href="group_1.html#K9">ord</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i><a class="txt" href="group_8.html#E1:14_2_1_2"><i><font color="Green">E18</font></i></a>, <a class="txt" href="group_8.html#E2:14_2_1_2"><i><font color="Green">E19</font></i></a></i>;<br/><b>hence </b><a NAME="E4:14_2_1_2"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> or <font color="Maroon">c<sub>4</sub></font> <a href="group_2.html#R1">=</a>  <a href="group_2.html#K5">(1).</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="ref" href="group_2.html#T82">GROUP_2:82</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>thus </b><a NAME="E6:14"/>
( ( for <font color="Olive">b<sub>1</sub></font> being <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> or <font color="Olive">b<sub>1</sub></font> <a href="group_2.html#R1">=</a>  <a href="group_2.html#K5">(1).</a> <font color="Maroon">c<sub>1</sub></font> ) ) iff ( <font color="Maroon">c<sub>1</sub></font> is  <a href="gr_cy_1.html#V1">cyclic</a> <a href="group_1.html#NM1">Group</a> &amp; <font color="Maroon">c<sub>1</sub></font> is <a href="group_1.html#V6">finite</a> &amp; ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>(  <a href="group_1.html#K9">ord</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="int_2.html#V1">prime</a> ) ) )
 <b>by </b><i><a class="txt" href="group_8.html#E4:14"><i><font color="Green">E13</font></i></a>, <a class="txt" href="group_8.html#E5:14"><i><font color="Green">E14</font></i></a></i>;<br/>


</div>
