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<b>let </b><font color="Maroon" title="c1">G</font> be   <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">A1</font>, <font color="Olive" title="b2">A2</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">G</font><br/> for <font color="Olive" title="b3">N</font> being <a href="group_3.html#V1" title="GROUP_3:attr.1">normal</a>  <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font>  st <font color="Olive" title="b1">A1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_6.html#NK4" title="GROUP_6:NK.4">Cosets</a> <font color="Olive" title="b3">N</font> &amp; <font color="Olive" title="b2">A2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_6.html#NK4" title="GROUP_6:NK.4">Cosets</a> <font color="Olive" title="b3">N</font> holds <br/><font color="Olive" title="b1">A1</font> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <font color="Olive" title="b2">A2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_6.html#NK4" title="GROUP_6:NK.4">Cosets</a> <font color="Olive" title="b3">N</font></span><br/><b>let </b><font color="Maroon" title="c2">A1</font>, <font color="Maroon" title="c3">A2</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">N</font> being <a href="group_3.html#V1" title="GROUP_3:attr.1">normal</a>  <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font>  st <font color="Maroon" title="c2">A1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_6.html#NK4" title="GROUP_6:NK.4">Cosets</a> <font color="Olive" title="b1">N</font> &amp; <font color="Maroon" title="c3">A2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_6.html#NK4" title="GROUP_6:NK.4">Cosets</a> <font color="Olive" title="b1">N</font> holds <br/><font color="Maroon" title="c2">A1</font> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <font color="Maroon" title="c3">A2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_6.html#NK4" title="GROUP_6:NK.4">Cosets</a> <font color="Olive" title="b1">N</font></span><br/><b>let </b><font color="Maroon" title="c4">N</font> be  <a href="group_3.html#V1" title="GROUP_3:attr.1">normal</a>  <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">A1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_6.html#NK4" title="GROUP_6:NK.4">Cosets</a> <font color="Maroon" title="c4">N</font> &amp; <font color="Maroon" title="c3">A2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_6.html#NK4" title="GROUP_6:NK.4">Cosets</a> <font color="Maroon" title="c4">N</font> implies <font color="Maroon" title="c2">A1</font> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <font color="Maroon" title="c3">A2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_6.html#NK4" title="GROUP_6:NK.4">Cosets</a> <font color="Maroon" title="c4">N</font> )</span><br/>







<b>assume </b><a NAME="E1:25"/><i><font color="Green" title="E18">A1</font></i>: 
( <font color="Maroon" title="c2">A1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_6.html#NK4" title="GROUP_6:NK.4">Cosets</a> <font color="Maroon" title="c4">N</font> &amp; <font color="Maroon" title="c3">A2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_6.html#NK4" title="GROUP_6:NK.4">Cosets</a> <font color="Maroon" title="c4">N</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">A1</font> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <font color="Maroon" title="c3">A2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_6.html#NK4" title="GROUP_6:NK.4">Cosets</a> <font color="Maroon" title="c4">N</font></span><br/>

<b>then consider </b><font color="Maroon" title="c5">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E3:25"/><i><font color="Green" title="E19">A2</font></i>: 
( <font color="Maroon" title="c2">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c4">N</font> &amp; <font color="Maroon" title="c2">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">N</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c5">a</font> )
 <b>by </b><i><a class="ref" href="group_6.html#T15" target="_self" title="GROUP_6:th.15">Th15</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c6">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E5:25"/><i><font color="Green" title="E20">A3</font></i>: 
( <font color="Maroon" title="c3">A2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c4">N</font> &amp; <font color="Maroon" title="c3">A2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">N</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c6">b</font> )
 <b>by </b><i><a class="txt" href="group_6.html#E1:25"><i><font color="Green" title="E18">A1</font></i></a>, <a class="ref" href="group_6.html#T15" target="_self" title="GROUP_6:th.15">Th15</a></i>;<br/>
<font color="Maroon" title="c2">A1</font> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <font color="Maroon" title="c3">A2</font> = 
<font color="Maroon" title="c5">a</font> <a href="group_2.html#K3" title="GROUP_2:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">N</font> <a href="group_2.html#K11" title="GROUP_2:func.11">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">b</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c4">N</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="group_6.html#E3:25"><i><font color="Green" title="E19">A2</font></i></a>, <a class="txt" href="group_6.html#E5:25"><i><font color="Green" title="E20">A3</font></i></a>, <a class="ref" href="group_3.html#T11" title="GROUP_3:th.11">GROUP_3:11</a></i>
<br/>.= 
<font color="Maroon" title="c5">a</font> <a href="group_2.html#K3" title="GROUP_2:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">b</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c4">N</font></span>)</span> <a href="group_2.html#K10" title="GROUP_2:func.10">*</a> <font color="Maroon" title="c4">N</font></span>)</span>
<b>by </b><i><a class="txt" href="group_6.html#E5:25"><i><font color="Green" title="E20">A3</font></i></a>, <a class="ref" href="group_3.html#T15" title="GROUP_3:th.15">GROUP_3:15</a></i>
<br/>.= 
<font color="Maroon" title="c5">a</font> <a href="group_2.html#K3" title="GROUP_2:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">b</font> <a href="group_2.html#K3" title="GROUP_2:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">N</font> <a href="group_4.html#K7" title="GROUP_4:func.7">*</a> <font color="Maroon" title="c4">N</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_4.html#T60" title="GROUP_4:th.60">GROUP_4:60</a></i>
<br/>.= 
<font color="Maroon" title="c5">a</font> <a href="group_2.html#K3" title="GROUP_2:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">b</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c4">N</font></span>)</span>
<b>by </b><i><a class="ref" href="group_2.html#T91" title="GROUP_2:th.91">GROUP_2:91</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">b</font></span>)</span> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c4">N</font>
<b>by </b><i><a class="ref" href="group_2.html#T127" title="GROUP_2:th.127">GROUP_2:127</a></i>
;<br/>
<b>hence </b><a NAME="E7:25"/>
<font color="Maroon" title="c2">A1</font> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <font color="Maroon" title="c3">A2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_6.html#NK4" title="GROUP_6:NK.4">Cosets</a> <font color="Maroon" title="c4">N</font>
 <b>by </b><i><a class="ref" href="group_2.html#D15" title="GROUP_2:def.15">GROUP_2:def 15</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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