<?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">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">H1</font> be    <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Maroon" title="c2">G</font>;<br/>







<b>thus </b><a NAME="E1:119"/>
<font color="Maroon" title="c4">H1</font> <a href="group_9.html#K17" title="GROUP_9:func.17">/\</a> <font color="Maroon" title="c3">N</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">H1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="119_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon" title="c5">A</font> = <font color="Maroon" title="c4">H1</font> <a href="group_9.html#K17" title="GROUP_9:func.17">/\</a> <font color="Maroon" title="c3">N</font> as    <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Maroon" title="c4">H1</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/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="119_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">H</font> be  <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a>  <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c4">H1</font>;<br/><b>assume </b><a NAME="E1:119_1_1"/><i><font color="Green" title="E95">A1</font></i>: 
<font color="Maroon" title="c6">H</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="c5">A</font>,the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c5">A</font> #)
 ;<br/><b>reconsider </b><font color="Maroon" title="c7">N'</font> =  <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> #) as  <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/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="119_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">H1</font>;<br/><b>thus </b><a NAME="E1:119_1_1_1"/>
<font color="Maroon" title="c8">b</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c6">H</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">H</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c8">b</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="119_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:119_1_1_1_1"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">b</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c6">H</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c10">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">H1</font><b> such that </b><br/><a NAME="E3:119_1_1_1_1"/><i><font color="Green" title="E96">A2</font></i>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c10">a</font>
 <b>and </b><br/><a NAME="E4:119_1_1_1_1"/><i><font color="Green" title="E97">A3</font></i>: 
<font color="Maroon" title="c10">a</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c6">H</font>
 <b>by </b><i><a class="ref" href="group_2.html#T125" title="GROUP_2:th.125">GROUP_2:125</a></i>;<br/>
<a NAME="E5:119_1_1_1_1"/>
<font color="Maroon" title="c10">a</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="c5">A</font>
 
<b>by </b><i><a class="txt" href="group_9.html#E1:119_1_1"><i><font color="Green" title="E95">A1</font></i></a>, <a class="txt" href="group_9.html#E4:119_1_1_1_1"><i><font color="Green" title="E97">A3</font></i></a>, <a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/>
<a NAME="E6:119_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c10">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="group_9.html#K15" title="GROUP_9:func.15">carr</a> <font color="Maroon" title="c4">H1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="group_9.html#K15" title="GROUP_9:func.15">carr</a> <font color="Maroon" title="c3">N</font></span>)</span>
 
<b>by </b><i><a class="ref" href="group_9.html#D28" target="_self" title="GROUP_9:def.28">Def28</a></i>;<br/>
<a NAME="E7:119_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c10">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <font color="Maroon" title="c7">N'</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E8:119_1_1_1_1"/><b>then </b><i><font color="Green" title="E98">A4</font></i>: 
<font color="Maroon" title="c10">a</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c7">N'</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/>
<b>reconsider </b><font color="Maroon" title="c11">a'</font> = <font color="Maroon" title="c10">a</font>, <font color="Maroon" title="c12">b'</font> = <font color="Maroon" title="c8">b</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G</font> <b>by </b><i><a class="ref" href="group_9.html#T2" target="_self" title="GROUP_9:th.2">Th2</a></i>;<br/>
<a NAME="E10:119_1_1_1_1"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">b'</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c11">a'</font>
 
<b>by </b><i><a class="txt" href="group_9.html#E3:119_1_1_1_1"><i><font color="Green" title="E96">A2</font></i></a>, <a class="ref" href="group_9.html#T3" target="_self" title="GROUP_9:th.3">Th3</a></i>;<br/>
<a NAME="E11:119_1_1_1_1"/><b>then </b>
( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">b'</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c7">N'</font> &amp; <font color="Maroon" title="c12">b'</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c7">N'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">N'</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c12">b'</font> )
 
<b>by </b><i><a class="txt" href="group_9.html#E8:119_1_1_1_1"><i><font color="Green" title="E98">A4</font></i></a>, <a class="ref" href="group_2.html#T125" title="GROUP_2:th.125">GROUP_2:125</a>, <a class="ref" href="group_3.html#T141" title="GROUP_3:th.141">GROUP_3:141</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c13">b1</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="E13:119_1_1_1_1"/><i><font color="Green" title="E99">A5</font></i>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">b1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c12">b'</font>
 <b>and </b><br/><a NAME="E14:119_1_1_1_1"/><i><font color="Green" title="E100">A6</font></i>: 
<font color="Maroon" title="c13">b1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c7">N'</font>
 <b>by </b><i><a class="ref" href="group_2.html#T126" title="GROUP_2:th.126">GROUP_2:126</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c14">x'</font> = <font color="Maroon" title="c9">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">H1</font> <b>by </b><i><a class="txt" href="group_9.html#E3:119_1_1_1_1"><i><font color="Green" title="E96">A2</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c15">x''</font> = <font color="Maroon" title="c9">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G</font> <b>by </b><i><a class="txt" href="group_9.html#E13:119_1_1_1_1"><i><font color="Green" title="E99">A5</font></i></a></i>;<br/>
<a NAME="E17:119_1_1_1_1"/><i><font color="Green" title="E101">A7</font></i>: 
<font color="Maroon" title="c13">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">x''</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">b'</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="group_9.html#E13:119_1_1_1_1"><i><font color="Green" title="E99">A5</font></i></a>, <a class="ref" href="group_1.html#T22" title="GROUP_1:th.22">GROUP_1:22</a></i>;<br/>
<a NAME="E18:119_1_1_1_1"/><i><font color="Green" title="E102">A8</font></i>: 
<font color="Maroon" title="c13">b1</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="c3">N</font>
 
<b>by </b><i><a class="txt" href="group_9.html#E14:119_1_1_1_1"><i><font color="Green" title="E100">A6</font></i></a>, <a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/>
<a NAME="E19:119_1_1_1_1"/>
<font color="Maroon" title="c12">b'</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> 
 
<b>by </b><i><a class="ref" href="group_9.html#T6" target="_self" title="GROUP_9:th.6">Th6</a></i>;<br/>
<a NAME="E20:119_1_1_1_1"/><b>then </b><i><font color="Green" title="E103">A9</font></i>: 
<font color="Maroon" title="c13">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">x'</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">b</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="group_9.html#E17:119_1_1_1_1"><i><font color="Green" title="E101">A7</font></i></a>, <a class="ref" href="group_9.html#T3" target="_self" title="GROUP_9:th.3">Th3</a></i>;<br/>
<a NAME="E21:119_1_1_1_1"/><b>then </b><i><font color="Green" title="E104">A10</font></i>: 
<font color="Maroon" title="c13">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="group_9.html#K15" title="GROUP_9:func.15">carr</a> <font color="Maroon" title="c4">H1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="group_9.html#K15" title="GROUP_9:func.15">carr</a> <font color="Maroon" title="c3">N</font></span>)</span>
 
<b>by </b><i><a class="txt" href="group_9.html#E18:119_1_1_1_1"><i><font color="Green" title="E102">A8</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c16">b1'</font> = <font color="Maroon" title="c13">b1</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">H1</font> <b>by </b><i><a class="txt" href="group_9.html#E20:119_1_1_1_1"><i><font color="Green" title="E103">A9</font></i></a></i>;<br/>
<a NAME="E23:119_1_1_1_1"/>
<font color="Maroon" title="c16">b1'</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="c5">A</font>
 
<b>by </b><i><a class="txt" href="group_9.html#E21:119_1_1_1_1"><i><font color="Green" title="E104">A10</font></i></a>, <a class="ref" href="group_9.html#D28" target="_self" title="GROUP_9:def.28">Def28</a></i>;<br/>
<a NAME="E24:119_1_1_1_1"/><b>then </b><i><font color="Green" title="E105">A11</font></i>: 
<font color="Maroon" title="c16">b1'</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c6">H</font>
 
<b>by </b><i><a class="txt" href="group_9.html#E1:119_1_1"><i><font color="Green" title="E95">A1</font></i></a>, <a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/>
<a NAME="E25:119_1_1_1_1"/>
<font color="Maroon" title="c16">b1'</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c8">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x</font>
 
<b>by </b><i><a class="txt" href="group_9.html#E13:119_1_1_1_1"><i><font color="Green" title="E99">A5</font></i></a>, <a class="ref" href="group_9.html#T3" target="_self" title="GROUP_9:th.3">Th3</a></i>;<br/>
<b>hence </b><a NAME="E26:119_1_1_1_1"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">H</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c8">b</font>
 <b>by </b><i><a class="txt" href="group_9.html#E24:119_1_1_1_1"><i><font color="Green" title="E105">A11</font></i></a>, <a class="ref" href="group_2.html#T126" title="GROUP_2:th.126">GROUP_2:126</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:119_1_1"/>
<font color="Maroon" title="c6">H</font> is <a href="group_3.html#V1" title="GROUP_3:attr.1">normal</a>
 <b>by </b><i><a class="ref" href="group_3.html#T141" title="GROUP_3:th.141">GROUP_3:141</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:119_1"/>
<font color="Maroon" title="c4">H1</font> <a href="group_9.html#K17" title="GROUP_9:func.17">/\</a> <font color="Maroon" title="c3">N</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">H1</font>
 <b>by </b><i><a class="ref" href="group_9.html#D10" target="_self" title="GROUP_9:def.10">Def10</a></i>;<br/>


</div><b>end;</b></div>

<b>hence </b><a NAME="E2:119"/>
<font color="Maroon" title="c3">N</font> <a href="group_9.html#K17" title="GROUP_9:func.17">/\</a> <font color="Maroon" title="c4">H1</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">H1</font>
 ;<br/>


</div>
