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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>, <font color="Maroon">c<sub>3</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be  <a href="group_3.html#V1">normal</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>







<b>assume </b><a NAME="E1:24"/><i><font color="Green">E19</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="group_6.html#K4" target="_self">Cosets</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="group_6.html#K4" target="_self">Cosets</a> <font color="Maroon">c<sub>4</sub></font> )
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>5</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="E3:24"/><i><font color="Green">E20</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="group_2.html#K13">*</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="ref" href="group_6.html#T15" target="_self">Th15</a></i>;<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="E5:24"/><i><font color="Green">E21</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="group_2.html#K13">*</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="group_6.html#E1:24"><i><font color="Green">E19</font></i></a>, <a class="ref" href="group_6.html#T15" target="_self">Th15</a></i>;<br/>
<font color="Maroon">c<sub>2</sub></font> <a href="group_2.html#K2">*</a> <font color="Maroon">c<sub>3</sub></font> = 
<font color="Maroon">c<sub>5</sub></font> <a href="group_2.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_2.html#K11">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="group_6.html#E3:24"><i><font color="Green">E20</font></i></a>, <a class="txt" href="group_6.html#E5:24"><i><font color="Green">E21</font></i></a>, <a class="ref" href="group_3.html#T11">GROUP_3:11</a></i>
<br/>.= 
<font color="Maroon">c<sub>5</sub></font> <a href="group_2.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="group_2.html#K10">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="group_6.html#E5:24"><i><font color="Green">E21</font></i></a>, <a class="ref" href="group_3.html#T15">GROUP_3:15</a></i>
<br/>.= 
<font color="Maroon">c<sub>5</sub></font> <a href="group_2.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="group_2.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_4.html#K7">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_4.html#T60">GROUP_4:60</a></i>
<br/>.= 
<font color="Maroon">c<sub>5</sub></font> <a href="group_2.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="group_2.html#K3">*</a> <span class="p2">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_4.html#T58">GROUP_4:58</a></i>
<br/>.= 
<font color="Maroon">c<sub>5</sub></font> <a href="group_2.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="ref" href="group_2.html#T127">GROUP_2:127</a></i>
;<br/>
<a NAME="E7:24"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="group_2.html#K2">*</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="group_2.html#K14">Left_Cosets</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="group_2.html#D15">GROUP_2:def 15</a></i>;<br/>
<b>hence </b><a NAME="E8:24"/>
<font color="Maroon">c<sub>2</sub></font> <a href="group_2.html#K2">*</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="group_6.html#K4" target="_self">Cosets</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>


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