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<div class="add">

<b>let </b><font color="Maroon" title="c1">G</font> be  <a href="struct_0.html#V8" title="STRUCT_0:attr.8">finite</a> <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a>;<br/>

<b>thus </b><a NAME="E1:41"/>
( <font color="Maroon" title="c1">G</font> is <a href="gr_cy_1.html#V1" title="GR_CY_1:attr.1">cyclic</a> implies  ex <font color="Olive" title="b1">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</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="c1">G</font>  ex <font color="Olive" title="b3">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="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Olive" title="b3">n</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:41_1"/>
<font color="Maroon" title="c1">G</font> is <a href="gr_cy_1.html#V1" title="GR_CY_1:attr.1">cyclic</a>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c2">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="E3:41_1"/><i><font color="Green" title="E29">A1</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">j2</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> st <font color="Olive" title="b1">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Olive" title="b2">j2</font>
 <b>by </b><i><a class="ref" href="gr_cy_1.html#T41" target="_self" title="GR_CY_1:th.41">Th41</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c2">a</font>
;<br/>

<b>let </b><font color="Maroon" title="c3">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font>;<br/>

<b>consider </b><font color="Maroon" title="c4">j2</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><b> such that </b><br/><a NAME="E5:41_1"/><i><font color="Green" title="E30">A2</font></i>: 
<font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c4">j2</font>
 <b>by </b><i><a class="txt" href="gr_cy_1.html#E3:41_1"><i><font color="Green" title="E29">A1</font></i></a></i>;<br/>
<b>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="E7:41_1"/><i><font color="Green" title="E31">A3</font></i>: 
( <font color="Maroon" title="c4">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">n</font> or <font color="Maroon" title="c4">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <font color="Maroon" title="c5">n</font> )
 <b>by </b><i><a class="ref" href="int_1.html#T8" title="INT_1:th.8">INT_1:8</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_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_1"/>
( <font color="Maroon" title="c4">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">n</font> or <font color="Maroon" title="c4">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <font color="Maroon" title="c5">n</font> )
 </span><b>by </b><i><a class="txt" href="gr_cy_1.html#E7:41_1"><i><font color="Green" title="E31">A3</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_1_1"><b>suppose </b></a><a NAME="E1:41_1_1_1"/>
<font color="Maroon" title="c4">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">n</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:41_1_1_1"/>
 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" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Olive" title="b1">n</font>
 <b>by </b><i><a class="txt" href="gr_cy_1.html#E5:41_1"><i><font color="Green" title="E30">A2</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_1_2"><b>suppose </b></a><a NAME="E1:41_1_1_2"/><i><font color="Green" title="E32">A4</font></i>: 
<font color="Maroon" title="c4">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <font color="Maroon" title="c5">n</font>
 ;<br/><div class="add"><a NAME="E2:41_1_1_2"/>
<font color="Maroon" title="c5">n</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><a href="group_1.html#K8" title="GROUP_1:func.8">ord</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="group_1.html#K8" title="GROUP_1:func.8">ord</a> <font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="ref" href="nat_d.html#T1" title="NAT_D:th.1">NAT_D:1</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c6">q'</font> = <span class="p1">(<span class="default"><a href="group_1.html#K8" title="GROUP_1:func.8">ord</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p2">(<span class="default"><a href="group_1.html#K8" title="GROUP_1:func.8">ord</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span> 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="int_1.html#T18" title="INT_1:th.18">INT_1:18</a></i>;<br/><b>take </b>
<font color="Maroon" title="c6">q'</font>
;<br/><font color="Maroon" title="c3">b</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> 
<b>by </b><i><a class="txt" href="gr_cy_1.html#E5:41_1"><i><font color="Green" title="E30">A2</font></i></a>, <a class="txt" href="gr_cy_1.html#E1:41_1_1_2"><i><font color="Green" title="E32">A4</font></i></a>, <a class="ref" href="group_1.html#T71" title="GROUP_1:th.71">GROUP_1:71</a></i>
<br/>.= 
<font color="Maroon" title="c2">a</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c6">q'</font>
<b>by </b><i><a class="ref" href="gr_cy_1.html#T30" target="_self" title="GR_CY_1:th.30">Th30</a></i>
;<br/><b>hence </b><a NAME="E5:41_1_1_2"/>
<font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c6">q'</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>

<b>given </b><font color="Maroon" title="c2">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><a NAME="E3:41"/><i><font color="Green" title="E29">A5</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="c2">a</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Olive" title="b2">n</font>
 ;<br/>

<a NAME="E4:41"/>
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">j2</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> st <font color="Olive" title="b1">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Olive" title="b2">j2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font>;<br/>

<b>consider </b><font color="Maroon" title="c4">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="E2:41_2"/><i><font color="Green" title="E30">A6</font></i>: 
<font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c4">n</font>
 <b>by </b><i><a class="txt" href="gr_cy_1.html#E3:41"><i><font color="Green" title="E29">A5</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">n</font> = <font color="Maroon" title="c4">n</font> as   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> ;<br/>
<b>take </b>
<font color="Maroon" title="c5">n</font>
;<br/>

<b>thus </b><a NAME="E4:41_2"/>
<font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c5">n</font>
 <b>by </b><i><a class="txt" href="gr_cy_1.html#E2:41_2"><i><font color="Green" title="E30">A6</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:41"/>
<font color="Maroon" title="c1">G</font> is <a href="gr_cy_1.html#V1" title="GR_CY_1:attr.1">cyclic</a>
 <b>by </b><i><a class="ref" href="gr_cy_1.html#T41" target="_self" title="GR_CY_1:th.41">Th41</a></i>;<br/>


</div>
