<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">G</font> be  <a href="group_1.html#V1">strict</a> <a href="group_1.html#NM1">Group</a>;<br/><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:7"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">G</font> is <a href="group_1.html#V6">finite</a>
 <b>and </b><br/><a NAME="E2:7"/><i><font color="Green">E49</font></i>: 
<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>
 ;<br/>

<a NAME="E3:7"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">G</font> is  <a href="gr_cy_1.html#V1">cyclic</a> <a href="group_1.html#NM1">Group</a>
 
<b>by </b><i>, <a class="ref" href="gr_cy_1.html#D9">GR_CY_1:def 9</a></i>;<br/>
<a NAME="E4:7"/>
for <font color="Olive">G1</font> being <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font>  ex <font color="Olive">p</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Olive">G1</font> <a href="group_2.html#R1">=</a>  <a href="group_4.html#K5">gr</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <font color="Olive">p</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">G1</font> be  <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font>;<br/>

<a NAME="E1:7_1"/>
<font color="Maroon">G1</font> is  <a href="gr_cy_1.html#V1">cyclic</a> <a href="group_1.html#NM1">Group</a>
 
<b>by </b><i>, , <a class="ref" href="gr_cy_1.html#T44">GR_CY_1:44</a></i>;<br/>
<b>then consider </b><font color="Maroon">b</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G1</font><b> such that </b><br/><a NAME="E3:7_1"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">G1</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">b</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="gr_cy_1.html#D9">GR_CY_1:def 9</a></i>;<br/>
<b>reconsider </b><font color="Maroon">b1</font> = <font color="Maroon">b</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font> <b>by </b><i><a class="ref" href="group_2.html#T51">GROUP_2:51</a></i>;<br/>
<b>consider </b><font color="Maroon">p</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E6:7_1"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">b1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">p</font>
 <b>by </b><i>, , </i>;<br/>
<b>take </b>
<font color="Maroon">p</font>
;<br/>

<b>thus </b><a NAME="E7:7_1"/>
<font color="Maroon">G1</font> <a href="group_2.html#R1">=</a>  <a href="group_4.html#K5">gr</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">p</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:7"/>
for <font color="Olive">G1</font> being <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font>  ex <font color="Olive">p</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Olive">G1</font> <a href="group_2.html#R1">=</a>  <a href="group_4.html#K5">gr</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <font color="Olive">p</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 ;<br/>


</div>
