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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> ;<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> be  <a href="group_9.html#V4" title="GROUP_9:attr.4">normal</a>  <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Maroon" title="c3">H1</font>;<br/>







<b>assume </b><a NAME="E1:140"/>
<font color="Maroon" title="c3">H1</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c4">N1</font> is <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c5">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:140"/><i><font color="Green" title="E115">A1</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">H1</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c4">N1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">e</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="group_6.html#D2" title="GROUP_6:def.2">GROUP_6:def 2</a></i>;<br/>
<b>set </b><font color="Maroon" title="c6">H</font> = <font color="Maroon" title="c3">H1</font>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">N</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="c4">N1</font>,the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c4">N1</font> #) as  <a href="group_3.html#V1" title="GROUP_3:attr.1">normal</a>  <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="txt" href="group_9.html#E20"><i><font color="Green" title="E19">Lm7</font></i></a></i>;<br/>
<a NAME="E5:140"/>
 <a href="group_9.html#K7" title="GROUP_9:func.7">Cosets</a> <font color="Maroon" title="c4">N1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_6.html#NK4" title="GROUP_6:NK.4">Cosets</a> <font color="Maroon" title="c7">N</font>
 
<b>by </b><i><a class="ref" href="group_9.html#D14" target="_self" title="GROUP_9:def.14">Def14</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c8">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E7:140"/><i><font color="Green" title="E116">A2</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">H1</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <font color="Maroon" title="c7">N</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c8">e</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="group_9.html#E3:140"><i><font color="Green" title="E115">A1</font></i></a></i>;<br/>
<a NAME="E8:140"/><i><font color="Green" title="E117">A3</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="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c8">e</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="txt" href="group_9.html#E7:140"><i><font color="Green" title="E116">A2</font></i></a>, <a class="ref" href="group_2.html#T167" title="GROUP_2:th.167">GROUP_2:167</a></i>;<br/>
<div><i><font color="Green" title="E118">A4</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="140_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:140_1"/>
not 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="c7">N</font>
 ;<br/><a NAME="E2:140_1"/><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="subset_1.html#K6" title="SUBSET_1:func.6">\</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">N</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="ref" href="xboole_1.html#T37" title="XBOOLE_1:th.37">XBOOLE_1:37</a></i>;<br/><b>then consider </b><font color="Maroon" title="c9">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:140_1"/><i><font color="Green" title="E119">A5</font></i>: 
<font color="Maroon" title="c9">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 <font color="Maroon" title="c3">H1</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">N</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c10">x</font> = <font color="Maroon" title="c9">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#E4:140_1"><i><font color="Green" title="E119">A5</font></i></a></i>;<br/><div><i><font color="Green" title="E120">A6</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="140_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:140_1_1"/>
<font color="Maroon" title="c10">x</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c7">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">e</font>
 ;<br/><a NAME="E2:140_1_1"/><b>then </b>
<font color="Maroon" title="c10">x</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c7">N</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="c3">H1</font>
 <b>by </b><i><a class="txt" href="group_9.html#E8:140"><i><font color="Green" title="E117">A3</font></i></a>, <a class="ref" href="zfmisc_1.html#T31" title="ZFMISC_1:th.31">ZFMISC_1:31</a></i>;<br/><b>then consider </b><font color="Maroon" title="c11">x'</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">H1</font><b> such that </b><br/><a NAME="E4:140_1_1"/><i><font color="Green" title="E121">A7</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#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c11">x'</font> &amp; <font color="Maroon" title="c11">x'</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c7">N</font> )
 <b>by </b><i><a class="ref" href="group_2.html#T125" title="GROUP_2:th.125">GROUP_2:125</a></i>;<br/><a NAME="E5:140_1_1"/>
<font color="Maroon" title="c11">x'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">x</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> 
 <b>by </b><i><a class="txt" href="group_9.html#E4:140_1_1"><i><font color="Green" title="E121">A7</font></i></a>, <a class="ref" href="group_1.html#T20" title="GROUP_1:th.20">GROUP_1:20</a></i>;<br/><a NAME="E6:140_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c10">x</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a>  <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c7">N</font>
 <b>by </b><i><a class="txt" href="group_9.html#E4:140_1_1"><i><font color="Green" title="E121">A7</font></i></a>, <a class="ref" href="group_2.html#T60" title="GROUP_2:th.60">GROUP_2:60</a></i>;<br/><a NAME="E7:140_1_1"/><b>then </b>
<font color="Maroon" title="c10">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c7">N</font>
 <b>by </b><i><a class="ref" href="group_1.html#T19" title="GROUP_1:th.19">GROUP_1:19</a></i>;<br/><a NAME="E8:140_1_1"/><b>then </b>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <font color="Maroon" title="c7">N</font>
 <b>by </b><i><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/><b>hence </b><a NAME="E9:140_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="group_9.html#E4:140_1"><i><font color="Green" title="E119">A5</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/></div><b>end;</b></div><a NAME="E7:140_1"/>
<font color="Maroon" title="c10">x</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c7">N</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_6.html#NK4" title="GROUP_6:NK.4">Cosets</a> <font color="Maroon" title="c7">N</font>
 <b>by </b><i><a class="ref" href="group_6.html#T16" title="GROUP_6:th.16">GROUP_6:16</a></i>;<br/><b>hence </b><a NAME="E8:140_1"/>
contradiction
 <b>by </b><i><a class="txt" href="group_9.html#E7:140"><i><font color="Green" title="E116">A2</font></i></a>, <a class="txt" href="group_9.html#E6:140_1"><i><font color="Green" title="E120">A6</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div>
<b>reconsider </b><font color="Maroon" title="c9">N'</font> = <font color="Maroon" title="c4">N1</font> 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/>
<a NAME="E11:140"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">N</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="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/>
<a NAME="E12:140"/><b>then </b>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c9">N'</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="c3">H1</font>
 
<b>by </b><i><a class="txt" href="group_9.html#E9:140"><i><font color="Green" title="E118">A4</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="E13:140"/>
 <a href="group_9.html#G1" title="GROUP_9:aggr.1">HGrWOpStr</a>(# 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="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c3">H1</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>  <a href="group_9.html#G1" title="GROUP_9:aggr.1">HGrWOpStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">N1</font>,the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c4">N1</font>,the <a href="group_9.html#U1" title="GROUP_9:sel.1">action</a> of <font color="Maroon" title="c4">N1</font> #)
 <b>by </b><i><a class="ref" href="group_9.html#T75" target="_self" title="GROUP_9:th.75">Th75</a></i>;<br/>


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