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

<span class="kw">let </span><font color="Maroon" title="c1">O</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">G</font> being  <a href="group_9.html#NM2" title="GROUP_9:NM.2">GroupWithOperators</a> of <font color="Maroon" title="c1">O</font><br/> for <font color="Olive" title="b2">A</font>, <font color="Olive" title="b3">B</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Olive" title="b1">G</font>  st <font color="Olive" title="b2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b3">B</font> holds <br/> <a href="group_9.html#K18" title="GROUP_9:func.18">the_stable_subgroup_of</a> <font color="Olive" title="b2">A</font> is    <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of  <a href="group_9.html#K18" title="GROUP_9:func.18">the_stable_subgroup_of</a> <font color="Olive" title="b3">B</font></span><br/><span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">G</font>  st <font color="Olive" title="b1">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">B</font> holds <br/> <a href="group_9.html#K18" title="GROUP_9:func.18">the_stable_subgroup_of</a> <font color="Olive" title="b1">A</font> is    <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of  <a href="group_9.html#K18" title="GROUP_9:func.18">the_stable_subgroup_of</a> <font color="Olive" title="b2">B</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">A</font>, <font color="Maroon" title="c4">B</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">B</font> implies  <a href="group_9.html#K18" title="GROUP_9:func.18">the_stable_subgroup_of</a> <font color="Maroon" title="c3">A</font> is    <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of  <a href="group_9.html#K18" title="GROUP_9:func.18">the_stable_subgroup_of</a> <font color="Maroon" title="c4">B</font> )</span><br/>







<span class="kw">assume </span><a NAME="E1:96"/><span class="lab"><font color="Green" title="E74">A1</font></span>: 
<font color="Maroon" title="c3">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="group_9.html#K18" title="GROUP_9:func.18">the_stable_subgroup_of</a> <font color="Maroon" title="c3">A</font> is    <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of  <a href="group_9.html#K18" title="GROUP_9:func.18">the_stable_subgroup_of</a> <font color="Maroon" title="c4">B</font></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="96_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:96_1"/>
( <font color="Maroon" title="c3">A</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  or  not <font color="Maroon" title="c3">A</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="96_1_1"><span class="kw">suppose </span></a><a NAME="E1:96_1_1"/><span class="lab"><font color="Green" title="E75">A2</font></span>: 
<font color="Maroon" title="c3">A</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="group_9.html#K18" title="GROUP_9:func.18">the_stable_subgroup_of</a> <font color="Maroon" title="c3">A</font> is    <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of  <a href="group_9.html#K18" title="GROUP_9:func.18">the_stable_subgroup_of</a> <font color="Maroon" title="c4">B</font></span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c5">H1</font> =  <a href="group_9.html#K4" title="GROUP_9:func.4">(1).</a> <font color="Maroon" title="c2">G</font>, <font color="Maroon" title="c6">H2</font> =  <a href="group_9.html#K4" title="GROUP_9:func.4">(1).</a> <span class="p1">(<span class="default"><a href="group_9.html#K18" title="GROUP_9:func.18">the_stable_subgroup_of</a> <font color="Maroon" title="c4">B</font></span>)</span> 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> <span class="kw">by</span> <span class="lab"><a class="ref" href="group_9.html#T11" target="_self" title="GROUP_9:th.11">Th11</a></span>;<br/>the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">H1</font> = 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<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></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_9.html#D8" target="_self" title="GROUP_9:def.8">Def8</a></span>
<br/>.= 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <span class="p3">(<span class="default"><a href="group_9.html#K18" title="GROUP_9:func.18">the_stable_subgroup_of</a> <font color="Maroon" title="c4">B</font></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_9.html#T4" target="_self" title="GROUP_9:th.4">Th4</a></span>
<br/>.= 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">H2</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_9.html#D8" target="_self" title="GROUP_9:def.8">Def8</a></span>
;<br/><a NAME="E4:96_1_1"/><span class="kw">then </span>
 <a href="group_9.html#K4" title="GROUP_9:func.4">(1).</a> <font color="Maroon" title="c2">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_9.html#K4" title="GROUP_9:func.4">(1).</a> <span class="p1">(<span class="default"><a href="group_9.html#K18" title="GROUP_9:func.18">the_stable_subgroup_of</a> <font color="Maroon" title="c4">B</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="group_9.html#E12"><span class="lab"><font color="Green" title="E12">Lm5</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E5:96_1_1"/>
 <a href="group_9.html#K18" title="GROUP_9:func.18">the_stable_subgroup_of</a> <font color="Maroon" title="c3">A</font> is    <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of  <a href="group_9.html#K18" title="GROUP_9:func.18">the_stable_subgroup_of</a> <font color="Maroon" title="c4">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:96_1_1"><span class="lab"><font color="Green" title="E75">A2</font></span></a>, <a class="txt" href="group_9.html#E79"><span class="lab"><font color="Green" title="E65">Lm25</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="96_1_2"><span class="kw">suppose </span></a><a NAME="E1:96_1_2"/><span class="lab"><font color="Green" title="E75">A3</font></span>: 
not <font color="Maroon" title="c3">A</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="group_9.html#K18" title="GROUP_9:func.18">the_stable_subgroup_of</a> <font color="Maroon" title="c3">A</font> is    <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of  <a href="group_9.html#K18" title="GROUP_9:func.18">the_stable_subgroup_of</a> <font color="Maroon" title="c4">B</font></span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">set </span><font color="Maroon" title="c5">D</font> =  <a href="group_9.html#K1" title="GROUP_9:func.1">the_stable_subset_generated_by</a> <font color="Maroon" title="c4">B</font>,the <a href="group_9.html#U1" title="GROUP_9:sel.1">action</a> of <font color="Maroon" title="c2">G</font>;<br/><span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">a</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> <font color="Maroon" title="c3">A</font> implies <font color="Maroon" title="c6">a</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> <font color="Maroon" title="c4">B</font> )</span><br/><span class="kw">assume </span><a NAME="E1:96_1_2_1"/>
<font color="Maroon" title="c6">a</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> <font color="Maroon" title="c3">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">a</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> <font color="Maroon" title="c4">B</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">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="c2">G</font>, <font color="Maroon" title="c8">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="c9">C</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">G</font><span class="kw"> such that </span><br/><a NAME="E3:96_1_2_1"/><span class="lab"><font color="Green" title="E76">A4</font></span>: 
<font color="Maroon" title="c9">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> <font color="Maroon" title="c3">A</font>,the <a href="group_9.html#U1" title="GROUP_9:sel.1">action</a> of <font color="Maroon" title="c2">G</font>
 <span class="kw">and </span><br/><a NAME="E4:96_1_2_1"/><span class="lab"><font color="Green" title="E77">A5</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">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="c8">I</font>
 <span class="kw">and </span><br/><a NAME="E5:96_1_2_1"/><span class="lab"><font color="Green" title="E78">A6</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c7">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">C</font>
 <span class="kw">and </span><br/><a NAME="E6:96_1_2_1"/><span class="lab"><font color="Green" title="E79">A7</font></span>: 
 <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Maroon" title="c8">I</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_9.html#T24" target="_self" title="GROUP_9:th.24">Th24</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_1_2_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c10">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">C</font> implies <font color="Maroon" title="c10">y</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> <font color="Maroon" title="c4">B</font>,the <a href="group_9.html#U1" title="GROUP_9:sel.1">action</a> of <font color="Maroon" title="c2">G</font> )</span><br/><span class="kw">assume </span><a NAME="E1:96_1_2_1_1"/><span class="lab"><font color="Green" title="E80">A8</font></span>: 
<font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c10">y</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> <font color="Maroon" title="c4">B</font>,the <a href="group_9.html#U1" title="GROUP_9:sel.1">action</a> of <font color="Maroon" title="c2">G</font></span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c11">b</font> = <font color="Maroon" title="c10">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G</font> ;<br/><span class="kw">consider </span><font color="Maroon" title="c12">F1</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="c13">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">A</font><span class="kw"> such that </span><br/><a NAME="E4:96_1_2_1_1"/><span class="lab"><font color="Green" title="E81">A9</font></span>: 
<span class="p1">(<span class="default"><a href="group_9.html#K2" title="GROUP_9:func.2">Product</a> <font color="Maroon" title="c12">F1</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="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:96_1_2"><span class="lab"><font color="Green" title="E75">A3</font></span></a>, <a class="txt" href="#E3:96_1_2_1"><span class="lab"><font color="Green" title="E76">A4</font></span></a>, <a class="txt" href="#E1:96_1_2_1_1"><span class="lab"><font color="Green" title="E80">A8</font></span></a>, <a class="txt" href="group_9.html#E85"><span class="lab"><font color="Green" title="E71">Lm31</font></span></a></span>;<br/><a NAME="E5:96_1_2_1_1"/>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:96_1_2"><span class="lab"><font color="Green" title="E75">A3</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E6:96_1_2_1_1"/>
<font color="Maroon" title="c10">y</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> <font color="Maroon" title="c4">B</font>,the <a href="group_9.html#U1" title="GROUP_9:sel.1">action</a> of <font color="Maroon" title="c2">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:96"><span class="lab"><font color="Green" title="E74">A1</font></span></a>, <a class="txt" href="#E4:96_1_2_1_1"><span class="lab"><font color="Green" title="E81">A9</font></span></a>, <a class="txt" href="group_9.html#E85"><span class="lab"><font color="Green" title="E71">Lm31</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E8:96_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">C</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> <font color="Maroon" title="c4">B</font>,the <a href="group_9.html#U1" title="GROUP_9:sel.1">action</a> of <font color="Maroon" title="c2">G</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/><a NAME="E9:96_1_2_1"/><span class="kw">then </span>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c7">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> <font color="Maroon" title="c4">B</font>,the <a href="group_9.html#U1" title="GROUP_9:sel.1">action</a> of <font color="Maroon" title="c2">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:96_1_2_1"><span class="lab"><font color="Green" title="E78">A6</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/><span class="kw">hence </span><a NAME="E10:96_1_2_1"/>
<font color="Maroon" title="c6">a</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> <font color="Maroon" title="c4">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:96_1_2_1"><span class="lab"><font color="Green" title="E77">A5</font></span></a>, <a class="txt" href="#E6:96_1_2_1"><span class="lab"><font color="Green" title="E79">A7</font></span></a>, <a class="ref" href="group_9.html#T24" target="_self" title="GROUP_9:th.24">Th24</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:96_1_2"/>
 <a href="group_9.html#K18" title="GROUP_9:func.18">the_stable_subgroup_of</a> <font color="Maroon" title="c3">A</font> is    <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of  <a href="group_9.html#K18" title="GROUP_9:func.18">the_stable_subgroup_of</a> <font color="Maroon" title="c4">B</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_9.html#T13" target="_self" title="GROUP_9:th.13">Th13</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

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