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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="E1:42_1_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">G</font> is <a href="gr_cy_1.html#V1" target="_self">cyclic</a>
 ;<br/>

<a NAME="E2:42_1_1"/>
for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font> holds  <font color="Olive">a</font> <a href="group_1.html#K1">*</a> <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> <a href="group_1.html#K1">*</a> <font color="Olive">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_1_1"><b>proof </b></a><div class="add">

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

<a NAME="E1:42_1_1_1"/>
 ex <font color="Olive">e</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 <br/>( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">e</font> <a href="group_1.html#K6">|^</a> <font color="Olive">j2</font> &amp;  ex <font color="Olive">j3</font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Olive">e</font> <a href="group_1.html#K6">|^</a> <font color="Olive">j3</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_1_1_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">e</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E2:42_1_1_1_1"/><i><font color="Green">E47</font></i>: 
for <font color="Olive">d</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font>  ex <font color="Olive">j3</font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Olive">d</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e</font> <a href="group_1.html#K6">|^</a> <font color="Olive">j3</font>
 <b>by </b><i>, </i>;<br/>
<b>take </b>
<font color="Maroon">e</font>
;<br/>

<a NAME="E3:42_1_1_1_1"/><i><font color="Green">E50</font></i>: 
 ex <font color="Olive">j2</font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e</font> <a href="group_1.html#K6">|^</a> <font color="Olive">j2</font>
 
<b>by </b><i/>;<br/>
<a NAME="E4:42_1_1_1_1"/>
 ex <font color="Olive">j3</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">e</font> <a href="group_1.html#K6">|^</a> <font color="Olive">j3</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E5:42_1_1_1_1"/>
 ex <font color="Olive">j2</font> being  <a href="int_1.html#NM1">Integer</a> st <br/>( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e</font> <a href="group_1.html#K6">|^</a> <font color="Olive">j2</font> &amp;  ex <font color="Olive">j3</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">e</font> <a href="group_1.html#K6">|^</a> <font color="Olive">j3</font> )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon">e</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font>, <font color="Maroon">j2</font> being   <a href="int_1.html#NM1">Integer</a>, <font color="Maroon">j3</font> being   <a href="int_1.html#NM1">Integer</a><b> such that </b><br/><a NAME="E3:42_1_1_1"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">j2</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">j3</font> )
 ;<br/>
<font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font> = 
<font color="Maroon">e</font> <a href="group_1.html#K6">|^</a> <span class="p1">(<span class="default"><font color="Maroon">j3</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">j2</font></span>)</span>
<b>by </b><i>, <a class="ref" href="group_1.html#T63">GROUP_1:63</a></i>
<br/>.= 
<font color="Maroon">b</font> <a href="group_1.html#K1">*</a> <font color="Maroon">a</font>
<b>by </b><i>, <a class="ref" href="group_1.html#T63">GROUP_1:63</a></i>
;<br/>
<b>hence </b><a NAME="E5:42_1_1_1"/>
<font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="group_1.html#K1">*</a> <font color="Maroon">a</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:42_1_1"/>
<font color="Maroon">G</font> is <a href="group_1.html#V7">commutative</a>
 <b>by </b><i><a class="ref" href="group_1.html#D16">GROUP_1:def 16</a></i>;<br/>


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