<?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#V2" title="GROUP_9:attr.2">strict</a> <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#V2" title="GROUP_9:attr.2">strict</a> <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">H</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> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c3">N</font>;<br/>







<b>reconsider </b><font color="Maroon" title="c5">N'</font> =  <a href="group_1.html#G1" title="GROUP_1:aggr.1">HGrStr</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="group_1.html#U1" title="GROUP_1:sel.1">mult</a> of <font color="Maroon" title="c3">N</font> #) as  <a href="group_1.html#V1" title="GROUP_1:attr.1">strict</a> <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> <b>by </b><i><a class="txt" href="group_9.html#E20"><i><font color="Green" title="E19">Lm7</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">H'</font> =  <a href="group_1.html#G1" title="GROUP_1:aggr.1">HGrStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">H</font>,the <a href="group_1.html#U1" title="GROUP_1:sel.1">mult</a> of <font color="Maroon" title="c4">H</font> #) as  <a href="group_1.html#V1" title="GROUP_1:attr.1">strict</a>  <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <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> <b>by </b><i><a class="txt" href="group_9.html#E40"><i><font color="Green" title="E38">Lm16</font></i></a></i>;<br/>
<a NAME="E3:144"/><i><font color="Green" title="E119">A1</font></i>: 
( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">H'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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> &amp; the <a href="group_1.html#U1" title="GROUP_1:sel.1">mult</a> of <font color="Maroon" title="c6">H'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="group_1.html#U1" title="GROUP_1:sel.1">mult</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> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">H'</font> )
 
<b>by </b><i><a class="ref" href="group_2.html#D5" title="GROUP_2:def.5">GROUP_2:def 5</a></i>;<br/>
<a NAME="E4:144"/><i><font color="Green" title="E120">A2</font></i>: 
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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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_6.html#K6" title="GROUP_6:func.6">./.</a> <font color="Maroon" title="c5">N'</font></span>)</span>
 
<b>by </b><i><a class="ref" href="group_9.html#D14" target="_self" title="GROUP_9:def.14">Def14</a></i>;<br/>
<a NAME="E5:144"/>
the <a href="group_1.html#U1" title="GROUP_1:sel.1">mult</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> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> the <a href="group_1.html#U1" title="GROUP_1:sel.1">mult</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="group_6.html#K6" title="GROUP_6:func.6">./.</a> <font color="Maroon" title="c5">N'</font></span>)</span>
 
<b>by </b><i><a class="ref" href="group_9.html#D15" target="_self" title="GROUP_9:def.15">Def15</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c7">H'</font> = <font color="Maroon" title="c6">H'</font> as  <a href="group_1.html#V1" title="GROUP_1:attr.1">strict</a>  <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c2">G</font> <a href="group_6.html#K6" title="GROUP_6:func.6">./.</a> <font color="Maroon" title="c5">N'</font> <b>by </b><i><a class="txt" href="group_9.html#E3:144"><i><font color="Green" title="E119">A1</font></i></a>, <a class="txt" href="group_9.html#E4:144"><i><font color="Green" title="E120">A2</font></i></a>, <a class="ref" href="group_2.html#D5" title="GROUP_2:def.5">GROUP_2:def 5</a></i>;<br/>
<b>assume </b><a NAME="E7:144"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="group_9.html#K12" title="GROUP_9:func.12">nat_hom</a> <font color="Maroon" title="c3">N</font></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">H</font>
 ;<br/>

<a NAME="E8:144"/><b>then </b><i><font color="Green" title="E121">A3</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="group_6.html#K10" title="GROUP_6:func.10">nat_hom</a> <font color="Maroon" title="c5">N'</font></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">H'</font>
 
<b>by </b><i><a class="ref" href="group_9.html#D23" target="_self" title="GROUP_9:def.23">Def23</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="144_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">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_6.html#K6" title="GROUP_6:func.6">./.</a> <font color="Maroon" title="c5">N'</font></span>)</span>;<br/><b>thus </b><a NAME="E1:144_1"/>
( <font color="Maroon" title="c8">h</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c7">H'</font> implies <font color="Maroon" title="c8">h</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="group_2.html#K6" title="GROUP_2:func.6">(Omega).</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="group_6.html#K6" title="GROUP_6:func.6">./.</a> <font color="Maroon" title="c5">N'</font></span>)</span> )
 <b>by </b><i><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/><b>assume </b><a NAME="E2:144_1"/>
<font color="Maroon" title="c8">h</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="group_2.html#K6" title="GROUP_2:func.6">(Omega).</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="group_6.html#K6" title="GROUP_6:func.6">./.</a> <font color="Maroon" title="c5">N'</font></span>)</span>
 ;<br/><a NAME="E3:144_1"/>
<font color="Maroon" title="c8">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_2.html#K14" title="GROUP_2:func.14">Left_Cosets</a> <font color="Maroon" title="c5">N'</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c9">g</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G</font><b> such that </b><br/><a NAME="E5:144_1"/><i><font color="Green" title="E122">A4</font></i>: 
<font color="Maroon" title="c8">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">g</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c5">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>;<br/><b>reconsider </b><font color="Maroon" title="c10">R</font> =  <a href="group_6.html#K10" title="GROUP_6:func.10">nat_hom</a> <font color="Maroon" title="c5">N'</font> as    <a href="relset_1.html#M2" title="RELSET_1:mode.2">Relation</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G</font>,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_6.html#K6" title="GROUP_6:func.6">./.</a> <font color="Maroon" title="c5">N'</font></span>)</span> ;<br/><b>consider </b><font color="Maroon" title="c11">h'</font> being   <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#K6" title="GROUP_6:func.6">./.</a> <font color="Maroon" title="c5">N'</font></span>)</span><b> such that </b><br/><a NAME="E8:144_1"/><i><font color="Green" title="E123">A5</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">g</font>,<font color="Maroon" title="c11">h'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">R</font> &amp; <font color="Maroon" title="c11">h'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">H'</font> )
 <b>by </b><i><a class="txt" href="group_9.html#E8:144"><i><font color="Green" title="E121">A3</font></i></a>, <a class="ref" href="relset_1.html#T53" title="RELSET_1:th.53">RELSET_1:53</a></i>;<br/><a NAME="E9:144_1"/>
<span class="p1">(<span class="default"><a href="group_6.html#K10" title="GROUP_6:func.10">nat_hom</a> <font color="Maroon" title="c5">N'</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">h'</font>
 <b>by </b><i><a class="txt" href="group_9.html#E8:144_1"><i><font color="Green" title="E123">A5</font></i></a>, <a class="ref" href="funct_1.html#T8" title="FUNCT_1:th.8">FUNCT_1:8</a></i>;<br/><a NAME="E10:144_1"/><b>then </b>
<font color="Maroon" title="c8">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">H'</font>
 <b>by </b><i><a class="txt" href="group_9.html#E5:144_1"><i><font color="Green" title="E122">A4</font></i></a>, <a class="txt" href="group_9.html#E8:144_1"><i><font color="Green" title="E123">A5</font></i></a>, <a class="ref" href="group_6.html#D9" title="GROUP_6:def.9">GROUP_6:def 9</a></i>;<br/><b>hence </b><a NAME="E11:144_1"/>
<font color="Maroon" title="c8">h</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c7">H'</font>
 <b>by </b><i><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/></div><b>end;</b></div>
<a NAME="E10:144"/><b>then </b>
<font color="Maroon" title="c7">H'</font> <a href="group_2.html#R1" title="GROUP_2:pred.1">=</a>  <a href="group_2.html#K6" title="GROUP_2:func.6">(Omega).</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="group_6.html#K6" title="GROUP_6:func.6">./.</a> <font color="Maroon" title="c5">N'</font></span>)</span>
 
<b>by </b><i><a class="ref" href="group_2.html#D6" title="GROUP_2:def.6">GROUP_2:def 6</a></i>;<br/>
<a NAME="E11:144"/><b>then </b>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">H</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="E12:144"/>
<font color="Maroon" title="c4">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_9.html#K5" title="GROUP_9:func.5">(Omega).</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="txt" href="group_9.html#E12"><i><font color="Green" title="E12">Lm5</font></i></a></i>;<br/>


</div>
