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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>thus </b><a NAME="E1:36"/>
( <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">j1</font> being  <a href="int_1.html#NM1">Integer</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">j1</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:36_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="E3:36_1"/><i><font color="Green">E42</font></i>: 
 <a href="group_1.html#G1">HGrStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font>,the <a href="group_1.html#U1">mult</a> of <font color="Maroon">G</font> #) <a href="hidden.html#R1">=</a>  <a href="group_4.html#K5">gr</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">a</font>
;<br/>

<a NAME="E4:36_1"/>
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">j1</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">j1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1"><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/>

<a NAME="E1:36_1_1"/>
<font color="Maroon">b</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"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i>, <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E2:36_1_1"/>
 ex <font color="Olive">j1</font> being  <a href="int_1.html#NM1">Integer</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">j1</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:36_1"/>
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">j1</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">j1</font>
 ;<br/>


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

<b>thus </b><a NAME="E2:36"/>
(  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">j1</font> being  <a href="int_1.html#NM1">Integer</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">j1</font> implies <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> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_2"><b>proof </b></a><div class="add">

<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="E1:36_2"/><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">j1</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">j1</font>
 ;<br/>

<a NAME="E2:36_2"/>
for <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font> holds  <font color="Olive">b</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"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_2_1"><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/>

<a NAME="E1:36_2_1"/>
 ex <font color="Olive">j1</font> being  <a href="int_1.html#NM1">Integer</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">j1</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E2:36_2_1"/>
<font color="Maroon">b</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"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E3:36_2"/><b>then </b>
 <a href="group_1.html#G1">HGrStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font>,the <a href="group_1.html#U1">mult</a> of <font color="Maroon">G</font> #) <a href="hidden.html#R1">=</a>  <a href="group_4.html#K5">gr</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="group_2.html#T71">GROUP_2:71</a></i>;<br/>
<b>hence </b><a NAME="E4:36_2"/>
<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><b>end;</b></div>


</div>
