<?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">H1</font>, <font color="Maroon" title="c4">H2</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>assume </b><a NAME="E1:106"/>
<font color="Maroon" title="c3">H1</font> <a href="group_9.html#K16" title="GROUP_9:func.16">*</a> <font color="Maroon" title="c4">H2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">H2</font> <a href="group_9.html#K16" title="GROUP_9:func.16">*</a> <font color="Maroon" title="c3">H1</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c5">H</font> being  <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><b> such that </b><br/><a NAME="E3:106"/><i><font color="Green" title="E82">A1</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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">H1</font></span>)</span> <a href="group_2.html#K2" title="GROUP_2:func.2">*</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">H2</font></span>)</span>
 <b>by </b><i><a class="ref" href="group_9.html#T17" target="_self" title="GROUP_9:th.17">Th17</a></i>;<br/>
<a NAME="E4:106"/>
<span class="p1">(<span class="default"><a href="group_9.html#K15" title="GROUP_9:func.15">carr</a> <font color="Maroon" title="c3">H1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</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">H2</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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">H1</font></span>)</span> <a href="group_2.html#K2" title="GROUP_2:func.2">*</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">H2</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="106_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">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="E1:106_1"/><i><font color="Green" title="E83">A2</font></i>: 
<font color="Maroon" title="c6">x</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="c3">H1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</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">H2</font></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c7">a</font> = <font color="Maroon" title="c6">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G</font> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="106_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="106_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:106_1_1_1"/>
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_9.html#K15" title="GROUP_9:func.15">carr</a> <font color="Maroon" title="c3">H1</font> or <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_9.html#K15" title="GROUP_9:func.15">carr</a> <font color="Maroon" title="c4">H2</font> )
 </span><b>by </b><i><a class="txt" href="group_9.html#E1:106_1"><i><font color="Green" title="E83">A2</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="106_1_1_1_1"><b>suppose </b></a><a NAME="E1:106_1_1_1_1"/><i><font color="Green" title="E84">A3</font></i>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_9.html#K15" title="GROUP_9:func.15">carr</a> <font color="Maroon" title="c3">H1</font>
 ;<br/><div class="add"><a NAME="E2:106_1_1_1_1"/>
 <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">G</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c4">H2</font>
 <b>by </b><i><a class="txt" href="group_9.html#E45"><i><font color="Green" title="E41">Lm18</font></i></a></i>;<br/><a NAME="E3:106_1_1_1_1"/><b>then </b>
(  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">G</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_9.html#K15" title="GROUP_9:func.15">carr</a> <font color="Maroon" title="c4">H2</font> &amp; <font color="Maroon" title="c7">a</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> <font color="Maroon" title="c2">G</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">a</font> )
 <b>by </b><i><a class="ref" href="group_1.html#D5" title="GROUP_1:def.5">GROUP_1:def 5</a>, <a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/><b>hence </b><a NAME="E4:106_1_1_1_1"/>
<font color="Maroon" title="c6">x</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="c3">H1</font></span>)</span> <a href="group_2.html#K2" title="GROUP_2:func.2">*</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">H2</font></span>)</span>
 <b>by </b><i><a class="txt" href="group_9.html#E1:106_1_1_1_1"><i><font color="Green" title="E84">A3</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="106_1_1_1_2"><b>suppose </b></a><a NAME="E1:106_1_1_1_2"/><i><font color="Green" title="E84">A4</font></i>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_9.html#K15" title="GROUP_9:func.15">carr</a> <font color="Maroon" title="c4">H2</font>
 ;<br/><div class="add"><a NAME="E2:106_1_1_1_2"/>
 <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">G</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">H1</font>
 <b>by </b><i><a class="txt" href="group_9.html#E45"><i><font color="Green" title="E41">Lm18</font></i></a></i>;<br/><a NAME="E3:106_1_1_1_2"/><b>then </b>
(  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">G</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_9.html#K15" title="GROUP_9:func.15">carr</a> <font color="Maroon" title="c3">H1</font> &amp; <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c7">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">a</font> )
 <b>by </b><i><a class="ref" href="group_1.html#D5" title="GROUP_1:def.5">GROUP_1:def 5</a>, <a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/><b>hence </b><a NAME="E4:106_1_1_1_2"/>
<font color="Maroon" title="c6">x</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="c3">H1</font></span>)</span> <a href="group_2.html#K2" title="GROUP_2:func.2">*</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">H2</font></span>)</span>
 <b>by </b><i><a class="txt" href="group_9.html#E1:106_1_1_1_2"><i><font color="Green" title="E84">A4</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:106_1"/>
<font color="Maroon" title="c6">x</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="c3">H1</font></span>)</span> <a href="group_2.html#K2" title="GROUP_2:func.2">*</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">H2</font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E5:106"/><b>then </b><i><font color="Green" title="E83">A5</font></i>: 
<font color="Maroon" title="c3">H1</font> <a href="group_9.html#K19" title="GROUP_9:func.19">"\/"</a> <font color="Maroon" title="c4">H2</font> is    <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Maroon" title="c5">H</font>
 
<b>by </b><i><a class="txt" href="group_9.html#E3:106"><i><font color="Green" title="E82">A1</font></i></a>, <a class="ref" href="group_9.html#D29" target="_self" title="GROUP_9:def.29">Def29</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="106_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G</font>;<br/><b>assume </b><a NAME="E1:106_2"/>
<font color="Maroon" title="c6">a</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c5">H</font>
 ;<br/><a NAME="E2:106_2"/><b>then </b>
<font color="Maroon" title="c6">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="c3">H1</font></span>)</span> <a href="group_2.html#K2" title="GROUP_2:func.2">*</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">H2</font></span>)</span>
 <b>by </b><i><a class="txt" href="group_9.html#E3:106"><i><font color="Green" title="E82">A1</font></i></a>, <a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/><b>then consider </b><font color="Maroon" title="c7">b</font>, <font color="Maroon" title="c8">c</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="E4:106_2"/><i><font color="Green" title="E84">A6</font></i>: 
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">b</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c8">c</font>
 <b>and </b><br/><a NAME="E5:106_2"/><i><font color="Green" title="E85">A7</font></i>: 
( <font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_9.html#K15" title="GROUP_9:func.15">carr</a> <font color="Maroon" title="c3">H1</font> &amp; <font color="Maroon" title="c8">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_9.html#K15" title="GROUP_9:func.15">carr</a> <font color="Maroon" title="c4">H2</font> )
 ;<br/><b>set </b><font color="Maroon" title="c9">A</font> = <span class="p1">(<span class="default"><a href="group_9.html#K15" title="GROUP_9:func.15">carr</a> <font color="Maroon" title="c3">H1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</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">H2</font></span>)</span>;<br/><b>set </b><font color="Maroon" title="c10">X</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">B</font> where <font color="Olive" title="b1">B</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">G</font> :  ex <font color="Olive" title="b2">H</font> being <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> st <br/>( <font color="Olive" title="b1">B</font> <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 <font color="Olive" title="b2">H</font> &amp; <span class="p1">(<span class="default"><a href="group_9.html#K15" title="GROUP_9:func.15">carr</a> <font color="Maroon" title="c3">H1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</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">H2</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="group_9.html#K15" title="GROUP_9:func.15">carr</a> <font color="Olive" title="b2">H</font> ) </span>}</span> ;<br/><a NAME="E6:106_2"/><i><font color="Green" title="E86">A8</font></i>: 
 <a href="group_9.html#K15" title="GROUP_9:func.15">carr</a> <span class="p1">(<span class="default"><a href="group_9.html#K5" title="GROUP_9:func.5">(Omega).</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">B</font> where <font color="Olive" title="b1">B</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">G</font> :  ex <font color="Olive" title="b2">H</font> being <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> st <br/>( <font color="Olive" title="b1">B</font> <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 <font color="Olive" title="b2">H</font> &amp; <span class="p1">(<span class="default"><a href="group_9.html#K15" title="GROUP_9:func.15">carr</a> <font color="Maroon" title="c3">H1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</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">H2</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="group_9.html#K15" title="GROUP_9:func.15">carr</a> <font color="Olive" title="b2">H</font> ) </span>}</span> 
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="106_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c11">Y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:106_2_1"/>
<font color="Maroon" title="c11">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">B</font> where <font color="Olive" title="b1">B</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">G</font> :  ex <font color="Olive" title="b2">H</font> being <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> st <br/>( <font color="Olive" title="b1">B</font> <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 <font color="Olive" title="b2">H</font> &amp; <span class="p1">(<span class="default"><a href="group_9.html#K15" title="GROUP_9:func.15">carr</a> <font color="Maroon" title="c3">H1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</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">H2</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="group_9.html#K15" title="GROUP_9:func.15">carr</a> <font color="Olive" title="b2">H</font> ) </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon" title="c12">B</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">G</font><b> such that </b><br/><a NAME="E3:106_2_1"/><i><font color="Green" title="E87">A9</font></i>: 
( <font color="Maroon" title="c11">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">B</font> &amp;  ex <font color="Olive" title="b1">H</font> being <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> st <br/>( <font color="Maroon" title="c12">B</font> <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 <font color="Olive" title="b1">H</font> &amp; <span class="p1">(<span class="default"><a href="group_9.html#K15" title="GROUP_9:func.15">carr</a> <font color="Maroon" title="c3">H1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</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">H2</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="group_9.html#K15" title="GROUP_9:func.15">carr</a> <font color="Olive" title="b1">H</font> ) )
 ;<br/><b>consider </b><font color="Maroon" title="c13">H'</font> being  <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><b> such that </b><br/><a NAME="E5:106_2_1"/><i><font color="Green" title="E88">A10</font></i>: 
( <font color="Maroon" title="c12">B</font> <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 <font color="Maroon" title="c13">H'</font> &amp; <span class="p1">(<span class="default"><a href="group_9.html#K15" title="GROUP_9:func.15">carr</a> <font color="Maroon" title="c3">H1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</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">H2</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="group_9.html#K15" title="GROUP_9:func.15">carr</a> <font color="Maroon" title="c13">H'</font> )
 <b>by </b><i><a class="txt" href="group_9.html#E3:106_2_1"><i><font color="Green" title="E87">A9</font></i></a></i>;<br/><a NAME="E6:106_2_1"/><i><font color="Green" title="E89">A11</font></i>: 
<font color="Maroon" title="c13">H'</font> is    <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="ref" href="group_9.html#D7" target="_self" title="GROUP_9:def.7">Def7</a></i>;<br/><a NAME="E7:106_2_1"/>
( <font color="Maroon" title="c7">b</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="c3">H1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</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">H2</font></span>)</span> &amp; <font color="Maroon" title="c8">c</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="c3">H1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</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">H2</font></span>)</span> )
 <b>by </b><i><a class="txt" href="group_9.html#E5:106_2"><i><font color="Green" title="E85">A7</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E8:106_2_1"/><b>then </b>
( <font color="Maroon" title="c7">b</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c13">H'</font> &amp; <font color="Maroon" title="c8">c</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c13">H'</font> )
 <b>by </b><i><a class="txt" href="group_9.html#E5:106_2_1"><i><font color="Green" title="E88">A10</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="E9:106_2_1"/><b>then </b>
<font color="Maroon" title="c7">b</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c8">c</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c13">H'</font>
 <b>by </b><i><a class="txt" href="group_9.html#E6:106_2_1"><i><font color="Green" title="E89">A11</font></i></a>, <a class="ref" href="group_2.html#T59" title="GROUP_2:th.59">GROUP_2:59</a></i>;<br/><b>hence </b><a NAME="E10:106_2_1"/>
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">Y</font>
 <b>by </b><i><a class="txt" href="group_9.html#E4:106_2"><i><font color="Green" title="E84">A6</font></i></a>, <a class="txt" href="group_9.html#E3:106_2_1"><i><font color="Green" title="E87">A9</font></i></a>, <a class="txt" href="group_9.html#E5:106_2_1"><i><font color="Green" title="E88">A10</font></i></a>, <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="E8:106_2"/><b>then </b>
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">B</font> where <font color="Olive" title="b1">B</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">G</font> :  ex <font color="Olive" title="b2">H</font> being <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> st <br/>( <font color="Olive" title="b1">B</font> <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 <font color="Olive" title="b2">H</font> &amp; <span class="p1">(<span class="default"><a href="group_9.html#K15" title="GROUP_9:func.15">carr</a> <font color="Maroon" title="c3">H1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</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">H2</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="group_9.html#K15" title="GROUP_9:func.15">carr</a> <font color="Olive" title="b2">H</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="group_9.html#E6:106_2"><i><font color="Green" title="E86">A8</font></i></a>, <a class="ref" href="setfam_1.html#D1" title="SETFAM_1:def.1">SETFAM_1:def 1</a></i>;<br/><a NAME="E9:106_2"/><b>then </b>
<font color="Maroon" title="c6">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 <span class="p1">(<span class="default"><a href="group_9.html#K18" title="GROUP_9:func.18">the_stable_subgroup_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_9.html#K15" title="GROUP_9:func.15">carr</a> <font color="Maroon" title="c3">H1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3">(<span class="default"><a href="group_9.html#K15" title="GROUP_9:func.15">carr</a> <font color="Maroon" title="c4">H2</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="group_9.html#T27" target="_self" title="GROUP_9:th.27">Th27</a></i>;<br/><b>hence </b><a NAME="E10:106_2"/>
<font color="Maroon" title="c6">a</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">H1</font> <a href="group_9.html#K19" title="GROUP_9:func.19">"\/"</a> <font color="Maroon" title="c4">H2</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="E7:106"/><b>then </b>
<font color="Maroon" title="c5">H</font> is    <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Maroon" title="c3">H1</font> <a href="group_9.html#K19" title="GROUP_9:func.19">"\/"</a> <font color="Maroon" title="c4">H2</font>
 
<b>by </b><i><a class="ref" href="group_9.html#T13" target="_self" title="GROUP_9:th.13">Th13</a></i>;<br/>
<a NAME="E8:106"/><b>then </b>
( <font color="Maroon" title="c5">H</font> is    <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c3">H1</font> <a href="group_9.html#K19" title="GROUP_9:func.19">"\/"</a> <font color="Maroon" title="c4">H2</font> &amp; <font color="Maroon" title="c3">H1</font> <a href="group_9.html#K19" title="GROUP_9:func.19">"\/"</a> <font color="Maroon" title="c4">H2</font> is    <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c5">H</font> )
 
<b>by </b><i><a class="txt" href="group_9.html#E5:106"><i><font color="Green" title="E83">A5</font></i></a>, <a class="ref" href="group_9.html#D7" target="_self" title="GROUP_9:def.7">Def7</a></i>;<br/>
<a NAME="E9:106"/><b>then </b>
( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">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="c3">H1</font> <a href="group_9.html#K19" title="GROUP_9:func.19">"\/"</a> <font color="Maroon" title="c4">H2</font></span>)</span> &amp; 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="c3">H1</font> <a href="group_9.html#K19" title="GROUP_9:func.19">"\/"</a> <font color="Maroon" title="c4">H2</font></span>)</span> <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 <font color="Maroon" title="c5">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/>
<b>hence </b><a NAME="E10:106"/>
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="c3">H1</font> <a href="group_9.html#K19" title="GROUP_9:func.19">"\/"</a> <font color="Maroon" title="c4">H2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">H1</font> <a href="group_9.html#K16" title="GROUP_9:func.16">*</a> <font color="Maroon" title="c4">H2</font>
 <b>by </b><i><a class="txt" href="group_9.html#E3:106"><i><font color="Green" title="E82">A1</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>


</div>
