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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="E1:42"/>
( <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  <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K8" title="GROUP_1:func.8">ord</a> <font color="Maroon" title="c1">G</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:42_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="E3:42_1"/><i><font color="Green" title="E30">A1</font></i>: 
 <a href="algstr_0.html#G3" title="ALGSTR_0:aggr.3">multMagma</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="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</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/>

<b>reconsider </b><font color="Maroon" title="c3">H</font> =  <a href="algstr_0.html#G3" title="ALGSTR_0:aggr.3">multMagma</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="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c1">G</font> #) as   <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a> ;<br/>
<a NAME="E5:42_1"/>
 <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K7" title="GROUP_1:func.7">Ord</a> <font color="Maroon" title="c3">H</font>
 
<b>by </b><i><a class="txt" href="gr_cy_1.html#E3:42_1"><i><font color="Green" title="E30">A1</font></i></a>, <a class="ref" href="gr_cy_1.html#T27" target="_self" title="GR_CY_1:th.27">Th27</a></i>;<br/>
<b>hence </b><a NAME="E6:42_1"/>
 <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K8" title="GROUP_1:func.8">ord</a> <font color="Maroon" title="c1">G</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="E3:42"/><i><font color="Green" title="E30">A2</font></i>: 
 <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K8" title="GROUP_1:func.8">ord</a> <font color="Maroon" title="c1">G</font>
 ;<br/>

<a NAME="E4:42"/>
 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  <a href="algstr_0.html#G3" title="ALGSTR_0:aggr.3">multMagma</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="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</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="Olive" title="b1">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="42_2"><b>proof </b></a><div class="add">

<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="E2:42_2"/><i><font color="Green" title="E31">A3</font></i>: 
 <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="txt" href="gr_cy_1.html#E3:42"><i><font color="Green" title="E30">A2</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c3">a</font>
;<br/>

<a NAME="E3:42_2"/>
 <a href="group_1.html#K8" title="GROUP_1:func.8">ord</a> <span class="p1">(<span class="default"><a href="group_4.html#K5" title="GROUP_4:func.5">gr</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="txt" href="gr_cy_1.html#E2:42_2"><i><font color="Green" title="E31">A3</font></i></a>, <a class="ref" href="gr_cy_1.html#T27" target="_self" title="GR_CY_1:th.27">Th27</a></i>;<br/>
<b>hence </b><a NAME="E4:42_2"/>
 <a href="algstr_0.html#G3" title="ALGSTR_0:aggr.3">multMagma</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="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</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="c3">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#T85" title="GROUP_2:th.85">GROUP_2:85</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:42"/>
<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#D9" target="_self" title="GR_CY_1:def.9">Def9</a></i>;<br/>


</div>
