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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="group_1.html#NM1">Group</a>;<br/>

<b>thus </b><a NAME="E1:39"/>
( <font color="Maroon">c<sub>1</sub></font> is  <a href="group_1.html#V7">commutative</a> <a href="group_1.html#NM1">Group</a> implies for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <span class="p1"><a href="group_5.html#K2" target="_self">[.</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="group_5.html#K2" target="_self">.]</a></span> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></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"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is  <a href="group_1.html#V7">commutative</a> <a href="group_1.html#NM1">Group</a>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<a NAME="E2:39_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="group_5.html#E1:39_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="group_5.html#E42"><i><font color="Green">Lemma24</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:39_1"/>
<span class="p1"><a href="group_5.html#K2" target="_self">[.</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="group_5.html#K2" target="_self">.]</a></span> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="group_5.html#T39" target="_self">Th39</a></i>;<br/>


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

<b>assume </b><a NAME="E2:39"/><i><font color="Green">E25</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <span class="p1"><a href="group_5.html#K2" target="_self">[.</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="group_5.html#K2" target="_self">.]</a></span> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<a NAME="E3:39"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="group_1.html#V7">commutative</a>
 
<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">c<sub>2</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="group_1.html#D16">GROUP_1:def 16</a><br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<a NAME="E1:39_2"/>
<span class="p1"><a href="group_5.html#K2" target="_self">[.</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="group_5.html#K2" target="_self">.]</a></span> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="group_5.html#E2:39"><i><font color="Green">E25</font></i></a></i>;<br/>
<b>hence </b><a NAME="E2:39_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="group_5.html#T39" target="_self">Th39</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:39"/>
<font color="Maroon">c<sub>1</sub></font> is  <a href="group_1.html#V7">commutative</a> <a href="group_1.html#NM1">Group</a>
 ;<br/>


</div>
