<?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> be    <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">N1</font>, <font color="Maroon" title="c5">N2</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="c3">H1</font>;<br/><b>let </b><font color="Maroon" title="c6">N1'</font>, <font color="Maroon" title="c7">N2'</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>assume </b><a NAME="E1:150"/><i><font color="Green" title="E125">A1</font></i>: 
( <font color="Maroon" title="c4">N1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">N1'</font> &amp; <font color="Maroon" title="c5">N2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">N2'</font> )
 ;<br/>

<a NAME="E2:150"/><i><font color="Green" title="E126">A2</font></i>: 
<font color="Maroon" title="c6">N1'</font> <a href="group_9.html#K19" title="GROUP_9:func.19">"\/"</a> <font color="Maroon" title="c7">N2'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_9.html#K18" title="GROUP_9:func.18">the_stable_subgroup_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">N1'</font> <a href="group_9.html#K16" title="GROUP_9:func.16">*</a> <font color="Maroon" title="c7">N2'</font></span>)</span>
 
<b>by </b><i><a class="ref" href="group_9.html#T29" target="_self" title="GROUP_9:th.29">Th29</a></i>;<br/>
<a NAME="E3:150"/><i><font color="Green" title="E127">A3</font></i>: 
<font color="Maroon" title="c4">N1</font> <a href="group_9.html#K19" title="GROUP_9:func.19">"\/"</a> <font color="Maroon" title="c5">N2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_9.html#K18" title="GROUP_9:func.18">the_stable_subgroup_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">N1</font> <a href="group_9.html#K16" title="GROUP_9:func.16">*</a> <font color="Maroon" title="c5">N2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="group_9.html#T29" target="_self" title="GROUP_9:th.29">Th29</a></i>;<br/>
<b>set </b><font color="Maroon" title="c8">S1</font> =  <a href="group_9.html#K18" title="GROUP_9:func.18">the_stable_subgroup_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">N1'</font> <a href="group_9.html#K16" title="GROUP_9:func.16">*</a> <font color="Maroon" title="c7">N2'</font></span>)</span>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">S2</font> =  <a href="group_9.html#K18" title="GROUP_9:func.18">the_stable_subgroup_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">N1</font> <a href="group_9.html#K16" title="GROUP_9:func.16">*</a> <font color="Maroon" title="c5">N2</font></span>)</span> as    <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Maroon" title="c2">G</font> <b>by </b><i><a class="ref" href="group_9.html#T11" target="_self" title="GROUP_9:th.11">Th11</a></i>;<br/>
<b>set </b><font color="Maroon" title="c10">X1</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; <font color="Maroon" title="c6">N1'</font> <a href="group_9.html#K16" title="GROUP_9:func.16">*</a> <font color="Maroon" title="c7">N2'</font> <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>set </b><font color="Maroon" title="c11">X2</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="c3">H1</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="c3">H1</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; <font color="Maroon" title="c4">N1</font> <a href="group_9.html#K16" title="GROUP_9:func.16">*</a> <font color="Maroon" title="c5">N2</font> <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><i><font color="Green" title="E128">A4</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="150_1"><b>now </b></a><div class="add"><b>set </b><font color="Maroon" title="c12">x'</font> =  <a href="group_9.html#K15" title="GROUP_9:func.15">carr</a> <font color="Maroon" title="c3">H1</font>;<br/><b>reconsider </b><font color="Maroon" title="c13">x</font> =  <a href="group_9.html#K15" title="GROUP_9:func.15">carr</a> <font color="Maroon" title="c3">H1</font> as    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/><b>take </b><font color="Maroon" title="c14">x</font> = <font color="Maroon" title="c13">x</font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="150_1_1"><b>now </b></a><div class="add"><b>set </b><font color="Maroon" title="c15">H</font> =  <a href="group_9.html#K5" title="GROUP_9:func.5">(Omega).</a> <font color="Maroon" title="c3">H1</font>;<br/><b>take </b><font color="Maroon" title="c16">H</font> =  <a href="group_9.html#K5" title="GROUP_9:func.5">(Omega).</a> <font color="Maroon" title="c3">H1</font>;<br/><b>thus </b><a NAME="E1:150_1_1"/>
 <a href="group_9.html#K15" title="GROUP_9:func.15">carr</a> <font color="Maroon" title="c3">H1</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="c16">H</font>
 ;<br/><b>thus </b><a NAME="E2:150_1_1"/>
<font color="Maroon" title="c4">N1</font> <a href="group_9.html#K16" title="GROUP_9:func.16">*</a> <font color="Maroon" title="c5">N2</font> <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="c16">H</font>
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:150_1"/>
<font color="Maroon" title="c14">x</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="c3">H1</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="c3">H1</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; <font color="Maroon" title="c4">N1</font> <a href="group_9.html#K16" title="GROUP_9:func.16">*</a> <font color="Maroon" title="c5">N2</font> <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><b>end;</b></div>
<a NAME="E6:150"/><i><font color="Green" title="E129">A5</font></i>: 
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"><font color="Maroon" title="c6">N1'</font> <a href="group_9.html#K16" title="GROUP_9:func.16">*</a> <font color="Maroon" title="c7">N2'</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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; <font color="Maroon" title="c6">N1'</font> <a href="group_9.html#K16" title="GROUP_9:func.16">*</a> <font color="Maroon" title="c7">N2'</font> <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="ref" href="group_9.html#T27" target="_self" title="GROUP_9:th.27">Th27</a></i>;<br/>
<a NAME="E7:150"/><i><font color="Green" title="E130">A6</font></i>: 
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"><font color="Maroon" title="c4">N1</font> <a href="group_9.html#K16" title="GROUP_9:func.16">*</a> <font color="Maroon" title="c5">N2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c3">H1</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="c3">H1</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; <font color="Maroon" title="c4">N1</font> <a href="group_9.html#K16" title="GROUP_9:func.16">*</a> <font color="Maroon" title="c5">N2</font> <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="ref" href="group_9.html#T27" target="_self" title="GROUP_9:th.27">Th27</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="150_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c12">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:150_2"/><i><font color="Green" title="E131">A7</font></i>: 
<font color="Maroon" title="c12">x</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"><font color="Maroon" title="c4">N1</font> <a href="group_9.html#K16" title="GROUP_9:func.16">*</a> <font color="Maroon" title="c5">N2</font></span>)</span></span>)</span>
 ;<br/><a NAME="E2:150_2"/>
 <a href="group_9.html#K18" title="GROUP_9:func.18">the_stable_subgroup_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">N1</font> <a href="group_9.html#K16" title="GROUP_9:func.16">*</a> <font color="Maroon" title="c5">N2</font></span>)</span> is    <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c3">H1</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="E3:150_2"/><b>then </b>
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"><font color="Maroon" title="c4">N1</font> <a href="group_9.html#K16" title="GROUP_9:func.16">*</a> <font color="Maroon" title="c5">N2</font></span>)</span></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="c3">H1</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>then reconsider </b><font color="Maroon" title="c13">g</font> = <font color="Maroon" title="c12">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">H1</font> <b>by </b><i><a class="txt" href="group_9.html#E1:150_2"><i><font color="Green" title="E131">A7</font></i></a></i>;<br/><a NAME="E5:150_2"/>
<font color="Maroon" title="c13">g</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="group_9.html#K18" title="GROUP_9:func.18">the_stable_subgroup_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">N1</font> <a href="group_9.html#K16" title="GROUP_9:func.16">*</a> <font color="Maroon" title="c5">N2</font></span>)</span>
 <b>by </b><i><a class="txt" href="group_9.html#E1:150_2"><i><font color="Green" title="E131">A7</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="c14">F</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">H1</font>, <font color="Maroon" title="c15">I</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> , <font color="Maroon" title="c16">C</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c3">H1</font><b> such that </b><br/><a NAME="E7:150_2"/><i><font color="Green" title="E132">A8</font></i>: 
( <font color="Maroon" title="c16">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_9.html#K1" title="GROUP_9:func.1">the_stable_subset_generated_by</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">N1</font> <a href="group_9.html#K16" title="GROUP_9:func.16">*</a> <font color="Maroon" title="c5">N2</font></span>)</span>,the <a href="group_9.html#U1" title="GROUP_9:sel.1">action</a> of <font color="Maroon" title="c3">H1</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">I</font> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c14">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c16">C</font> &amp;  <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">F</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Maroon" title="c15">I</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">g</font> )
 <b>by </b><i><a class="ref" href="group_9.html#T24" target="_self" title="GROUP_9:th.24">Th24</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c17">g</font> = <font color="Maroon" title="c13">g</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/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="150_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c18">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:150_2_1"/><i><font color="Green" title="E133">A9</font></i>: 
<font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_9.html#K1" title="GROUP_9:func.1">the_stable_subset_generated_by</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">N1</font> <a href="group_9.html#K16" title="GROUP_9:func.16">*</a> <font color="Maroon" title="c5">N2</font></span>)</span>,the <a href="group_9.html#U1" title="GROUP_9:sel.1">action</a> of <font color="Maroon" title="c3">H1</font>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c19">a</font> = <font color="Maroon" title="c18">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">H1</font> ;<br/><a NAME="E3:150_2_1"/>
( <font color="Maroon" title="c4">N1</font> is    <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c3">H1</font> &amp; <font color="Maroon" title="c5">N2</font> is    <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c3">H1</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="E4:150_2_1"/><b>then </b>
(  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c3">H1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c4">N1</font> &amp;  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c3">H1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c5">N2</font> )
 <b>by </b><i><a class="ref" href="group_2.html#T55" title="GROUP_2:th.55">GROUP_2:55</a></i>;<br/><a NAME="E5:150_2_1"/><b>then </b>
(  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c3">H1</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">N1</font> &amp;  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c3">H1</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="c5">N2</font> &amp;  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c3">H1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c3">H1</font></span>)</span> <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="c3">H1</font></span>)</span> )
 <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/><a NAME="E6:150_2_1"/><b>then </b><i><font color="Green" title="E134">A10</font></i>: 
 <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c3">H1</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">N1</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="c5">N2</font></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon" title="c20">F</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">O</font>, <font color="Maroon" title="c21">h</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">N1</font> <a href="group_9.html#K16" title="GROUP_9:func.16">*</a> <font color="Maroon" title="c5">N2</font><b> such that </b><br/><a NAME="E8:150_2_1"/><i><font color="Green" title="E135">A11</font></i>: 
<span class="p1">(<span class="default"><a href="group_9.html#K2" title="GROUP_9:func.2">Product</a> <font color="Maroon" title="c20">F</font>,the <a href="group_9.html#U1" title="GROUP_9:sel.1">action</a> of <font color="Maroon" title="c3">H1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">a</font>
 <b>by </b><i><a class="txt" href="group_9.html#E1:150_2_1"><i><font color="Green" title="E133">A9</font></i></a>, <a class="txt" href="group_9.html#E85"><i><font color="Green" title="E74">Lm31</font></i></a></i>;<br/><div><i><font color="Green" title="E136">A12</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="150_2_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c22">o</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">O</font>;<br/><b>let </b><font color="Maroon" title="c23">f1</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">H1</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">H1</font>;<br/><b>let </b><font color="Maroon" title="c24">f2</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</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 <font color="Maroon" title="c2">G</font>;<br/><b>assume </b><a NAME="E1:150_2_1_1"/><i><font color="Green" title="E137">A13</font></i>: 
( <font color="Maroon" title="c23">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="group_9.html#U1" title="GROUP_9:sel.1">action</a> of <font color="Maroon" title="c3">H1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">o</font> &amp; <font color="Maroon" title="c24">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="group_9.html#U1" title="GROUP_9:sel.1">action</a> of <font color="Maroon" title="c2">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">o</font> )
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="150_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:150_2_1_1_1"/>
( <font color="Maroon" title="c22">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">O</font> or  not <font color="Maroon" title="c22">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">O</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="150_2_1_1_1_1"><b>suppose </b></a><a NAME="E1:150_2_1_1_1_1"/>
<font color="Maroon" title="c22">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">O</font>
 ;<br/><div class="add"><a NAME="E2:150_2_1_1_1_1"/><b>then </b>
( <font color="Maroon" title="c3">H1</font> <a href="group_9.html#K3" title="GROUP_9:func.3">^</a> <font color="Maroon" title="c22">o</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c23">f1</font> &amp; <font color="Maroon" title="c2">G</font> <a href="group_9.html#K3" title="GROUP_9:func.3">^</a> <font color="Maroon" title="c22">o</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c24">f2</font> )
 <b>by </b><i><a class="txt" href="group_9.html#E1:150_2_1_1"><i><font color="Green" title="E137">A13</font></i></a>, <a class="ref" href="group_9.html#D6" target="_self" title="GROUP_9:def.6">Def6</a></i>;<br/><b>hence </b><a NAME="E3:150_2_1_1_1_1"/>
<font color="Maroon" title="c23">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">f2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">H1</font>
 <b>by </b><i><a class="ref" href="group_9.html#D7" target="_self" title="GROUP_9:def.7">Def7</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="150_2_1_1_1_2"><b>suppose </b></a><a NAME="E1:150_2_1_1_1_2"/>
not <font color="Maroon" title="c22">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">O</font>
 ;<br/><div class="add"><a NAME="E2:150_2_1_1_1_2"/><b>then </b>
( not <font color="Maroon" title="c22">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> the <a href="group_9.html#U1" title="GROUP_9:sel.1">action</a> of <font color="Maroon" title="c3">H1</font> &amp; not <font color="Maroon" title="c22">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> the <a href="group_9.html#U1" title="GROUP_9:sel.1">action</a> of <font color="Maroon" title="c2">G</font> )
 ;<br/><b>hence </b><a NAME="E3:150_2_1_1_1_2"/>
<font color="Maroon" title="c23">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">f2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">H1</font>
 <b>by </b><i><a class="txt" href="group_9.html#E1:150_2_1_1"><i><font color="Green" title="E137">A13</font></i></a>, <a class="ref" href="funct_1.html#D4" title="FUNCT_1:def.4">FUNCT_1:def 4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E10:150_2_1"/>
<font color="Maroon" title="c3">H1</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="E11:150_2_1"/><b>then </b><i><font color="Green" title="E137">A14</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">H1</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 <font color="Maroon" title="c2">G</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="E12:150_2_1"/><b>then </b><i><font color="Green" title="E138">A15</font></i>: 
 <a href="group_9.html#K2" title="GROUP_9:func.2">Product</a> <font color="Maroon" title="c20">F</font>,the <a href="group_9.html#U1" title="GROUP_9:sel.1">action</a> of <font color="Maroon" title="c3">H1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="group_9.html#K2" title="GROUP_9:func.2">Product</a> <font color="Maroon" title="c20">F</font>,the <a href="group_9.html#U1" title="GROUP_9:sel.1">action</a> of <font color="Maroon" title="c2">G</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">H1</font>
 <b>by </b><i><a class="txt" href="group_9.html#E9:150_2_1"><i><font color="Green" title="E136">A12</font></i></a>, <a class="ref" href="group_9.html#T84" target="_self" title="GROUP_9:th.84">Th84</a></i>;<br/><a NAME="E13:150_2_1"/><i><font color="Green" title="E139">A16</font></i>: 
<font color="Maroon" title="c21">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">N1</font> <a href="group_9.html#K16" title="GROUP_9:func.16">*</a> <font color="Maroon" title="c5">N2</font>
 <b>by </b><i><a class="txt" href="group_9.html#E6:150_2_1"><i><font color="Green" title="E134">A10</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c22">a</font> = <font color="Maroon" title="c19">a</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#E11:150_2_1"><i><font color="Green" title="E137">A14</font></i></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c23">h</font> = <font color="Maroon" title="c21">h</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">N1'</font> <a href="group_9.html#K16" title="GROUP_9:func.16">*</a> <font color="Maroon" title="c7">N2'</font> <b>by </b><i><a class="txt" href="group_9.html#E1:150"><i><font color="Green" title="E125">A1</font></i></a>, <a class="ref" href="group_9.html#T85" target="_self" title="GROUP_9:th.85">Th85</a></i>;<br/><a NAME="E16:150_2_1"/>
( <span class="p1">(<span class="default"><a href="group_9.html#K2" title="GROUP_9:func.2">Product</a> <font color="Maroon" title="c20">F</font>,the <a href="group_9.html#U1" title="GROUP_9:sel.1">action</a> of <font color="Maroon" title="c2">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">a</font> &amp; not <font color="Maroon" title="c6">N1'</font> <a href="group_9.html#K16" title="GROUP_9:func.16">*</a> <font color="Maroon" title="c7">N2'</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )
 <b>by </b><i><a class="txt" href="group_9.html#E1:150"><i><font color="Green" title="E125">A1</font></i></a>, <a class="txt" href="group_9.html#E8:150_2_1"><i><font color="Green" title="E135">A11</font></i></a>, <a class="txt" href="group_9.html#E12:150_2_1"><i><font color="Green" title="E138">A15</font></i></a>, <a class="txt" href="group_9.html#E13:150_2_1"><i><font color="Green" title="E139">A16</font></i></a>, <a class="ref" href="group_9.html#T85" target="_self" title="GROUP_9:th.85">Th85</a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>;<br/><b>hence </b><a NAME="E17:150_2_1"/>
<font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_9.html#K1" title="GROUP_9:func.1">the_stable_subset_generated_by</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">N1'</font> <a href="group_9.html#K16" title="GROUP_9:func.16">*</a> <font color="Maroon" title="c7">N2'</font></span>)</span>,the <a href="group_9.html#U1" title="GROUP_9:sel.1">action</a> of <font color="Maroon" title="c2">G</font>
 <b>by </b><i><a class="txt" href="group_9.html#E85"><i><font color="Green" title="E74">Lm31</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E10:150_2"/><b>then </b>
 <a href="group_9.html#K1" title="GROUP_9:func.1">the_stable_subset_generated_by</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">N1</font> <a href="group_9.html#K16" title="GROUP_9:func.16">*</a> <font color="Maroon" title="c5">N2</font></span>)</span>,the <a href="group_9.html#U1" title="GROUP_9:sel.1">action</a> of <font color="Maroon" title="c3">H1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="group_9.html#K1" title="GROUP_9:func.1">the_stable_subset_generated_by</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">N1'</font> <a href="group_9.html#K16" title="GROUP_9:func.16">*</a> <font color="Maroon" title="c7">N2'</font></span>)</span>,the <a href="group_9.html#U1" title="GROUP_9:sel.1">action</a> of <font color="Maroon" title="c2">G</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/><a NAME="E11:150_2"/><b>then </b><i><font color="Green" title="E133">A17</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c14">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="group_9.html#K1" title="GROUP_9:func.1">the_stable_subset_generated_by</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">N1'</font> <a href="group_9.html#K16" title="GROUP_9:func.16">*</a> <font color="Maroon" title="c7">N2'</font></span>)</span>,the <a href="group_9.html#U1" title="GROUP_9:sel.1">action</a> of <font color="Maroon" title="c2">G</font>
 <b>by </b><i><a class="txt" href="group_9.html#E7:150_2"><i><font color="Green" title="E132">A8</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><a NAME="E12:150_2"/>
<font color="Maroon" title="c3">H1</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="E13:150_2"/><b>then </b>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">H1</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 <font color="Maroon" title="c2">G</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="E14:150_2"/><b>then </b>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c14">F</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 <font color="Maroon" title="c2">G</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c18">F</font> = <font color="Maroon" title="c14">F</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G</font> <b>by </b><i><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></i>;<br/><a NAME="E16:150_2"/>
 <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">F</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Maroon" title="c15">I</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">g</font>
 <b>by </b><i><a class="txt" href="group_9.html#E7:150_2"><i><font color="Green" title="E132">A8</font></i></a>, <a class="ref" href="group_9.html#T83" target="_self" title="GROUP_9:th.83">Th83</a></i>;<br/><a NAME="E17:150_2"/><b>then </b><i><font color="Green" title="E134">A18</font></i>: 
<font color="Maroon" title="c17">g</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="group_9.html#K18" title="GROUP_9:func.18">the_stable_subgroup_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">N1'</font> <a href="group_9.html#K16" title="GROUP_9:func.16">*</a> <font color="Maroon" title="c7">N2'</font></span>)</span>
 <b>by </b><i><a class="txt" href="group_9.html#E7:150_2"><i><font color="Green" title="E132">A8</font></i></a>, <a class="txt" href="group_9.html#E11:150_2"><i><font color="Green" title="E133">A17</font></i></a>, <a class="ref" href="group_9.html#T24" target="_self" title="GROUP_9:th.24">Th24</a></i>;<br/><b>assume </b><a NAME="E18:150_2"/>
not <font color="Maroon" title="c12">x</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"><font color="Maroon" title="c6">N1'</font> <a href="group_9.html#K16" title="GROUP_9:func.16">*</a> <font color="Maroon" title="c7">N2'</font></span>)</span></span>)</span>
 ;<br/><b>hence </b><a NAME="E19:150_2"/>
contradiction
 <b>by </b><i><a class="txt" href="group_9.html#E17:150_2"><i><font color="Green" title="E134">A18</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="E9:150"/><b>then </b><i><font color="Green" title="E131">A19</font></i>: 
 <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="c3">H1</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="c3">H1</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; <font color="Maroon" title="c4">N1</font> <a href="group_9.html#K16" title="GROUP_9:func.16">*</a> <font color="Maroon" title="c5">N2</font> <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>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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; <font color="Maroon" title="c6">N1'</font> <a href="group_9.html#K16" title="GROUP_9:func.16">*</a> <font color="Maroon" title="c7">N2'</font> <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#E6:150"><i><font color="Green" title="E129">A5</font></i></a>, <a class="txt" href="group_9.html#E7:150"><i><font color="Green" title="E130">A6</font></i></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="150_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c12">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:150_3"/>
<font color="Maroon" title="c12">x</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="c3">H1</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="c3">H1</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; <font color="Maroon" title="c4">N1</font> <a href="group_9.html#K16" title="GROUP_9:func.16">*</a> <font color="Maroon" title="c5">N2</font> <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="c13">B</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c3">H1</font><b> such that </b><br/><a NAME="E3:150_3"/><i><font color="Green" title="E132">A20</font></i>: 
( <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">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="c3">H1</font> st <br/>( <font color="Maroon" title="c13">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; <font color="Maroon" title="c4">N1</font> <a href="group_9.html#K16" title="GROUP_9:func.16">*</a> <font color="Maroon" title="c5">N2</font> <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/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="150_3_1"><b>now </b></a><div class="add"><b>consider </b><font color="Maroon" title="c14">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="c3">H1</font><b> such that </b><br/><a NAME="E2:150_3_1"/><i><font color="Green" title="E133">A21</font></i>: 
( <font color="Maroon" title="c13">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="c14">H</font> &amp; <font color="Maroon" title="c4">N1</font> <a href="group_9.html#K16" title="GROUP_9:func.16">*</a> <font color="Maroon" title="c5">N2</font> <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="c14">H</font> )
 <b>by </b><i><a class="txt" href="group_9.html#E3:150_3"><i><font color="Green" title="E132">A20</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c15">H</font> = <font color="Maroon" title="c14">H</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="c2">G</font> <b>by </b><i><a class="ref" href="group_9.html#T11" target="_self" title="GROUP_9:th.11">Th11</a></i>;<br/><b>take </b><font color="Maroon" title="c16">H</font> = <font color="Maroon" title="c15">H</font>;<br/><b>thus </b><a NAME="E4:150_3_1"/>
( <font color="Maroon" title="c13">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="c16">H</font> &amp; <font color="Maroon" title="c6">N1'</font> <a href="group_9.html#K16" title="GROUP_9:func.16">*</a> <font color="Maroon" title="c7">N2'</font> <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="c16">H</font> )
 <b>by </b><i><a class="txt" href="group_9.html#E1:150"><i><font color="Green" title="E125">A1</font></i></a>, <a class="txt" href="group_9.html#E2:150_3_1"><i><font color="Green" title="E133">A21</font></i></a>, <a class="ref" href="group_9.html#T85" target="_self" title="GROUP_9:th.85">Th85</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:150_3"/>
<font color="Maroon" title="c12">x</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; <font color="Maroon" title="c6">N1'</font> <a href="group_9.html#K16" title="GROUP_9:func.16">*</a> <font color="Maroon" title="c7">N2'</font> <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#E3:150_3"><i><font color="Green" title="E132">A20</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E11:150"/><b>then </b>
<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="c3">H1</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="c3">H1</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; <font color="Maroon" title="c4">N1</font> <a href="group_9.html#K16" title="GROUP_9:func.16">*</a> <font color="Maroon" title="c5">N2</font> <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>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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; <font color="Maroon" title="c6">N1'</font> <a href="group_9.html#K16" title="GROUP_9:func.16">*</a> <font color="Maroon" title="c7">N2'</font> <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="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<a NAME="E12:150"/><b>then </b>
 <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; <font color="Maroon" title="c6">N1'</font> <a href="group_9.html#K16" title="GROUP_9:func.16">*</a> <font color="Maroon" title="c7">N2'</font> <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>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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="c3">H1</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="c3">H1</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; <font color="Maroon" title="c4">N1</font> <a href="group_9.html#K16" title="GROUP_9:func.16">*</a> <font color="Maroon" title="c5">N2</font> <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#E5:150"><i><font color="Green" title="E128">A4</font></i></a>, <a class="ref" href="setfam_1.html#T7" title="SETFAM_1:th.7">SETFAM_1:7</a></i>;<br/>
<a NAME="E13:150"/><b>then </b>
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"><font color="Maroon" title="c6">N1'</font> <a href="group_9.html#K16" title="GROUP_9:func.16">*</a> <font color="Maroon" title="c7">N2'</font></span>)</span></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 <font color="Maroon" title="c9">S2</font>
 
<b>by </b><i><a class="txt" href="group_9.html#E6:150"><i><font color="Green" title="E129">A5</font></i></a>, <a class="txt" href="group_9.html#E7:150"><i><font color="Green" title="E130">A6</font></i></a>, <a class="txt" href="group_9.html#E9:150"><i><font color="Green" title="E131">A19</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>
<b>hence </b><a NAME="E14:150"/>
<font color="Maroon" title="c6">N1'</font> <a href="group_9.html#K19" title="GROUP_9:func.19">"\/"</a> <font color="Maroon" title="c7">N2'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">N1</font> <a href="group_9.html#K19" title="GROUP_9:func.19">"\/"</a> <font color="Maroon" title="c5">N2</font>
 <b>by </b><i><a class="txt" href="group_9.html#E2:150"><i><font color="Green" title="E126">A2</font></i></a>, <a class="txt" href="group_9.html#E3:150"><i><font color="Green" title="E127">A3</font></i></a>, <a class="txt" href="group_9.html#E12"><i><font color="Green" title="E12">Lm5</font></i></a></i>;<br/>


</div>
