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<div class="add">

<b>let </b><font color="Maroon">G</font> be   <a href="group_1.html#NM1">Group</a>;<br/>

<b>assume </b><a NAME="E1:37"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">G</font> is <a href="group_1.html#V6">finite</a>
 ;<br/>

<b>thus </b><a NAME="E2:37"/>
( <font color="Maroon">G</font> is  <a href="gr_cy_1.html#V1" target="_self">cyclic</a> <a href="group_1.html#NM1">Group</a> implies  ex <font color="Olive">a</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font> st <br/>for <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font>  ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font> <a href="group_1.html#K6">|^</a> <font color="Olive">n</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:37_1"/>
<font color="Maroon">G</font> is  <a href="gr_cy_1.html#V1" target="_self">cyclic</a> <a href="group_1.html#NM1">Group</a>
 ;<br/>

<b>then consider </b><font color="Maroon">a</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E3:37_1"/><i><font color="Green">E47</font></i>: 
for <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font>  ex <font color="Olive">j2</font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <font color="Olive">j2</font>
 <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">a</font>
;<br/>

<b>let </b><font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font>;<br/>

<b>consider </b><font color="Maroon">j2</font> being   <a href="int_1.html#NM1">Integer</a><b> such that </b><br/><a NAME="E5:37_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">j2</font>
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E7:37_1"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">j2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> or <font color="Maroon">j2</font> <a href="hidden.html#R1">=</a>  <a href="binop_2.html#K7">-</a> <font color="Maroon">n</font> )
 <b>by </b><i><a class="ref" href="int_1.html#T8">INT_1:8</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:37_1_1"/>
( <font color="Maroon">j2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> or <font color="Maroon">j2</font> <a href="hidden.html#R1">=</a>  <a href="binop_2.html#K7">-</a> <font color="Maroon">n</font> )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_1"><b>suppose </b></a><a NAME="E1:37_1_1_1"/>
<font color="Maroon">j2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:37_1_1_1"/>
 ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <font color="Olive">n</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_2"><b>suppose </b></a><a NAME="E1:37_1_1_2"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">j2</font> <a href="hidden.html#R1">=</a>  <a href="binop_2.html#K7">-</a> <font color="Maroon">n</font>
 ;<br/><div class="add"><a NAME="E2:37_1_1_2"/>
 <a href="group_1.html#K9">ord</a> <font color="Maroon">G</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>by </b><i>, <a class="ref" href="group_1.html#T90">GROUP_1:90</a></i>;<br/><a NAME="E3:37_1_1_2"/><b>then </b>
 <a href="group_1.html#K9">ord</a> <font color="Maroon">G</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><a NAME="E4:37_1_1_2"/><b>then </b>
<font color="Maroon">n</font> <a href="nat_1.html#K6">mod</a> <span class="p1">(<span class="default"><a href="group_1.html#K9">ord</a> <font color="Maroon">G</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="group_1.html#K9">ord</a> <font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T46">NAT_1:46</a></i>;<br/><b>then reconsider </b><font color="Maroon">q'</font> = <span class="p1">(<span class="default"><a href="group_1.html#K9">ord</a> <font color="Maroon">G</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K6">mod</a> <span class="p2">(<span class="default"><a href="group_1.html#K9">ord</a> <font color="Maroon">G</font></span>)</span></span>)</span> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T18">INT_1:18</a></i>;<br/><b>take </b>
<font color="Maroon">q'</font>
;<br/><font color="Maroon">b</font> = 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">n</font></span>)</span> <a href="group_1.html#K3">"</a> 
<b>by </b><i>, , <a class="ref" href="group_1.html#T71">GROUP_1:71</a></i>
<br/>.= 
<font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">q'</font>
<b>by </b><i>, </i>
;<br/><b>hence </b><a NAME="E7:37_1_1_2"/>
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">q'</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

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

<b>given </b><font color="Maroon">a</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font><b> such that </b><a NAME="E3:37"/><i><font color="Green">E62</font></i>: 
for <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font>  ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <font color="Olive">n</font>
 ;<br/>

<a NAME="E4:37"/>
for <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font>  ex <font color="Olive">j2</font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <font color="Olive">j2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font>;<br/>

<b>consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E2:37_2"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">n</font>
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">n</font> = <font color="Maroon">n</font> as   <a href="int_1.html#NM1">Integer</a> ;<br/>
<b>take </b>
<font color="Maroon">n</font>
;<br/>

<b>thus </b><a NAME="E4:37_2"/>
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">n</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:37"/>
<font color="Maroon">G</font> is  <a href="gr_cy_1.html#V1" target="_self">cyclic</a> <a href="group_1.html#NM1">Group</a>
 <b>by </b><i/>;<br/>


</div>
