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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">B</font>, <font color="Olive" title="b3">A</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">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 <span class="p1">(<span class="default"><a href="group_4.html#K5" title="GROUP_4:func.5">gr</a> <font color="Olive" title="b3">A</font></span>)</span> holds <br/> <a href="group_9.html#K18" title="GROUP_9:func.18">the_stable_subgroup_of</a> <font color="Olive" title="b3">A</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> <font color="Olive" title="b2">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">B</font>, <font color="Olive" title="b2">A</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">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 <span class="p1">(<span class="default"><a href="group_4.html#K5" title="GROUP_4:func.5">gr</a> <font color="Olive" title="b2">A</font></span>)</span> 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> <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> <font color="Olive" title="b1">B</font></span><br/><b>let </b><font color="Maroon" title="c3">B</font>, <font color="Maroon" title="c4">A</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">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 <span class="p1">(<span class="default"><a href="group_4.html#K5" title="GROUP_4:func.5">gr</a> <font color="Maroon" title="c4">A</font></span>)</span> implies  <a href="group_9.html#K18" title="GROUP_9:func.18">the_stable_subgroup_of</a> <font color="Maroon" title="c4">A</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> <font color="Maroon" title="c3">B</font> )</span><br/>







<b>assume </b><a NAME="E1:103"/><i><font color="Green" title="E79">A1</font></i>: 
<font color="Maroon" title="c3">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 <span class="p1">(<span class="default"><a href="group_4.html#K5" title="GROUP_4:func.5">gr</a> <font color="Maroon" title="c4">A</font></span>)</span>
 ; <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="c4">A</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> <font color="Maroon" title="c3">B</font></span><br/>

<a NAME="E2:103"/><b>then </b><i><font color="Green" title="E80">A2</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_4.html#K5" title="GROUP_4:func.5">gr</a> <font color="Maroon" title="c4">A</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <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="c3">B</font></span>)</span>
 
<b>by </b><i><a class="ref" href="group_9.html#D29" target="_self" title="GROUP_9:def.29">Def29</a></i>;<br/>
<a NAME="E3:103"/>
<font color="Maroon" title="c4">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="group_4.html#K5" title="GROUP_4:func.5">gr</a> <font color="Maroon" title="c4">A</font></span>)</span>
 
<b>by </b><i><a class="ref" href="group_4.html#D5" title="GROUP_4:def.5">GROUP_4:def 5</a></i>;<br/>
<a NAME="E4:103"/><b>then </b><i><font color="Green" title="E81">A3</font></i>: 
<font color="Maroon" title="c4">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="group_9.html#K18" title="GROUP_9:func.18">the_stable_subgroup_of</a> <font color="Maroon" title="c3">B</font></span>)</span>
 
<b>by </b><i><a class="txt" href="group_9.html#E2:103"><i><font color="Green" title="E80">A2</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="103_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">H</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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">A</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="c5">H</font> implies  <a href="group_9.html#K18" title="GROUP_9:func.18">the_stable_subgroup_of</a> <font color="Maroon" title="c3">B</font> is    <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Maroon" title="c5">H</font> )</span><br/><b>assume </b><a NAME="E1:103_1"/><i><font color="Green" title="E82">A4</font></i>: 
<font color="Maroon" title="c4">A</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="c5">H</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">B</font> is    <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Maroon" title="c5">H</font></span><br/><b>reconsider </b><font color="Maroon" title="c6">H'</font> =  <a href="algstr_0.html#G3" title="ALGSTR_0:aggr.3">multMagma</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">H</font>,the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c5">H</font> #) as  <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a>  <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c2">G</font> <b>by </b><i><a class="txt" href="group_9.html#E40"><i><font color="Green" title="E38">Lm16</font></i></a></i>;<br/><a NAME="E3:103_1"/>
 <a href="group_4.html#K5" title="GROUP_4:func.5">gr</a> <font color="Maroon" title="c4">A</font> is    <a href="group_6.html#M1" title="GROUP_6:mode.1">Subgroup</a> of <font color="Maroon" title="c6">H'</font>
 <b>by </b><i><a class="txt" href="group_9.html#E1:103_1"><i><font color="Green" title="E82">A4</font></i></a>, <a class="ref" href="group_4.html#D5" title="GROUP_4:def.5">GROUP_4:def 5</a></i>;<br/><a NAME="E4:103_1"/><b>then </b>
<font color="Maroon" title="c3">B</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="c6">H'</font>
 <b>by </b><i><a class="txt" href="group_9.html#E1:103"><i><font color="Green" title="E79">A1</font></i></a>, <a class="ref" href="group_2.html#D5" title="GROUP_2:def.5">GROUP_2:def 5</a></i>;<br/><b>hence </b><a NAME="E5:103_1"/>
 <a href="group_9.html#K18" title="GROUP_9:func.18">the_stable_subgroup_of</a> <font color="Maroon" title="c3">B</font> is    <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Maroon" title="c5">H</font>
 <b>by </b><i><a class="ref" href="group_9.html#D29" target="_self" title="GROUP_9:def.29">Def29</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="E6:103"/>
 <a href="group_9.html#K18" title="GROUP_9:func.18">the_stable_subgroup_of</a> <font color="Maroon" title="c4">A</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> <font color="Maroon" title="c3">B</font>
 <b>by </b><i><a class="txt" href="group_9.html#E4:103"><i><font color="Green" title="E81">A3</font></i></a>, <a class="ref" href="group_9.html#D29" target="_self" title="GROUP_9:def.29">Def29</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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