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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="group_1.html#NM1">Group</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be  <a href="group_1.html#V1">strict</a> <a href="group_1.html#NM1">Group</a>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:100"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="group_6.html#R1" target="_self">are_isomorphic</a> 
 <b>and </b><br/><a NAME="E2:100"/><i><font color="Green">E72</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>consider </b><font color="Maroon">c<sub>3</sub></font> being   <a href="group_6.html#NM2">Homomorphism</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E4:100"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="group_6.html#NR3" target="_self">is_isomorphism</a> 
 <b>by </b><i><a class="txt" href="group_6.html#E1:100"><i><font color="Green">E71</font></i></a>, <a class="ref" href="group_6.html#D15" target="_self">Def15</a></i>;<br/>
<a NAME="E5:100"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="group_6.html#NR2" target="_self">is_epimorphism</a> 
 
<b>by </b><i><a class="txt" href="group_6.html#E4:100"><i><font color="Green">E73</font></i></a>, <a class="ref" href="group_6.html#D14" target="_self">Def14</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="100_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font>;<br/><b>consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E2:100_1"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="group_6.html#E5:100"><i><font color="Green">E74</font></i></a>, <a class="ref" href="group_6.html#T68" target="_self">Th68</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E4:100_1"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="group_6.html#E5:100"><i><font color="Green">E74</font></i></a>, <a class="ref" href="group_6.html#T68" target="_self">Th68</a></i>;<br/><b>thus </b><font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>5</sub></font> = 
<font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="group_6.html#E2:100_1"><i><font color="Green">E75</font></i></a>, <a class="txt" href="group_6.html#E4:100_1"><i><font color="Green">E76</font></i></a>, <a class="ref" href="group_6.html#D7" target="_self">Def7</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="group_6.html#E2:100"><i><font color="Green">E72</font></i></a>, <a class="txt" href="group_6.html#E72"><i><font color="Green">Lemma49</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>5</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="group_6.html#E2:100_1"><i><font color="Green">E75</font></i></a>, <a class="txt" href="group_6.html#E4:100_1"><i><font color="Green">E76</font></i></a>, <a class="ref" href="group_6.html#D7" target="_self">Def7</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:100"/>
<font color="Maroon">c<sub>2</sub></font> is  <a href="group_1.html#V7">commutative</a> <a href="group_1.html#NM1">Group</a>
 <b>by </b><i><a class="ref" href="group_1.html#D16">GROUP_1:def 16</a></i>;<br/>


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