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<div class="add">

<b>set </b><font color="Maroon" title="c2">Gp2</font> =  <a href="matrix_2.html#K13" title="MATRIX_2:func.13">Group_of_Perm</a> 2;<br/>
<b>consider </b><font color="Maroon" title="c3">G</font> being  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="group_9.html#L1" title="GROUP_9:struct.1">HGrWOpStr</a> of <font color="Maroon" title="c1">O</font><b> such that </b><br/><a NAME="E2:33_1_1"/><i><font color="Green" title="E21">A1</font></i>: 
( <font color="Maroon" title="c3">G</font> is <a href="group_9.html#V2" title="GROUP_9:attr.2">strict</a> &amp; <font color="Maroon" title="c3">G</font> is <a href="group_9.html#V3" title="GROUP_9:attr.3">distributive</a> &amp; <font color="Maroon" title="c3">G</font> is <a href="group_1.html#V3" title="GROUP_1:attr.3">Group-like</a> &amp; <font color="Maroon" title="c3">G</font> is <a href="group_1.html#V4" title="GROUP_1:attr.4">associative</a> &amp;  <a href="matrix_2.html#K13" title="MATRIX_2:func.13">Group_of_Perm</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#G1" title="GROUP_1:aggr.1">HGrStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">G</font>,the <a href="group_1.html#U1" title="GROUP_1:sel.1">mult</a> of <font color="Maroon" title="c3">G</font> #) )
 <b>by </b><i><a class="txt" href="group_9.html#E8"><i><font color="Green" title="E8">Lm3</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c4">G</font> = <font color="Maroon" title="c3">G</font> as  <a href="group_9.html#V2" title="GROUP_9:attr.2">strict</a> <a href="group_9.html#NM2" title="GROUP_9:NM.2">GroupWithOperators</a> of <font color="Maroon" title="c1">O</font> <b>by </b><i><a class="txt" href="group_9.html#E2:33_1_1"><i><font color="Green" title="E21">A1</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c4">G</font>
;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:33_1_1_1"/><i><font color="Green" title="E22">A2</font></i>: 
not <font color="Maroon" title="c4">G</font> is <a href="group_9.html#V6" title="GROUP_9:attr.6">simple</a>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_1_1_1_1"/>
( <font color="Maroon" title="c4">G</font> is <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a> or  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#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="c4">G</font> st <br/>( <font color="Olive" title="b1">H</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="group_9.html#K5" title="GROUP_9:func.5">(Omega).</a> <font color="Maroon" title="c4">G</font> &amp; <font color="Olive" title="b1">H</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="group_9.html#K4" title="GROUP_9:func.4">(1).</a> <font color="Maroon" title="c4">G</font> ) )
 </span><b>by </b><i><a class="txt" href="group_9.html#E1:33_1_1_1"><i><font color="Green" title="E22">A2</font></i></a>, <a class="ref" href="group_9.html#D13" target="_self" title="GROUP_9:def.13">Def13</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:33_1_1_1_1_1"/>
<font color="Maroon" title="c4">G</font> is <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a>
 ;<br/><div class="add"><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="E3:33_1_1_1_1_1"/><i><font color="Green" title="E23">A3</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">G</font> <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/><a NAME="E4:33_1_1_1_1_1"/>
 <a href="matrix_2.html#K13" title="MATRIX_2:func.13">Group_of_Perm</a> 2 is <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a>
 <b>by </b><i><a class="txt" href="group_9.html#E2:33_1_1"><i><font color="Green" title="E21">A1</font></i></a>, <a class="txt" href="group_9.html#E3:33_1_1_1_1_1"><i><font color="Green" title="E23">A3</font></i></a>, <a class="ref" href="group_6.html#D2" title="GROUP_6:def.2">GROUP_6:def 2</a></i>;<br/><b>hence </b><a NAME="E5:33_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="group_9.html#D12" target="_self" title="GROUP_9:def.12">Def12</a>, <a class="txt" href="group_9.html#E18"><i><font color="Green" title="E17">Lm6</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:33_1_1_1_1_2"/>
 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#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="c4">G</font> st <br/>( <font color="Olive" title="b1">H</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="group_9.html#K5" title="GROUP_9:func.5">(Omega).</a> <font color="Maroon" title="c4">G</font> &amp; <font color="Olive" title="b1">H</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="group_9.html#K4" title="GROUP_9:func.4">(1).</a> <font color="Maroon" title="c4">G</font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c5">H</font> being  <a href="group_9.html#V2" title="GROUP_9:attr.2">strict</a> <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="c4">G</font><b> such that </b><br/><a NAME="E3:33_1_1_1_1_2"/><i><font color="Green" title="E23">A4</font></i>: 
( <font color="Maroon" title="c5">H</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="group_9.html#K5" title="GROUP_9:func.5">(Omega).</a> <font color="Maroon" title="c4">G</font> &amp; <font color="Maroon" title="c5">H</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="group_9.html#K4" title="GROUP_9:func.4">(1).</a> <font color="Maroon" title="c4">G</font> )
 ;<br/><b>reconsider </b><font color="Maroon" title="c6">H'</font> =  <a href="group_1.html#G1" title="GROUP_1:aggr.1">HGrStr</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="group_1.html#U1" title="GROUP_1:sel.1">mult</a> of <font color="Maroon" title="c5">H</font> #) as  <a href="group_1.html#V1" title="GROUP_1:attr.1">strict</a> <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="c4">G</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/><b>reconsider </b><font color="Maroon" title="c7">G'</font> = <font color="Maroon" title="c4">G</font> as   <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a> ;<br/><b>reconsider </b><font color="Maroon" title="c8">H'</font> = <font color="Maroon" title="c6">H'</font> as  <a href="group_1.html#V1" title="GROUP_1:attr.1">strict</a> <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="c7">G'</font> ;<br/><b>set </b><font color="Maroon" title="c9">H''</font> = <font color="Maroon" title="c8">H'</font>;<br/><a NAME="E7:33_1_1_1_1_2"/>
( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">H'</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">G'</font> &amp; the <a href="group_1.html#U1" title="GROUP_1:sel.1">mult</a> of <font color="Maroon" title="c8">H'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="group_1.html#U1" title="GROUP_1:sel.1">mult</a> of <font color="Maroon" title="c7">G'</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">H'</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="c10">H''</font> = <font color="Maroon" title="c8">H'</font> as  <a href="group_1.html#V1" title="GROUP_1:attr.1">strict</a>  <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of  <a href="matrix_2.html#K13" title="MATRIX_2:func.13">Group_of_Perm</a> 2 <b>by </b><i><a class="txt" href="group_9.html#E2:33_1_1"><i><font color="Green" title="E21">A1</font></i></a>, <a class="ref" href="group_2.html#D5" title="GROUP_2:def.5">GROUP_2:def 5</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_1_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c11">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="matrix_2.html#K13" title="MATRIX_2:func.13">Group_of_Perm</a> 2</span>)</span>;<br/><b>reconsider </b><font color="Maroon" title="c12">A'</font> = <font color="Maroon" title="c11">A</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c7">G'</font> <b>by </b><i><a class="txt" href="group_9.html#E2:33_1_1"><i><font color="Green" title="E21">A1</font></i></a></i>;<br/><font color="Maroon" title="c11">A</font> <a href="group_2.html#K10" title="GROUP_2:func.10">*</a> <font color="Maroon" title="c10">H''</font> = 
<font color="Maroon" title="c12">A'</font> <a href="group_2.html#K10" title="GROUP_2:func.10">*</a> <font color="Maroon" title="c8">H'</font>
<b>by </b><i><a class="txt" href="group_9.html#E2:33_1_1"><i><font color="Green" title="E21">A1</font></i></a>, <a class="txt" href="group_9.html#E21"><i><font color="Green" title="E20">Lm8</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c8">H'</font> <a href="group_2.html#K11" title="GROUP_2:func.11">*</a> <font color="Maroon" title="c12">A'</font>
<b>by </b><i><a class="ref" href="group_3.html#T143" title="GROUP_3:th.143">GROUP_3:143</a></i>
;<br/><b>hence </b><a NAME="E3:33_1_1_1_1_2_1"/>
<font color="Maroon" title="c11">A</font> <a href="group_2.html#K10" title="GROUP_2:func.10">*</a> <font color="Maroon" title="c10">H''</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">H''</font> <a href="group_2.html#K11" title="GROUP_2:func.11">*</a> <font color="Maroon" title="c11">A</font>
 <b>by </b><i><a class="txt" href="group_9.html#E2:33_1_1"><i><font color="Green" title="E21">A1</font></i></a>, <a class="txt" href="group_9.html#E21"><i><font color="Green" title="E20">Lm8</font></i></a></i>;<br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c11">H''</font> = <font color="Maroon" title="c10">H''</font> as  <a href="group_1.html#V1" title="GROUP_1:attr.1">strict</a> <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  <a href="matrix_2.html#K13" title="MATRIX_2:func.13">Group_of_Perm</a> 2 <b>by </b><i><a class="ref" href="group_3.html#T143" title="GROUP_3:th.143">GROUP_3:143</a></i>;<br/><div><i><font color="Green" title="E24">A5</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_1_1_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:33_1_1_1_1_2_2"/>
<font color="Maroon" title="c11">H''</font> <a href="group_2.html#R1" title="GROUP_2:pred.1">=</a>  <a href="group_2.html#K5" title="GROUP_2:func.5">(1).</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K13" title="MATRIX_2:func.13">Group_of_Perm</a> 2</span>)</span>
 ;<br/><a NAME="E2:33_1_1_1_1_2_2"/><b>then </b><i><font color="Green" title="E25">A6</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">H''</font> <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"><span class="p2">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <span class="p3">(<span class="default"><a href="matrix_2.html#K13" title="MATRIX_2:func.13">Group_of_Perm</a> 2</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_2.html#D7" title="GROUP_2:def.7">GROUP_2:def 7</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c12">e</font> =  <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K13" title="MATRIX_2:func.13">Group_of_Perm</a> 2</span>)</span> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">G</font> <b>by </b><i><a class="txt" href="group_9.html#E2:33_1_1"><i><font color="Green" title="E21">A1</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_1_1_2_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c13">h</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">G</font>;<br/><b>reconsider </b><font color="Maroon" title="c14">h'</font> = <font color="Maroon" title="c13">h</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="matrix_2.html#K13" title="MATRIX_2:func.13">Group_of_Perm</a> 2</span>)</span> <b>by </b><i><a class="txt" href="group_9.html#E2:33_1_1"><i><font color="Green" title="E21">A1</font></i></a></i>;<br/><font color="Maroon" title="c13">h</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c12">e</font> = 
<font color="Maroon" title="c14">h'</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <span class="p2">(<span class="default"><a href="matrix_2.html#K13" title="MATRIX_2:func.13">Group_of_Perm</a> 2</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="group_9.html#E2:33_1_1"><i><font color="Green" title="E21">A1</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c14">h'</font>
<b>by </b><i><a class="ref" href="group_1.html#D5" title="GROUP_1:def.5">GROUP_1:def 5</a></i>
;<br/><b>hence </b><a NAME="E3:33_1_1_1_1_2_2_1"/>
<font color="Maroon" title="c13">h</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c12">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">h</font>
 ;<br/><font color="Maroon" title="c12">e</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c13">h</font> = 
<span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <span class="p2">(<span class="default"><a href="matrix_2.html#K13" title="MATRIX_2:func.13">Group_of_Perm</a> 2</span>)</span></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c14">h'</font>
<b>by </b><i><a class="txt" href="group_9.html#E2:33_1_1"><i><font color="Green" title="E21">A1</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c14">h'</font>
<b>by </b><i><a class="ref" href="group_1.html#D5" title="GROUP_1:def.5">GROUP_1:def 5</a></i>
;<br/><b>hence </b><a NAME="E5:33_1_1_1_1_2_2_1"/>
<font color="Maroon" title="c12">e</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c13">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">h</font>
 ;<br/></div><b>end;</b></div><a NAME="E5:33_1_1_1_1_2_2"/><b>then </b>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">H</font> <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"><span class="p2">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c4">G</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="group_9.html#E2:33_1_1_1_1_2_2"><i><font color="Green" title="E25">A6</font></i></a>, <a class="ref" href="group_1.html#D5" title="GROUP_1:def.5">GROUP_1:def 5</a></i>;<br/><b>hence </b><a NAME="E6:33_1_1_1_1_2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="group_9.html#E3:33_1_1_1_1_2"><i><font color="Green" title="E23">A4</font></i></a>, <a class="ref" href="group_9.html#D8" target="_self" title="GROUP_9:def.8">Def8</a></i>;<br/></div><b>end;</b></div><a NAME="E12:33_1_1_1_1_2"/>
<font color="Maroon" title="c11">H''</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="group_2.html#K6" title="GROUP_2:func.6">(Omega).</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K13" title="MATRIX_2:func.13">Group_of_Perm</a> 2</span>)</span>
 <b>by </b><i><a class="txt" href="group_9.html#E2:33_1_1"><i><font color="Green" title="E21">A1</font></i></a>, <a class="txt" href="group_9.html#E3:33_1_1_1_1_2"><i><font color="Green" title="E23">A4</font></i></a>, <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="E13:33_1_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="group_9.html#E11:33_1_1_1_1_2"><i><font color="Green" title="E24">A5</font></i></a>, <a class="ref" href="group_9.html#D12" target="_self" title="GROUP_9:def.12">Def12</a>, <a class="txt" href="group_9.html#E18"><i><font color="Green" title="E17">Lm6</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E5:33_1_1"/>
( <font color="Maroon" title="c4">G</font> is <a href="group_9.html#V2" title="GROUP_9:attr.2">strict</a> &amp; <font color="Maroon" title="c4">G</font> is <a href="group_9.html#V6" title="GROUP_9:attr.6">simple</a> )
 ;<br/>


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