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<b>let </b><font color="Maroon" title="c1">O</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">G</font> be   <a href="group_9.html#NM2" title="GROUP_9:NM.2">GroupWithOperators</a> of <font color="Maroon" title="c1">O</font>;<br/><b>let </b><font color="Maroon" title="c3">N</font> be  <a href="group_9.html#V4" title="GROUP_9:attr.4">normal</a>  <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Maroon" title="c2">G</font>;<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="c2">G</font>;<br/>







<b>assume </b><a NAME="E1:121"/><i><font color="Green" title="E97">A1</font></i>: 
<font color="Maroon" title="c4">N'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#G3" title="ALGSTR_0:aggr.3">multMagma</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">N</font>,the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c3">N</font> #)
 ;<br/>

<a NAME="E2:121"/><b>then </b>
 <a href="group_6.html#NK4" title="GROUP_6:NK.4">Cosets</a> <font color="Maroon" title="c4">N'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_9.html#K7" title="GROUP_9:func.7">Cosets</a> <font color="Maroon" title="c3">N</font>
 
<b>by </b><i><a class="ref" href="group_9.html#D14" target="_self" title="GROUP_9:def.14">Def14</a></i>;<br/>
<b>hence </b><a NAME="E3:121"/>
<font color="Maroon" title="c2">G</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <font color="Maroon" title="c4">N'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#G3" title="ALGSTR_0:aggr.3">multMagma</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c3">N</font></span>)</span>,the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c3">N</font></span>)</span> #)
 <b>by </b><i><a class="txt" href="group_9.html#E1:121"><i><font color="Green" title="E97">A1</font></i></a>, <a class="ref" href="group_9.html#D15" target="_self" title="GROUP_9:def.15">Def15</a></i>;<br/>

<b>reconsider </b><font color="Maroon" title="c5">e</font> =  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <font color="Maroon" title="c4">N'</font></span>)</span> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c3">N</font></span>)</span> <b>by </b><i><a class="txt" href="group_9.html#E1:121"><i><font color="Green" title="E97">A1</font></i></a>, <a class="ref" href="group_9.html#D14" target="_self" title="GROUP_9:def.14">Def14</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="121_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">h</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c3">N</font></span>)</span>;<br/><b>reconsider </b><font color="Maroon" title="c7">h'</font> = <font color="Maroon" title="c6">h</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <font color="Maroon" title="c4">N'</font></span>)</span> <b>by </b><i><a class="txt" href="group_9.html#E1:121"><i><font color="Green" title="E97">A1</font></i></a>, <a class="ref" href="group_9.html#D14" target="_self" title="GROUP_9:def.14">Def14</a></i>;<br/><b>thus </b><font color="Maroon" title="c6">h</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">e</font> = 
<font color="Maroon" title="c7">h'</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <font color="Maroon" title="c4">N'</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="group_9.html#E1:121"><i><font color="Green" title="E97">A1</font></i></a>, <a class="ref" href="group_9.html#D15" target="_self" title="GROUP_9:def.15">Def15</a></i>
<br/>.= 
<font color="Maroon" title="c6">h</font>
<b>by </b><i><a class="ref" href="group_1.html#D5" title="GROUP_1:def.5">GROUP_1:def 5</a></i>
;<br/><b>thus </b><font color="Maroon" title="c5">e</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">h</font> = 
<span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <font color="Maroon" title="c4">N'</font></span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c7">h'</font>
<b>by </b><i><a class="txt" href="group_9.html#E1:121"><i><font color="Green" title="E97">A1</font></i></a>, <a class="ref" href="group_9.html#D15" target="_self" title="GROUP_9:def.15">Def15</a></i>
<br/>.= 
<font color="Maroon" title="c6">h</font>
<b>by </b><i><a class="ref" href="group_1.html#D5" title="GROUP_1:def.5">GROUP_1:def 5</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:121"/>
 <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <font color="Maroon" title="c4">N'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c3">N</font></span>)</span>
 <b>by </b><i><a class="ref" href="group_1.html#T10" title="GROUP_1:th.10">GROUP_1:10</a></i>;<br/>


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