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<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>thus </b><a NAME="E1:39"/>
( <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> 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">j1</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</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#K6" title="GROUP_1:func.6">|^</a> <font color="Olive" title="b3">j1</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:39_1"/>
<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>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="E3:39_1"/><i><font color="Green" title="E29">A1</font></i>: 
 <a href="group_1.html#G1" title="GROUP_1:aggr.1">HGrStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font>,the <a href="group_1.html#U1" title="GROUP_1:sel.1">mult</a> of <font color="Maroon" title="c1">G</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_4.html#K5" title="GROUP_4:func.5">gr</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="ref" href="gr_cy_1.html#D9" target="_self" title="GR_CY_1:def.9">Def9</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c2">a</font>
;<br/>

<a NAME="E4:39_1"/>
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">j1</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#K6" title="GROUP_1:func.6">|^</a> <font color="Olive" title="b2">j1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1"><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/>

<a NAME="E1:39_1_1"/>
<font color="Maroon" title="c3">b</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="group_4.html#K5" title="GROUP_4:func.5">gr</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="txt" href="gr_cy_1.html#E3:39_1"><i><font color="Green" title="E29">A1</font></i></a>, <a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/>
<b>hence </b><a NAME="E2:39_1_1"/>
 ex <font color="Olive" title="b1">j1</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</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#K6" title="GROUP_1:func.6">|^</a> <font color="Olive" title="b1">j1</font>
 <b>by </b><i><a class="ref" href="gr_cy_1.html#T25" target="_self" title="GR_CY_1:th.25">Th25</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:39_1"/>
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">j1</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#K6" title="GROUP_1:func.6">|^</a> <font color="Olive" title="b2">j1</font>
 ;<br/>


</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="E3:39"/><i><font color="Green" title="E29">A2</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">j1</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#K6" title="GROUP_1:func.6">|^</a> <font color="Olive" title="b2">j1</font>
 ;<br/>

<a NAME="E4:39"/>
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> holds  <font color="Olive" title="b1">b</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="group_4.html#K5" title="GROUP_4:func.5">gr</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_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/>

<a NAME="E1:39_2"/>
 ex <font color="Olive" title="b1">j1</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</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#K6" title="GROUP_1:func.6">|^</a> <font color="Olive" title="b1">j1</font>
 
<b>by </b><i><a class="txt" href="gr_cy_1.html#E3:39"><i><font color="Green" title="E29">A2</font></i></a></i>;<br/>
<b>hence </b><a NAME="E2:39_2"/>
<font color="Maroon" title="c3">b</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="group_4.html#K5" title="GROUP_4:func.5">gr</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="ref" href="gr_cy_1.html#T25" target="_self" title="GR_CY_1:th.25">Th25</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:39"/><b>then </b>
 <a href="group_1.html#G1" title="GROUP_1:aggr.1">HGrStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font>,the <a href="group_1.html#U1" title="GROUP_1:sel.1">mult</a> of <font color="Maroon" title="c1">G</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_4.html#K5" title="GROUP_4:func.5">gr</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="ref" href="group_2.html#T71" title="GROUP_2:th.71">GROUP_2:71</a></i>;<br/>
<b>hence </b><a NAME="E6:39"/>
<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>
 <b>by </b><i><a class="ref" href="gr_cy_1.html#D9" target="_self" title="GR_CY_1:def.9">Def9</a></i>;<br/>


</div>
