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

<b>let </b><font color="Maroon" title="c1">G</font> be   <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a>;<br/><b>let </b><font color="Maroon" title="c2">H</font> be  <a href="group_1.html#V1" title="GROUP_1:attr.1">strict</a>  <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:41"/><i><font color="Green" title="E35">A1</font></i>: 
<font color="Maroon" title="c1">G</font> is <a href="group_1.html#V6" title="GROUP_1:attr.6">finite</a>
 <b>and </b><br/><a NAME="E2:41"/><i><font color="Green" title="E36">A2</font></i>: 
<font color="Maroon" title="c1">G</font> is  <a href="gr_cy_1.html#V1" title="GR_CY_1:attr.1">cyclic</a> <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a>
 ;<br/>

<b>consider </b><font color="Maroon" title="c3">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E4:41"/><i><font color="Green" title="E37">A3</font></i>: 
for <font color="Olive" title="b1">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font>  ex <font color="Olive" title="b2">n</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">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Olive" title="b2">n</font>
 <b>by </b><i><a class="txt" href="gr_cy_1.html#E1:41"><i><font color="Green" title="E35">A1</font></i></a>, <a class="txt" href="gr_cy_1.html#E2:41"><i><font color="Green" title="E36">A2</font></i></a>, <a class="ref" href="gr_cy_1.html#T42" target="_self" title="GR_CY_1:th.42">Th42</a></i>;<br/>
<a NAME="E5:41"/><i><font color="Green" title="E38">A4</font></i>: 
<font color="Maroon" title="c2">H</font> is <a href="group_1.html#V6" title="GROUP_1:attr.6">finite</a>
 
<b>by </b><i><a class="txt" href="gr_cy_1.html#E1:41"><i><font color="Green" title="E35">A1</font></i></a>, <a class="ref" href="group_2.html#T48" title="GROUP_2:th.48">GROUP_2:48</a></i>;<br/>
<a NAME="E6:41"/><b>then </b><i><font color="Green" title="E39">A5</font></i>: 
 <a href="group_1.html#K9" title="GROUP_1:func.9">ord</a> <font color="Maroon" title="c2">H</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<b>by </b><i><a class="ref" href="group_1.html#T90" title="GROUP_1:th.90">GROUP_1:90</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:41_1"/>
(  <a href="group_1.html#K9" title="GROUP_1:func.9">ord</a> <font color="Maroon" title="c2">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or  <a href="group_1.html#K9" title="GROUP_1:func.9">ord</a> <font color="Maroon" title="c2">H</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 )
 </span><b>by </b><i><a class="txt" href="gr_cy_1.html#E6:41"><i><font color="Green" title="E39">A5</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_1"><b>suppose </b></a><a NAME="E1:41_1_1"/>
 <a href="group_1.html#K9" title="GROUP_1:func.9">ord</a> <font color="Maroon" title="c2">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:41_1_1"/><b>then </b>
<font color="Maroon" title="c2">H</font> <a href="group_2.html#R1" title="GROUP_2:pred.1">=</a>  <a href="group_2.html#K5" title="GROUP_2:func.5">(1).</a> <font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="txt" href="gr_cy_1.html#E5:41"><i><font color="Green" title="E38">A4</font></i></a>, <a class="ref" href="group_2.html#T82" title="GROUP_2:th.82">GROUP_2:82</a></i>;<br/><b>hence </b><a NAME="E3:41_1_1"/>
<font color="Maroon" title="c2">H</font> is  <a href="gr_cy_1.html#V1" title="GR_CY_1:attr.1">cyclic</a> <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_2"><b>suppose </b></a><a NAME="E1:41_1_2"/>
 <a href="group_1.html#K9" title="GROUP_1:func.9">ord</a> <font color="Maroon" title="c2">H</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c4">h</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">H</font><b> such that </b><br/><a NAME="E3:41_1_2"/><i><font color="Green" title="E40">A6</font></i>: 
<font color="Maroon" title="c4">h</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c2">H</font>
 <b>by </b><i><a class="ref" href="gr_cy_1.html#T31" target="_self" title="GR_CY_1:th.31">Th31</a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <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>means </b>( $1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 &amp; <font color="Maroon" title="c3">a</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> $1 is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">H</font> );<br/><a NAME="E4:41_1_2"/>
<font color="Maroon" title="c4">h</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="ref" href="group_2.html#T51" title="GROUP_2:th.51">GROUP_2:51</a></i>;<br/><b>then consider </b><font color="Maroon" title="c5">n</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="E6:41_1_2"/><i><font color="Green" title="E41">A7</font></i>: 
<font color="Maroon" title="c4">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c5">n</font>
 <b>by </b><i><a class="txt" href="gr_cy_1.html#E4:41"><i><font color="Green" title="E37">A3</font></i></a></i>;<br/><a NAME="E7:41_1_2"/>
<font color="Maroon" title="c5">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:41_1_2_1"/>
<font color="Maroon" title="c5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/>

<b>then </b><font color="Maroon" title="c4">h</font> = 
 <a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">G</font>
<b>by </b><i><a class="txt" href="gr_cy_1.html#E6:41_1_2"><i><font color="Green" title="E41">A7</font></i></a>, <a class="ref" href="group_1.html#T59" title="GROUP_1:th.59">GROUP_1:59</a></i>
<br/>.= 
 <a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c2">H</font>
<b>by </b><i><a class="ref" href="group_2.html#T53" title="GROUP_2:th.53">GROUP_2:53</a></i>
;<br/>
<b>hence </b><a NAME="E3:41_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="gr_cy_1.html#E3:41_1_2"><i><font color="Green" title="E40">A6</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E8:41_1_2"/><b>then </b>
<font color="Maroon" title="c5">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 ;<br/><a NAME="E9:41_1_2"/><b>then </b><i><font color="Green" title="E42">A8</font></i>: 
 ex <font color="Olive" title="b1">n</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="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 <b>by </b><i><a class="txt" href="gr_cy_1.html#E6:41_1_2"><i><font color="Green" title="E41">A7</font></i></a></i>;<br/><a NAME="E10:41_1_2"/>
 ex <font color="Olive" title="b1">h1</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">H</font> st <br/>for <font color="Olive" title="b2">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">H</font>  ex <font color="Olive" title="b3">s</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="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">h1</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Olive" title="b3">s</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_2_2"><b>proof </b></a><div class="add">

<a NAME="E1:41_1_2_2"/>
 ex <font color="Olive" title="b1">k</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/>( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] &amp; ( for <font color="Olive" title="b2">n</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="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">n</font>] holds <br/><font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">n</font> ) )
 
<b>from </b><i><a class="ref" href="nat_1.html#S5" title="NAT_1:sch.5">NAT_1:sch 5</a>(<a class="txt" href="gr_cy_1.html#E9:41_1_2"><i><font color="Green" title="E42">A8</font></i></a>);<br/></i>
<b>then consider </b><font color="Maroon" title="c6">n</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="E3:41_1_2_2"/><i><font color="Green" title="E43">A9</font></i>: 
( <font color="Maroon" title="c6">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 &amp; <font color="Maroon" title="c3">a</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c6">n</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">H</font> )
 <b>and </b><br/><a NAME="E4:41_1_2_2"/><i><font color="Green" title="E44">A10</font></i>: 
for <font color="Olive" title="b1">k</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">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 &amp; <font color="Maroon" title="c3">a</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Olive" title="b1">k</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">H</font> holds <br/><font color="Maroon" title="c6">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font>
 ;<br/>
<b>consider </b><font color="Maroon" title="c7">h1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">H</font><b> such that </b><br/><a NAME="E6:41_1_2_2"/><i><font color="Green" title="E45">A11</font></i>: 
<font color="Maroon" title="c7">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c6">n</font>
 <b>by </b><i><a class="txt" href="gr_cy_1.html#E3:41_1_2_2"><i><font color="Green" title="E43">A9</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c7">h1</font>
;<br/>

<a NAME="E7:41_1_2_2"/>
for <font color="Olive" title="b1">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">H</font>  ex <font color="Olive" title="b2">s</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">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">h1</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Olive" title="b2">s</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_2_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">H</font>;<br/>

<a NAME="E1:41_1_2_2_1"/>
<font color="Maroon" title="c8">b</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font>
 
<b>by </b><i><a class="ref" href="group_2.html#T51" title="GROUP_2:th.51">GROUP_2:51</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c9">p</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="E3:41_1_2_2_1"/><i><font color="Green" title="E46">A12</font></i>: 
<font color="Maroon" title="c8">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c9">p</font>
 <b>by </b><i><a class="txt" href="gr_cy_1.html#E4:41"><i><font color="Green" title="E37">A3</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c10">s</font>, <font color="Maroon" title="c11">r</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E5:41_1_2_2_1"/><i><font color="Green" title="E47">A13</font></i>: 
<font color="Maroon" title="c9">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c10">s</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c11">r</font>
 <b>and </b><br/><a NAME="E6:41_1_2_2_1"/><i><font color="Green" title="E48">A14</font></i>: 
<font color="Maroon" title="c11">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">n</font>
 <b>by </b><i><a class="txt" href="gr_cy_1.html#E3:41_1_2_2"><i><font color="Green" title="E43">A9</font></i></a>, <a class="ref" href="nat_1.html#T17" title="NAT_1:th.17">NAT_1:17</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c12">s</font> = <font color="Maroon" title="c10">s</font>, <font color="Maroon" title="c13">r</font> = <font color="Maroon" title="c11">r</font> as    <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>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E8:41_1_2_2_1"/><i><font color="Green" title="E49">A15</font></i>: 
<font color="Maroon" title="c13">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_2_2_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:41_1_2_2_1_1"/>
<font color="Maroon" title="c13">r</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 ;<br/>

<a NAME="E2:41_1_2_2_1_1"/><b>then </b><i><font color="Green" title="E50">A16</font></i>: 
<font color="Maroon" title="c13">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 
;<br/>
<a NAME="E3:41_1_2_2_1_1"/><i><font color="Green" title="E51">A17</font></i>: 
<font color="Maroon" title="c3">a</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c9">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c12">s</font></span>)</span></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c13">r</font></span>)</span>
 
<b>by </b><i><a class="txt" href="gr_cy_1.html#E5:41_1_2_2_1"><i><font color="Green" title="E47">A13</font></i></a>, <a class="ref" href="group_1.html#T48" title="GROUP_1:th.48">GROUP_1:48</a></i>;<br/>
<a NAME="E4:41_1_2_2_1_1"/>
<font color="Maroon" title="c7">h1</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c12">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c12">s</font>
 
<b>by </b><i><a class="txt" href="gr_cy_1.html#E6:41_1_2_2"><i><font color="Green" title="E45">A11</font></i></a>, <a class="ref" href="group_4.html#T3" title="GROUP_4:th.3">GROUP_4:3</a></i>;<br/>
<b>then </b><span class="p1">(<span class="default"><font color="Maroon" title="c7">h1</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c12">s</font></span>)</span> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a>  = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c12">s</font></span>)</span> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> 
<b>by </b><i><a class="ref" href="group_2.html#T57" title="GROUP_2:th.57">GROUP_2:57</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c12">s</font></span>)</span></span>)</span> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> 
<b>by </b><i><a class="ref" href="group_1.html#T50" title="GROUP_1:th.50">GROUP_1:50</a></i>
;<br/>
<b>then reconsider </b><font color="Maroon" title="c14">g</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c12">s</font></span>)</span></span>)</span> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a>  as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">H</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c15">b</font> = <font color="Maroon" title="c3">a</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c9">p</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">H</font> <b>by </b><i><a class="txt" href="gr_cy_1.html#E3:41_1_2_2_1"><i><font color="Green" title="E46">A12</font></i></a></i>;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c12">s</font></span>)</span></span>)</span> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c9">p</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c12">s</font></span>)</span></span>)</span> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c12">s</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c13">r</font></span>)</span>
<b>by </b><i><a class="txt" href="gr_cy_1.html#E3:41_1_2_2_1_1"><i><font color="Green" title="E51">A17</font></i></a>, <a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">GROUP_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c13">r</font></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D6" title="GROUP_1:def.6">GROUP_1:def 6</a></i>
<br/>.= 
<font color="Maroon" title="c3">a</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c13">r</font>
<b>by </b><i><a class="ref" href="group_1.html#D5" title="GROUP_1:def.5">GROUP_1:def 5</a></i>
;<br/>
<a NAME="E9:41_1_2_2_1_1"/><b>then </b>
<font color="Maroon" title="c14">g</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c15">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c13">r</font>
 
<b>by </b><i><a class="ref" href="group_2.html#T52" title="GROUP_2:th.52">GROUP_2:52</a></i>;<br/>
<b>hence </b><a NAME="E10:41_1_2_2_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="gr_cy_1.html#E4:41_1_2_2"><i><font color="Green" title="E44">A10</font></i></a>, <a class="txt" href="gr_cy_1.html#E6:41_1_2_2_1"><i><font color="Green" title="E48">A14</font></i></a>, <a class="txt" href="gr_cy_1.html#E2:41_1_2_2_1_1"><i><font color="Green" title="E50">A16</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>take </b>
<font color="Maroon" title="c12">s</font>
;<br/>

<font color="Maroon" title="c3">a</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c9">p</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c12">s</font>
<b>by </b><i><a class="txt" href="gr_cy_1.html#E5:41_1_2_2_1"><i><font color="Green" title="E47">A13</font></i></a>, <a class="txt" href="gr_cy_1.html#E8:41_1_2_2_1"><i><font color="Green" title="E49">A15</font></i></a>, <a class="ref" href="group_1.html#T50" title="GROUP_1:th.50">GROUP_1:50</a></i>
<br/>.= 
<font color="Maroon" title="c7">h1</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c12">s</font>
<b>by </b><i><a class="txt" href="gr_cy_1.html#E6:41_1_2_2"><i><font color="Green" title="E45">A11</font></i></a>, <a class="ref" href="group_4.html#T3" title="GROUP_4:th.3">GROUP_4:3</a></i>
;<br/>
<b>hence </b><a NAME="E10:41_1_2_2_1"/>
<font color="Maroon" title="c8">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">h1</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c12">s</font>
 <b>by </b><i><a class="txt" href="gr_cy_1.html#E3:41_1_2_2_1"><i><font color="Green" title="E46">A12</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:41_1_2_2"/>
for <font color="Olive" title="b1">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">H</font>  ex <font color="Olive" title="b2">s</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">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">h1</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Olive" title="b2">s</font>
 ;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E11:41_1_2"/>
<font color="Maroon" title="c2">H</font> is  <a href="gr_cy_1.html#V1" title="GR_CY_1:attr.1">cyclic</a> <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a>
 <b>by </b><i><a class="txt" href="gr_cy_1.html#E5:41"><i><font color="Green" title="E38">A4</font></i></a>, <a class="ref" href="gr_cy_1.html#T42" target="_self" title="GR_CY_1:th.42">Th42</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
