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<b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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/><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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/><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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/>







<b>assume </b><a NAME="E1:96"/><i><font color="Green" title="E74">A1</font></i>: 
<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:()"><i><font color="Red">::  thesis: </font></i></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"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></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"><b>suppose </b></a><a NAME="E1:96_1_1"/><i><font color="Green" title="E75">A2</font></i>: 
<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:()"><i><font color="Red">::  thesis: </font></i></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"><b>reconsider </b><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> <b>by </b><i><a class="ref" href="group_9.html#T11" target="_self" title="GROUP_9:th.11">Th11</a></i>;<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>
<b>by </b><i><a class="ref" href="group_9.html#D8" target="_self" title="GROUP_9:def.8">Def8</a></i>
<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>
<b>by </b><i><a class="ref" href="group_9.html#T4" target="_self" title="GROUP_9:th.4">Th4</a></i>
<br/>.= 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">H2</font>
<b>by </b><i><a class="ref" href="group_9.html#D8" target="_self" title="GROUP_9:def.8">Def8</a></i>
;<br/><a NAME="E4:96_1_1"/><b>then </b>
 <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>
 <b>by </b><i><a class="txt" href="group_9.html#E12"><i><font color="Green" title="E12">Lm5</font></i></a></i>;<br/><b>hence </b><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>
 <b>by </b><i><a class="txt" href="#E1:96_1_1"><i><font color="Green" title="E75">A2</font></i></a>, <a class="txt" href="group_9.html#E79"><i><font color="Green" title="E65">Lm25</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="96_1_2"><b>suppose </b></a><a NAME="E1:96_1_2"/><i><font color="Green" title="E75">A3</font></i>: 
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:()"><i><font color="Red">::  thesis: </font></i></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"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">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="c5">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/><b>assume </b><a NAME="E1:96_1_2_1"/>
<font color="Maroon" title="c5">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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">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/><b>then consider </b><font color="Maroon" title="c6">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="c7">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="c8">C</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="E3:96_1_2_1"/><i><font color="Green" title="E76">A4</font></i>: 
( <font color="Maroon" title="c8">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> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">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="c7">I</font> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">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="c6">F</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Maroon" title="c7">I</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</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>set </b><font color="Maroon" title="c9">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/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_1_2_1_1"><b>now </b></a><div class="add"><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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="c8">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/><b>assume </b><a NAME="E1:96_1_2_1_1"/><i><font color="Green" title="E77">A5</font></i>: 
<font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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/><b>then reconsider </b><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/><b>consider </b><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><b> such that </b><br/><a NAME="E4:96_1_2_1_1"/><i><font color="Green" title="E78">A6</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="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>
 <b>by </b><i><a class="txt" href="#E1:96_1_2"><i><font color="Green" title="E75">A3</font></i></a>, <a class="txt" href="#E3:96_1_2_1"><i><font color="Green" title="E76">A4</font></i></a>, <a class="txt" href="#E1:96_1_2_1_1"><i><font color="Green" title="E77">A5</font></i></a>, <a class="txt" href="group_9.html#E85"><i><font color="Green" title="E71">Lm31</font></i></a></i>;<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>
 <b>by </b><i><a class="txt" href="#E1:96_1_2"><i><font color="Green" title="E75">A3</font></i></a></i>;<br/><b>hence </b><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>
 <b>by </b><i><a class="txt" href="#E1:96"><i><font color="Green" title="E74">A1</font></i></a>, <a class="txt" href="#E4:96_1_2_1_1"><i><font color="Green" title="E78">A6</font></i></a>, <a class="txt" href="group_9.html#E85"><i><font color="Green" title="E71">Lm31</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><a NAME="E5:96_1_2_1"/><b>then </b>
<font color="Maroon" title="c8">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>
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/><a NAME="E6:96_1_2_1"/><b>then </b>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">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>
 <b>by </b><i><a class="txt" href="#E3:96_1_2_1"><i><font color="Green" title="E76">A4</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><b>hence </b><a NAME="E7:96_1_2_1"/>
<font color="Maroon" title="c5">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>
 <b>by </b><i><a class="txt" href="#E3:96_1_2_1"><i><font color="Green" title="E76">A4</font></i></a>, <a class="ref" href="group_9.html#T24" target="_self" title="GROUP_9:th.24">Th24</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>hence </b><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>
 <b>by </b><i><a class="ref" href="group_9.html#T13" target="_self" title="GROUP_9:th.13">Th13</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

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