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

<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">H</font>, <font color="Maroon" title="c4">K</font>, <font color="Maroon" title="c5">H'</font>, <font color="Maroon" title="c6">K'</font> be  <a href="group_9.html#V2" title="GROUP_9:attr.2">strict</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="c7">JH</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="c5">H'</font> <a href="group_9.html#K19" title="GROUP_9:func.19">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">H</font> <a href="group_9.html#K17" title="GROUP_9:func.17">/\</a> <font color="Maroon" title="c4">K</font></span>)</span>;<br/><b>let </b><font color="Maroon" title="c8">HK</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="c3">H</font> <a href="group_9.html#K17" title="GROUP_9:func.17">/\</a> <font color="Maroon" title="c4">K</font>;<br/>









<b>assume </b><a NAME="E1:155"/><i><font color="Green" title="E130">A1</font></i>: 
( <font color="Maroon" title="c5">H'</font> is  <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="c3">H</font> &amp; <font color="Maroon" title="c6">K'</font> is  <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="c4">K</font> )
 ;<br/>

<b>assume </b><a NAME="E2:155"/><i><font color="Green" title="E131">A2</font></i>: 
( <font color="Maroon" title="c7">JH</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">H'</font> <a href="group_9.html#K19" title="GROUP_9:func.19">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">H</font> <a href="group_9.html#K17" title="GROUP_9:func.17">/\</a> <font color="Maroon" title="c6">K'</font></span>)</span> &amp; <font color="Maroon" title="c8">HK</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">H'</font> <a href="group_9.html#K17" title="GROUP_9:func.17">/\</a> <font color="Maroon" title="c4">K</font></span>)</span> <a href="group_9.html#K19" title="GROUP_9:func.19">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">K'</font> <a href="group_9.html#K17" title="GROUP_9:func.17">/\</a> <font color="Maroon" title="c3">H</font></span>)</span> )
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c9">GG</font> = <font color="Maroon" title="c3">H</font> as   <a href="group_9.html#NM2" title="GROUP_9:NM.2">GroupWithOperators</a> of <font color="Maroon" title="c1">O</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c10">N'</font> = <font color="Maroon" title="c5">H'</font> as  <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="c9">GG</font> <b>by </b><i><a class="txt" href="group_9.html#E1:155"><i><font color="Green" title="E130">A1</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">N</font> = <font color="Maroon" title="c10">N'</font> as    <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Maroon" title="c9">GG</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c12">HH</font> = <font color="Maroon" title="c9">GG</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c10">N'</font> as   <a href="group_9.html#NM2" title="GROUP_9:NM.2">GroupWithOperators</a> of <font color="Maroon" title="c1">O</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c13">f</font> =  <a href="group_9.html#K12" title="GROUP_9:func.12">nat_hom</a> <font color="Maroon" title="c10">N'</font> as   <a href="group_9.html#NM4" title="GROUP_9:NM.4">Homomorphism</a> of <font color="Maroon" title="c9">GG</font>,<font color="Maroon" title="c12">HH</font> ;<br/>
<b>set </b><font color="Maroon" title="c14">G'</font> = <font color="Maroon" title="c3">H</font> <a href="group_9.html#K17" title="GROUP_9:func.17">/\</a> <font color="Maroon" title="c4">K</font>;<br/>
<b>reconsider </b><font color="Maroon" title="c15">G'</font> = <font color="Maroon" title="c3">H</font> <a href="group_9.html#K17" title="GROUP_9:func.17">/\</a> <font color="Maroon" title="c4">K</font> as  <a href="group_9.html#V2" title="GROUP_9:attr.2">strict</a>  <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Maroon" title="c9">GG</font> <b>by </b><i><a class="txt" href="group_9.html#E104"><i><font color="Green" title="E93">Lm34</font></i></a></i>;<br/>
<b>set </b><font color="Maroon" title="c16">L</font> = <font color="Maroon" title="c3">H</font> <a href="group_9.html#K17" title="GROUP_9:func.17">/\</a> <font color="Maroon" title="c6">K'</font>;<br/>
<a NAME="E9:155"/><i><font color="Green" title="E132">A3</font></i>: 
<font color="Maroon" title="c3">H</font> <a href="group_9.html#K17" title="GROUP_9:func.17">/\</a> <font color="Maroon" title="c6">K'</font> is  <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="c15">G'</font>
 
<b>by </b><i><a class="txt" href="group_9.html#E1:155"><i><font color="Green" title="E130">A1</font></i></a>, <a class="ref" href="group_9.html#T60" target="_self" title="GROUP_9:th.60">Th60</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c17">L</font> = <font color="Maroon" title="c3">H</font> <a href="group_9.html#K17" title="GROUP_9:func.17">/\</a> <font color="Maroon" title="c6">K'</font> as    <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Maroon" title="c9">GG</font> <b>by </b><i><a class="ref" href="group_9.html#T11" target="_self" title="GROUP_9:th.11">Th11</a></i>;<br/>
<i><font color="Green" title="E133">A4</font></i>: <font color="Maroon" title="c17">L</font> <a href="group_9.html#K19" title="GROUP_9:func.19">"\/"</a> <font color="Maroon" title="c11">N</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">H</font> <a href="group_9.html#K17" title="GROUP_9:func.17">/\</a> <font color="Maroon" title="c6">K'</font></span>)</span> <a href="group_9.html#K19" title="GROUP_9:func.19">"\/"</a> <font color="Maroon" title="c5">H'</font>
<b>by </b><i><a class="ref" href="group_9.html#T86" target="_self" title="GROUP_9:th.86">Th86</a></i>
<br/>.= 
<font color="Maroon" title="c7">JH</font>
<b>by </b><i><a class="txt" href="group_9.html#E2:155"><i><font color="Green" title="E131">A2</font></i></a></i>
;<br/>
<b>set </b><font color="Maroon" title="c18">N1</font> = <font color="Maroon" title="c15">G'</font> <a href="group_9.html#K17" title="GROUP_9:func.17">/\</a> <font color="Maroon" title="c11">N</font>;<br/>
<a NAME="E12:155"/>
<font color="Maroon" title="c15">G'</font> <a href="group_9.html#K17" title="GROUP_9:func.17">/\</a> <font color="Maroon" title="c11">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">H</font> <a href="group_9.html#K17" title="GROUP_9:func.17">/\</a> <font color="Maroon" title="c4">K</font></span>)</span> <a href="group_9.html#K17" title="GROUP_9:func.17">/\</a> <font color="Maroon" title="c5">H'</font>
 
<b>by </b><i><a class="ref" href="group_9.html#T39" target="_self" title="GROUP_9:th.39">Th39</a></i>;<br/>
<b>then </b><i><font color="Green" title="E134">A5</font></i>: <font color="Maroon" title="c17">L</font> <a href="group_9.html#K19" title="GROUP_9:func.19">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">G'</font> <a href="group_9.html#K17" title="GROUP_9:func.17">/\</a> <font color="Maroon" title="c11">N</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">H</font> <a href="group_9.html#K17" title="GROUP_9:func.17">/\</a> <font color="Maroon" title="c6">K'</font></span>)</span> <a href="group_9.html#K19" title="GROUP_9:func.19">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">H</font> <a href="group_9.html#K17" title="GROUP_9:func.17">/\</a> <font color="Maroon" title="c4">K</font></span>)</span> <a href="group_9.html#K17" title="GROUP_9:func.17">/\</a> <font color="Maroon" title="c5">H'</font></span>)</span>
<b>by </b><i><a class="ref" href="group_9.html#T86" target="_self" title="GROUP_9:th.86">Th86</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">H'</font> <a href="group_9.html#K17" title="GROUP_9:func.17">/\</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="group_9.html#K17" title="GROUP_9:func.17">/\</a> <font color="Maroon" title="c4">K</font></span>)</span> <a href="group_9.html#K19" title="GROUP_9:func.19">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">K'</font> <a href="group_9.html#K17" title="GROUP_9:func.17">/\</a> <font color="Maroon" title="c3">H</font></span>)</span>
<b>by </b><i><a class="ref" href="group_9.html#T20" target="_self" title="GROUP_9:th.20">Th20</a></i>
<br/>.= 
<font color="Maroon" title="c8">HK</font>
<b>by </b><i><a class="txt" href="group_9.html#E1:155"><i><font color="Green" title="E130">A1</font></i></a>, <a class="txt" href="group_9.html#E2:155"><i><font color="Green" title="E131">A2</font></i></a>, <a class="txt" href="group_9.html#E72"><i><font color="Green" title="E61">Lm22</font></i></a></i>
;<br/>
<a NAME="E14:155"/><i><font color="Green" title="E135">A6</font></i>: 
<font color="Maroon" title="c11">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_9.html#K13" title="GROUP_9:func.13">Ker</a> <font color="Maroon" title="c13">f</font>
 
<b>by </b><i><a class="ref" href="group_9.html#T48" target="_self" title="GROUP_9:th.48">Th48</a></i>;<br/>
<font color="Maroon" title="c15">G'</font> <a href="group_9.html#K19" title="GROUP_9:func.19">"\/"</a> <font color="Maroon" title="c11">N</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">H</font> <a href="group_9.html#K17" title="GROUP_9:func.17">/\</a> <font color="Maroon" title="c4">K</font></span>)</span> <a href="group_9.html#K19" title="GROUP_9:func.19">"\/"</a> <font color="Maroon" title="c5">H'</font>
<b>by </b><i><a class="ref" href="group_9.html#T86" target="_self" title="GROUP_9:th.86">Th86</a></i>
<br/>.= 
<font color="Maroon" title="c5">H'</font> <a href="group_9.html#K19" title="GROUP_9:func.19">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">H</font> <a href="group_9.html#K17" title="GROUP_9:func.17">/\</a> <font color="Maroon" title="c4">K</font></span>)</span>

;<br/>
<b>hence </b><a NAME="E16:155"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">H'</font> <a href="group_9.html#K19" title="GROUP_9:func.19">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">H</font> <a href="group_9.html#K17" title="GROUP_9:func.17">/\</a> <font color="Maroon" title="c4">K</font></span>)</span></span>)</span> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c7">JH</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">H</font> <a href="group_9.html#K17" title="GROUP_9:func.17">/\</a> <font color="Maroon" title="c4">K</font></span>)</span> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c8">HK</font> <a href="group_9.html#R3" title="GROUP_9:pred.3">are_isomorphic</a> 
 <b>by </b><i><a class="txt" href="group_9.html#E9:155"><i><font color="Green" title="E132">A3</font></i></a>, <a class="txt" href="group_9.html#E14:155"><i><font color="Green" title="E135">A6</font></i></a>, <a class="txt" href="group_9.html#E13:155"><i><font color="Green" title="E134">A5</font></i></a>, <a class="txt" href="group_9.html#E11:155"><i><font color="Green" title="E133">A4</font></i></a>, <a class="ref" href="group_9.html#T90" target="_self" title="GROUP_9:th.90">Th90</a></i>;<br/>


</div>
