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

<b>let </b><font color="Maroon" title="c1">G</font> be  <a href="struct_0.html#V8" title="STRUCT_0:attr.8">finite</a> <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a>;<br/><b>let </b><font color="Maroon" title="c2">p</font> be  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a> <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/>



<b>consider </b><font color="Maroon" title="c3">P</font> being    <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E2:54"/><i><font color="Green" title="E40">A1</font></i>: 
<font color="Maroon" title="c3">P</font> <a href="group_10.html#R5" title="GROUP_10:pred.5">is_Sylow_p-subgroup_of_prime</a> <font color="Maroon" title="c2">p</font>
 <b>by </b><i><a class="ref" href="group_10.html#T12" target="_self" title="GROUP_10:th.12">Th12</a></i>;<br/>
<b>set </b><font color="Maroon" title="c4">T</font> =  <a href="group_10.html#K17" title="GROUP_10:func.17">the_left_operation_of</a> <font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c2">p</font>;<br/>
<b>set </b><font color="Maroon" title="c5">E</font> =  <a href="group_10.html#K15" title="GROUP_10:func.15">the_sylow_p-subgroups_of_prime</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c1">G</font>;<br/>
<a NAME="E3:54"/>
( <font color="Maroon" title="c3">P</font> <a href="group_10.html#R4" title="GROUP_10:pred.4">is_p-group_of_prime</a> <font color="Maroon" title="c2">p</font> &amp; not <font color="Maroon" title="c2">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a>  <a href="group_2.html#K17" title="GROUP_2:func.17">index</a> <font color="Maroon" title="c3">P</font> )
 
<b>by </b><i><a class="ref" href="group_10.html#D19" target="_self" title="GROUP_10:def.19">Def19</a>, <a class="txt" href="group_10.html#E2:54"><i><font color="Green" title="E40">A1</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c6">H</font> being  <a href="struct_0.html#V8" title="STRUCT_0:attr.8">finite</a> <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a><b> such that </b><br/><a NAME="E5:54"/><i><font color="Green" title="E41">A2</font></i>: 
( <font color="Maroon" title="c3">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">H</font> &amp; <font color="Maroon" title="c6">H</font> <a href="group_10.html#R3" title="GROUP_10:pred.3">is_p-group_of_prime</a> <font color="Maroon" title="c2">p</font> )
 <b>by </b><i><a class="ref" href="group_10.html#D18" target="_self" title="GROUP_10:def.18">Def18</a></i>;<br/>
<b>set </b><font color="Maroon" title="c7">P'</font> =  <a href="group_2.html#K6" title="GROUP_2:func.6">(Omega).</a> <font color="Maroon" title="c3">P</font>;<br/>
<a NAME="E6:54"/>
( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">P</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="c1">G</font> &amp; the <a href="group_1.html#U1" title="GROUP_1:sel.1">mult</a> of <font color="Maroon" title="c3">P</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="c1">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="c3">P</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="c8">P'</font> =  <a href="group_2.html#K6" title="GROUP_2:func.6">(Omega).</a> <font color="Maroon" title="c3">P</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 <font color="Maroon" title="c1">G</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>reconsider </b><font color="Maroon" title="c9">H'</font> = <font color="Maroon" title="c8">P'</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="group_3.html#K1" title="GROUP_3:func.1">Subgroups</a> <font color="Maroon" title="c1">G</font> <b>by </b><i><a class="ref" href="group_3.html#D1" title="GROUP_3:def.1">GROUP_3:def 1</a></i>;<br/>
<a NAME="E9:54"/><i><font color="Green" title="E42">A3</font></i>: 
 <a href="group_1.html#K9" title="GROUP_1:func.9">ord</a> <font color="Maroon" title="c3">P</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 
<b>by </b><i><a class="txt" href="group_10.html#E5:54"><i><font color="Green" title="E41">A2</font></i></a>, <a class="ref" href="group_1.html#T90" title="GROUP_1:th.90">GROUP_1:90</a></i>;<br/>
<a NAME="E10:54"/>
<font color="Maroon" title="c3">P</font> is <a href="struct_0.html#V8" title="STRUCT_0:attr.8">finite</a>
 
<b>by </b><i><a class="ref" href="group_2.html#T48" title="GROUP_2:th.48">GROUP_2:48</a></i>;<br/>
<a NAME="E11:54"/><b>then </b><i><font color="Green" title="E43">A4</font></i>: 
 ex <font color="Olive" title="b1">B</font> being <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <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 <font color="Maroon" title="c3">P</font> &amp;  <a href="group_1.html#K9" title="GROUP_1:func.9">ord</a> <font color="Maroon" title="c3">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Olive" title="b1">B</font> )
 
<b>by </b><i><a class="ref" href="group_1.html#D15" title="GROUP_1:def.15">GROUP_1:def 15</a></i>;<br/>
<a NAME="E12:54"/>
<font color="Maroon" title="c8">P'</font> is <a href="struct_0.html#V8" title="STRUCT_0:attr.8">finite</a>
 
<b>by </b><i><a class="ref" href="group_2.html#T48" title="GROUP_2:th.48">GROUP_2:48</a></i>;<br/>
<a NAME="E13:54"/><b>then </b><i><font color="Green" title="E44">A5</font></i>: 
 ex <font color="Olive" title="b1">B</font> being <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <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 <font color="Maroon" title="c8">P'</font> &amp;  <a href="group_1.html#K9" title="GROUP_1:func.9">ord</a> <font color="Maroon" title="c8">P'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Olive" title="b1">B</font> )
 
<b>by </b><i><a class="ref" href="group_1.html#D15" title="GROUP_1:def.15">GROUP_1:def 15</a></i>;<br/>
<span class="p1">(<span class="default"><a href="group_1.html#K9" title="GROUP_1:func.9">ord</a> <font color="Maroon" title="c3">P</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><a href="group_2.html#K17" title="GROUP_2:func.17">index</a> <font color="Maroon" title="c3">P</font></span>)</span> = 
 <a href="group_1.html#K9" title="GROUP_1:func.9">ord</a> <font color="Maroon" title="c1">G</font>
<b>by </b><i><a class="ref" href="group_2.html#T177" title="GROUP_2:th.177">GROUP_2:177</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_1.html#K9" title="GROUP_1:func.9">ord</a> <font color="Maroon" title="c3">P</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><a href="group_2.html#K17" title="GROUP_2:func.17">index</a> <font color="Maroon" title="c8">P'</font></span>)</span>
<b>by </b><i><a class="txt" href="group_10.html#E11:54"><i><font color="Green" title="E43">A4</font></i></a>, <a class="txt" href="group_10.html#E13:54"><i><font color="Green" title="E44">A5</font></i></a>, <a class="ref" href="group_2.html#T177" title="GROUP_2:th.177">GROUP_2:177</a></i>
;<br/>
<a NAME="E15:54"/><b>then </b><i><font color="Green" title="E45">A6</font></i>: 
 <a href="group_2.html#K17" title="GROUP_2:func.17">index</a> <font color="Maroon" title="c8">P'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_2.html#K17" title="GROUP_2:func.17">index</a> <font color="Maroon" title="c3">P</font>
 
<b>by </b><i><a class="txt" href="group_10.html#E9:54"><i><font color="Green" title="E42">A3</font></i></a>, <a class="ref" href="xcmplx_1.html#T5" title="XCMPLX_1:th.5">XCMPLX_1:5</a></i>;<br/>
<a NAME="E16:54"/>
 ex <font color="Olive" title="b1">H</font> being <a href="struct_0.html#V8" title="STRUCT_0:attr.8">finite</a> <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a> st <br/>( <font color="Maroon" title="c8">P'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">H</font> &amp; <font color="Olive" title="b1">H</font> <a href="group_10.html#R3" title="GROUP_10:pred.3">is_p-group_of_prime</a> <font color="Maroon" title="c2">p</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_2"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon" title="c10">H''</font> =  <a href="group_2.html#K6" title="GROUP_2:func.6">(Omega).</a> <font color="Maroon" title="c6">H</font>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">H''</font> =  <a href="group_2.html#K6" title="GROUP_2:func.6">(Omega).</a> <font color="Maroon" title="c6">H</font> as  <a href="struct_0.html#V8" title="STRUCT_0:attr.8">finite</a> <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a> <b>by </b><i><a class="ref" href="group_2.html#T48" title="GROUP_2:th.48">GROUP_2:48</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c11">H''</font>
;<br/>

<b>thus </b><a NAME="E2:54_2"/>
<font color="Maroon" title="c8">P'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">H''</font>
 <b>by </b><i><a class="txt" href="group_10.html#E5:54"><i><font color="Green" title="E41">A2</font></i></a></i>;<br/>

<a NAME="E3:54_2"/><i><font color="Green" title="E46">A7</font></i>: 
 ex <font color="Olive" title="b1">B</font> being <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <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 <font color="Maroon" title="c6">H</font> &amp;  <a href="group_1.html#K9" title="GROUP_1:func.9">ord</a> <font color="Maroon" title="c6">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Olive" title="b1">B</font> )
 
<b>by </b><i><a class="ref" href="group_1.html#D15" title="GROUP_1:def.15">GROUP_1:def 15</a></i>;<br/>
<a NAME="E4:54_2"/><i><font color="Green" title="E47">A8</font></i>: 
 ex <font color="Olive" title="b1">B</font> being <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <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 <font color="Maroon" title="c11">H''</font> &amp;  <a href="group_1.html#K9" title="GROUP_1:func.9">ord</a> <font color="Maroon" title="c11">H''</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Olive" title="b1">B</font> )
 
<b>by </b><i><a class="ref" href="group_1.html#D15" title="GROUP_1:def.15">GROUP_1:def 15</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c12">r</font> being  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <b> such that </b><br/><a NAME="E6:54_2"/><i><font color="Green" title="E48">A9</font></i>: 
 <a href="group_1.html#K9" title="GROUP_1:func.9">ord</a> <font color="Maroon" title="c6">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c12">r</font>
 <b>by </b><i><a class="ref" href="group_10.html#D17" target="_self" title="GROUP_10:def.17">Def17</a>, <a class="txt" href="group_10.html#E5:54"><i><font color="Green" title="E41">A2</font></i></a></i>;<br/>
<b>thus </b><a NAME="E7:54_2"/>
<font color="Maroon" title="c11">H''</font> <a href="group_10.html#R3" title="GROUP_10:pred.3">is_p-group_of_prime</a> <font color="Maroon" title="c2">p</font>
 <b>by </b><i><a class="txt" href="group_10.html#E6:54_2"><i><font color="Green" title="E48">A9</font></i></a>, <a class="txt" href="group_10.html#E3:54_2"><i><font color="Green" title="E46">A7</font></i></a>, <a class="txt" href="group_10.html#E4:54_2"><i><font color="Green" title="E47">A8</font></i></a>, <a class="ref" href="group_10.html#D17" target="_self" title="GROUP_10:def.17">Def17</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E17:54"/><b>then </b>
( <font color="Maroon" title="c8">P'</font> <a href="group_10.html#R4" title="GROUP_10:pred.4">is_p-group_of_prime</a> <font color="Maroon" title="c2">p</font> &amp; not <font color="Maroon" title="c2">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a>  <a href="group_2.html#K17" title="GROUP_2:func.17">index</a> <font color="Maroon" title="c8">P'</font> )
 
<b>by </b><i><a class="ref" href="group_10.html#D19" target="_self" title="GROUP_10:def.19">Def19</a>, <a class="txt" href="group_10.html#E2:54"><i><font color="Green" title="E40">A1</font></i></a>, <a class="txt" href="group_10.html#E15:54"><i><font color="Green" title="E45">A6</font></i></a>, <a class="ref" href="group_10.html#D18" target="_self" title="GROUP_10:def.18">Def18</a></i>;<br/>
<a NAME="E18:54"/><b>then </b><i><font color="Green" title="E46">A10</font></i>: 
( <font color="Maroon" title="c9">H'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">P'</font> &amp; <font color="Maroon" title="c8">P'</font> <a href="group_10.html#R5" title="GROUP_10:pred.5">is_Sylow_p-subgroup_of_prime</a> <font color="Maroon" title="c2">p</font> )
 
<b>by </b><i><a class="ref" href="group_10.html#D19" target="_self" title="GROUP_10:def.19">Def19</a></i>;<br/>
<a NAME="E19:54"/><b>then </b><i><font color="Green" title="E47">A11</font></i>: 
<font color="Maroon" title="c8">P'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_10.html#K15" title="GROUP_10:func.15">the_sylow_p-subgroups_of_prime</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c1">G</font>
 
;<br/>
<b>reconsider </b><font color="Maroon" title="c10">T</font> =  <a href="group_10.html#K17" title="GROUP_10:func.17">the_left_operation_of</a> <font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c2">p</font> as   <a href="group_10.html#NM1" title="GROUP_10:NM.1">LeftOperation</a> of <font color="Maroon" title="c3">P</font>,<span class="p1">(<span class="default"><a href="group_10.html#K15" title="GROUP_10:func.15">the_sylow_p-subgroups_of_prime</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c1">G</font></span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon" title="c11">P'</font> = <font color="Maroon" title="c8">P'</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="group_10.html#K15" title="GROUP_10:func.15">the_sylow_p-subgroups_of_prime</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c1">G</font> <b>by </b><i><a class="txt" href="group_10.html#E19:54"><i><font color="Green" title="E47">A11</font></i></a></i>;<br/>
<a NAME="E22:54"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_10.html#K12" title="GROUP_10:func.12">the_fixed_points_of</a> <font color="Maroon" title="c10">T</font> iff <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">P'</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c12">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:54_3_1"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_10.html#K12" title="GROUP_10:func.12">the_fixed_points_of</a> <font color="Maroon" title="c10">T</font>
 ;<br/><a NAME="E2:54_3_1"/><b>then </b>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x'</font> where <font color="Olive" title="b1">x'</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="group_10.html#K15" title="GROUP_10:func.15">the_sylow_p-subgroups_of_prime</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c1">G</font> : <font color="Olive" title="b1">x'</font> <a href="group_10.html#R1" title="GROUP_10:pred.1">is_fixed_under</a> <font color="Maroon" title="c10">T</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="group_10.html#D13" target="_self" title="GROUP_10:def.13">Def13</a></i>;<br/><b>then consider </b><font color="Maroon" title="c13">Q</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="group_10.html#K15" title="GROUP_10:func.15">the_sylow_p-subgroups_of_prime</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E4:54_3_1"/><i><font color="Green" title="E48">A12</font></i>: 
( <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">Q</font> &amp; <font color="Maroon" title="c13">Q</font> <a href="group_10.html#R1" title="GROUP_10:pred.1">is_fixed_under</a> <font color="Maroon" title="c10">T</font> )
 ;<br/><a NAME="E5:54_3_1"/>
<font color="Maroon" title="c13">Q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">H''</font> where <font color="Olive" title="b1">H''</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="group_3.html#K1" title="GROUP_3:func.1">Subgroups</a> <font color="Maroon" title="c1">G</font> :  ex <font color="Olive" title="b2">P</font> being <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 <font color="Maroon" title="c1">G</font> st <br/>( <font color="Olive" title="b2">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">H''</font> &amp; <font color="Olive" title="b2">P</font> <a href="group_10.html#R5" title="GROUP_10:pred.5">is_Sylow_p-subgroup_of_prime</a> <font color="Maroon" title="c2">p</font> ) </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon" title="c14">H''</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="group_3.html#K1" title="GROUP_3:func.1">Subgroups</a> <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E7:54_3_1"/><i><font color="Green" title="E49">A13</font></i>: 
( <font color="Maroon" title="c13">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">H''</font> &amp;  ex <font color="Olive" title="b1">P</font> being <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 <font color="Maroon" title="c1">G</font> st <br/>( <font color="Olive" title="b1">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">H''</font> &amp; <font color="Olive" title="b1">P</font> <a href="group_10.html#R5" title="GROUP_10:pred.5">is_Sylow_p-subgroup_of_prime</a> <font color="Maroon" title="c2">p</font> ) )
 ;<br/><b>consider </b><font color="Maroon" title="c15">Q'</font> being  <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 <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E9:54_3_1"/><i><font color="Green" title="E50">A14</font></i>: 
( <font color="Maroon" title="c15">Q'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">H''</font> &amp; <font color="Maroon" title="c15">Q'</font> <a href="group_10.html#R5" title="GROUP_10:pred.5">is_Sylow_p-subgroup_of_prime</a> <font color="Maroon" title="c2">p</font> )
 <b>by </b><i><a class="txt" href="group_10.html#E7:54_3_1"><i><font color="Green" title="E49">A13</font></i></a></i>;<br/><b>set </b><font color="Maroon" title="c16">N</font> =  <a href="group_3.html#K11" title="GROUP_3:func.11">Normalizator</a> <font color="Maroon" title="c15">Q'</font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_3_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c17">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:54_3_1_1"/><i><font color="Green" title="E51">A15</font></i>: 
<font color="Maroon" title="c17">y</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">P</font>
 ;<br/><a NAME="E2:54_3_1_1"/>
 ex <font color="Olive" title="b1">h</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> st <br/>( <font color="Maroon" title="c17">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">h</font> &amp; <font color="Maroon" title="c15">Q'</font> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Olive" title="b1">h</font> <a href="group_2.html#R1" title="GROUP_2:pred.1">=</a> <font color="Maroon" title="c15">Q'</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_3_1_1_1"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon" title="c18">h</font> = <font color="Maroon" title="c17">y</font>;<br/>
<a NAME="E1:54_3_1_1_1"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">P</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="c1">G</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="c19">h</font> = <font color="Maroon" title="c17">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> <b>by </b><i><a class="txt" href="group_10.html#E1:54_3_1_1"><i><font color="Green" title="E51">A15</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c19">h</font>
;<br/>

<b>thus </b><a NAME="E3:54_3_1_1_1"/>
<font color="Maroon" title="c17">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">h</font>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c20">h'</font> = <font color="Maroon" title="c19">h</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">P</font> <b>by </b><i><a class="txt" href="group_10.html#E1:54_3_1_1"><i><font color="Green" title="E51">A15</font></i></a></i>;<br/>
<a NAME="E5:54_3_1_1_1"/><i><font color="Green" title="E52">A16</font></i>: 
<font color="Maroon" title="c15">Q'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">T</font> <a href="group_10.html#K1" title="GROUP_10:func.1">^</a> <font color="Maroon" title="c20">h'</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">Q'</font>
 
<b>by </b><i><a class="txt" href="group_10.html#E7:54_3_1"><i><font color="Green" title="E49">A13</font></i></a>, <a class="txt" href="group_10.html#E9:54_3_1"><i><font color="Green" title="E50">A14</font></i></a>, <a class="txt" href="group_10.html#E4:54_3_1"><i><font color="Green" title="E48">A12</font></i></a>, <a class="ref" href="group_10.html#D12" target="_self" title="GROUP_10:def.12">Def12</a></i>;<br/>
<a NAME="E6:54_3_1_1_1"/><i><font color="Green" title="E53">A17</font></i>: 
<font color="Maroon" title="c10">T</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">h'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_10.html#K16" title="GROUP_10:func.16">the_left_translation_of</a> <font color="Maroon" title="c20">h'</font>,<font color="Maroon" title="c2">p</font>
 
<b>by </b><i><a class="ref" href="group_10.html#D22" target="_self" title="GROUP_10:def.22">Def22</a></i>;<br/>
<a NAME="E7:54_3_1_1_1"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">T</font> <a href="group_10.html#K1" title="GROUP_10:func.1">^</a> <font color="Maroon" title="c20">h'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_10.html#K15" title="GROUP_10:func.15">the_sylow_p-subgroups_of_prime</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c1">G</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E8:54_3_1_1_1"/><b>then </b>
<font color="Maroon" title="c15">Q'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">T</font> <a href="group_10.html#K1" title="GROUP_10:func.1">^</a> <font color="Maroon" title="c20">h'</font></span>)</span>
 
<b>by </b><i><a class="txt" href="group_10.html#E9:54_3_1"><i><font color="Green" title="E50">A14</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c21">Q1</font> = <font color="Maroon" title="c15">Q'</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="group_10.html#K15" title="GROUP_10:func.15">the_sylow_p-subgroups_of_prime</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c1">G</font> ;<br/>
<b>consider </b><font color="Maroon" title="c22">Q2</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="group_10.html#K15" title="GROUP_10:func.15">the_sylow_p-subgroups_of_prime</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c1">G</font>, <font color="Maroon" title="c23">H1</font>, <font color="Maroon" title="c24">H2</font> being  <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 <font color="Maroon" title="c1">G</font>, <font color="Maroon" title="c25">g</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E11:54_3_1_1_1"/><i><font color="Green" title="E54">A18</font></i>: 
( <font color="Maroon" title="c22">Q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="group_10.html#K16" title="GROUP_10:func.16">the_left_translation_of</a> <font color="Maroon" title="c20">h'</font>,<font color="Maroon" title="c2">p</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">Q1</font> &amp; <font color="Maroon" title="c21">Q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">H1</font> &amp; <font color="Maroon" title="c22">Q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">H2</font> &amp; <font color="Maroon" title="c20">h'</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">g</font> &amp; <font color="Maroon" title="c24">H2</font> <a href="group_2.html#R1" title="GROUP_2:pred.1">=</a> <font color="Maroon" title="c23">H1</font> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Maroon" title="c25">g</font> )
 <b>by </b><i><a class="ref" href="group_10.html#D21" target="_self" title="GROUP_10:def.21">Def21</a></i>;<br/>
<a NAME="E12:54_3_1_1_1"/><i><font color="Green" title="E55">A19</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c19">h</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c19">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c1">G</font>
 
<b>by </b><i><a class="ref" href="group_1.html#D6" title="GROUP_1:def.6">GROUP_1:def 6</a></i>;<br/>
<a NAME="E13:54_3_1_1_1"/>
<font color="Maroon" title="c15">Q'</font> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">h</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span> <a href="group_2.html#R1" title="GROUP_2:pred.1">=</a> <font color="Maroon" title="c15">Q'</font>
 
<b>by </b><i><a class="txt" href="group_10.html#E6:54_3_1_1_1"><i><font color="Green" title="E53">A17</font></i></a>, <a class="txt" href="group_10.html#E5:54_3_1_1_1"><i><font color="Green" title="E52">A16</font></i></a>, <a class="txt" href="group_10.html#E11:54_3_1_1_1"><i><font color="Green" title="E54">A18</font></i></a>, <a class="ref" href="group_2.html#T57" title="GROUP_2:th.57">GROUP_2:57</a></i>;<br/>
<a NAME="E14:54_3_1_1_1"/><b>then </b>
<font color="Maroon" title="c15">Q'</font> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">h</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c19">h</font></span>)</span> <a href="group_2.html#R1" title="GROUP_2:pred.1">=</a> <font color="Maroon" title="c15">Q'</font> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Maroon" title="c19">h</font>
 
<b>by </b><i><a class="ref" href="group_3.html#T72" title="GROUP_3:th.72">GROUP_3:72</a></i>;<br/>
<b>hence </b><a NAME="E15:54_3_1_1_1"/>
<font color="Maroon" title="c15">Q'</font> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Maroon" title="c19">h</font> <a href="group_2.html#R1" title="GROUP_2:pred.1">=</a> <font color="Maroon" title="c15">Q'</font>
 <b>by </b><i><a class="txt" href="group_10.html#E12:54_3_1_1_1"><i><font color="Green" title="E55">A19</font></i></a>, <a class="ref" href="group_3.html#T73" title="GROUP_3:th.73">GROUP_3:73</a></i>;<br/>


</div><b>end;</b></div><a NAME="E3:54_3_1_1"/><b>then </b>
<font color="Maroon" title="c17">y</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="group_3.html#K11" title="GROUP_3:func.11">Normalizator</a> <font color="Maroon" title="c15">Q'</font>
 <b>by </b><i><a class="ref" href="group_3.html#T160" title="GROUP_3:th.160">GROUP_3:160</a></i>;<br/><b>hence </b><a NAME="E4:54_3_1_1"/>
<font color="Maroon" title="c17">y</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 <span class="p1">(<span class="default"><a href="group_3.html#K11" title="GROUP_3:func.11">Normalizator</a> <font color="Maroon" title="c15">Q'</font></span>)</span>
 <b>by </b><i><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/></div><b>end;</b></div><a NAME="E11:54_3_1"/><b>then </b><i><font color="Green" title="E51">A20</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">P</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_3.html#K11" title="GROUP_3:func.11">Normalizator</a> <font color="Maroon" title="c15">Q'</font></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/><div><i><font color="Green" title="E52">A21</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="54_3_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c17">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:54_3_1_2"/><i><font color="Green" title="E53">A22</font></i>: 
<font color="Maroon" title="c17">y</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="c15">Q'</font>
 ;<br/><a NAME="E2:54_3_1_2"/>
 ex <font color="Olive" title="b1">h</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> st <br/>( <font color="Maroon" title="c17">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">h</font> &amp; <font color="Maroon" title="c15">Q'</font> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Olive" title="b1">h</font> <a href="group_2.html#R1" title="GROUP_2:pred.1">=</a> <font color="Maroon" title="c15">Q'</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_3_1_2_1"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon" title="c18">h</font> = <font color="Maroon" title="c17">y</font>;<br/>
<a NAME="E1:54_3_1_2_1"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c15">Q'</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="c1">G</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="c19">h</font> = <font color="Maroon" title="c17">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> <b>by </b><i><a class="txt" href="group_10.html#E1:54_3_1_2"><i><font color="Green" title="E53">A22</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c19">h</font>
;<br/>

<b>thus </b><a NAME="E3:54_3_1_2_1"/>
<font color="Maroon" title="c17">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">h</font>
 ;<br/>

<a NAME="E4:54_3_1_2_1"/>
for <font color="Olive" title="b1">g</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> holds <br/> ( <font color="Olive" title="b1">g</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c15">Q'</font> iff <font color="Olive" title="b1">g</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c15">Q'</font> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Maroon" title="c19">h</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_3_1_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c20">g</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font>;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:54_3_1_2_1_1_1"/><i><font color="Green" title="E54">A23</font></i>: 
<font color="Maroon" title="c20">g</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c15">Q'</font>
 ;<br/><a NAME="E2:54_3_1_2_1_1_1"/>
 ex <font color="Olive" title="b1">g'</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> st <br/>( <font color="Maroon" title="c20">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g'</font> <a href="group_3.html#K2" title="GROUP_3:func.2">|^</a> <font color="Maroon" title="c19">h</font> &amp; <font color="Olive" title="b1">g'</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c15">Q'</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_3_1_2_1_1_1_1"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon" title="c21">g'</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c19">h</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c20">g</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">h</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span>;<br/>
<b>take </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c19">h</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c20">g</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">h</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span>
;<br/>

<b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">h</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c20">g</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">h</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span></span>)</span> <a href="group_3.html#K2" title="GROUP_3:func.2">|^</a> <font color="Maroon" title="c19">h</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">h</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c19">h</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c20">g</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c19">h</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span></span>)</span></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c19">h</font>
<b>by </b><i><a class="ref" href="group_3.html#D2" title="GROUP_3:def.2">GROUP_3:def 2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">h</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">h</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c20">g</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c19">h</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c19">h</font>
<b>by </b><i><a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">GROUP_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c19">h</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c19">h</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">g</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c19">h</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span></span>)</span></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c19">h</font>
<b>by </b><i><a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">GROUP_1:def 4</a></i>
<br/>.= 
<span class="p1">(<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="c1">G</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">g</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c19">h</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span></span>)</span></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c19">h</font>
<b>by </b><i><a class="ref" href="group_1.html#D6" title="GROUP_1:def.6">GROUP_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c20">g</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">h</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c19">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/>.= 
<font color="Maroon" title="c20">g</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">h</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c19">h</font></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">GROUP_1:def 4</a></i>
<br/>.= 
<font color="Maroon" title="c20">g</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> <font color="Maroon" title="c1">G</font></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D6" title="GROUP_1:def.6">GROUP_1:def 6</a></i>
<br/>.= 
<font color="Maroon" title="c20">g</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/>

<a NAME="E2:54_3_1_2_1_1_1_1"/><i><font color="Green" title="E55">A24</font></i>: 
<font color="Maroon" title="c19">h</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c15">Q'</font>
 
<b>by </b><i><a class="txt" href="group_10.html#E1:54_3_1_2"><i><font color="Green" title="E53">A22</font></i></a>, <a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/>
<a NAME="E3:54_3_1_2_1_1_1_1"/><b>then </b><i><font color="Green" title="E56">A25</font></i>: 
<font color="Maroon" title="c19">h</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a>  <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c15">Q'</font>
 
<b>by </b><i><a class="ref" href="group_2.html#T60" title="GROUP_2:th.60">GROUP_2:60</a></i>;<br/>
<a NAME="E4:54_3_1_2_1_1_1_1"/>
<font color="Maroon" title="c19">h</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c20">g</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c15">Q'</font>
 
<b>by </b><i><a class="txt" href="group_10.html#E2:54_3_1_2_1_1_1_1"><i><font color="Green" title="E55">A24</font></i></a>, <a class="txt" href="group_10.html#E1:54_3_1_2_1_1_1"><i><font color="Green" title="E54">A23</font></i></a>, <a class="ref" href="group_2.html#T59" title="GROUP_2:th.59">GROUP_2:59</a></i>;<br/>
<b>hence </b><a NAME="E5:54_3_1_2_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c19">h</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c20">g</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">h</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c15">Q'</font>
 <b>by </b><i><a class="txt" href="group_10.html#E3:54_3_1_2_1_1_1_1"><i><font color="Green" title="E56">A25</font></i></a>, <a class="ref" href="group_2.html#T59" title="GROUP_2:th.59">GROUP_2:59</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:54_3_1_2_1_1_1"/>
<font color="Maroon" title="c20">g</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c15">Q'</font> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Maroon" title="c19">h</font>
 <b>by </b><i><a class="ref" href="group_3.html#T70" title="GROUP_3:th.70">GROUP_3:70</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:54_3_1_2_1_1"/>
<font color="Maroon" title="c20">g</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c15">Q'</font> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Maroon" title="c19">h</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c21">g'</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E4:54_3_1_2_1_1"/><i><font color="Green" title="E54">A26</font></i>: 
( <font color="Maroon" title="c20">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">g'</font> <a href="group_3.html#K2" title="GROUP_3:func.2">|^</a> <font color="Maroon" title="c19">h</font> &amp; <font color="Maroon" title="c21">g'</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c15">Q'</font> )
 <b>by </b><i><a class="ref" href="group_3.html#T70" title="GROUP_3:th.70">GROUP_3:70</a></i>;<br/>
<a NAME="E5:54_3_1_2_1_1"/><i><font color="Green" title="E55">A27</font></i>: 
<font color="Maroon" title="c19">h</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c15">Q'</font>
 
<b>by </b><i><a class="txt" href="group_10.html#E1:54_3_1_2"><i><font color="Green" title="E53">A22</font></i></a>, <a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/>
<a NAME="E6:54_3_1_2_1_1"/><b>then </b>
<font color="Maroon" title="c19">h</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a>  <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c15">Q'</font>
 
<b>by </b><i><a class="ref" href="group_2.html#T60" title="GROUP_2:th.60">GROUP_2:60</a></i>;<br/>
<a NAME="E7:54_3_1_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c19">h</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c21">g'</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c15">Q'</font>
 
<b>by </b><i><a class="txt" href="group_10.html#E4:54_3_1_2_1_1"><i><font color="Green" title="E54">A26</font></i></a>, <a class="ref" href="group_2.html#T59" title="GROUP_2:th.59">GROUP_2:59</a></i>;<br/>
<a NAME="E8:54_3_1_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">h</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c21">g'</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c19">h</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c15">Q'</font>
 
<b>by </b><i><a class="txt" href="group_10.html#E5:54_3_1_2_1_1"><i><font color="Green" title="E55">A27</font></i></a>, <a class="ref" href="group_2.html#T59" title="GROUP_2:th.59">GROUP_2:59</a></i>;<br/>
<b>hence </b><a NAME="E9:54_3_1_2_1_1"/>
<font color="Maroon" title="c20">g</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c15">Q'</font>
 <b>by </b><i><a class="txt" href="group_10.html#E4:54_3_1_2_1_1"><i><font color="Green" title="E54">A26</font></i></a>, <a class="ref" href="group_3.html#D2" title="GROUP_3:def.2">GROUP_3:def 2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:54_3_1_2_1"/>
<font color="Maroon" title="c15">Q'</font> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Maroon" title="c19">h</font> <a href="group_2.html#R1" title="GROUP_2:pred.1">=</a> <font color="Maroon" title="c15">Q'</font>
 <b>by </b><i><a class="ref" href="group_2.html#D6" title="GROUP_2:def.6">GROUP_2:def 6</a></i>;<br/>


</div><b>end;</b></div><a NAME="E3:54_3_1_2"/><b>then </b>
<font color="Maroon" title="c17">y</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="group_3.html#K11" title="GROUP_3:func.11">Normalizator</a> <font color="Maroon" title="c15">Q'</font>
 <b>by </b><i><a class="ref" href="group_3.html#T160" title="GROUP_3:th.160">GROUP_3:160</a></i>;<br/><b>hence </b><a NAME="E4:54_3_1_2"/>
<font color="Maroon" title="c17">y</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 <span class="p1">(<span class="default"><a href="group_3.html#K11" title="GROUP_3:func.11">Normalizator</a> <font color="Maroon" title="c15">Q'</font></span>)</span>
 <b>by </b><i><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/></div><b>end;</b></div><a NAME="E13:54_3_1"/><b>then </b><i><font color="Green" title="E53">A28</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c15">Q'</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_3.html#K11" title="GROUP_3:func.11">Normalizator</a> <font color="Maroon" title="c15">Q'</font></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c17">N</font> =  <a href="group_3.html#K11" title="GROUP_3:func.11">Normalizator</a> <font color="Maroon" title="c15">Q'</font> as  <a href="struct_0.html#V8" title="STRUCT_0:attr.8">finite</a> <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a> <b>by </b><i><a class="ref" href="group_2.html#T48" title="GROUP_2:th.48">GROUP_2:48</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c18">P''</font> = <font color="Maroon" title="c11">P'</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 <font color="Maroon" title="c17">N</font> <b>by </b><i><a class="txt" href="group_10.html#E11:54_3_1"><i><font color="Green" title="E51">A20</font></i></a>, <a class="ref" href="group_2.html#T66" title="GROUP_2:th.66">GROUP_2:66</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c19">Q''</font> = <font color="Maroon" title="c15">Q'</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 <font color="Maroon" title="c17">N</font> <b>by </b><i><a class="txt" href="group_10.html#E13:54_3_1"><i><font color="Green" title="E53">A28</font></i></a>, <a class="ref" href="group_2.html#T66" title="GROUP_2:th.66">GROUP_2:66</a></i>;<br/><a NAME="E17:54_3_1"/><i><font color="Green" title="E54">A29</font></i>: 
<font color="Maroon" title="c18">P''</font> <a href="group_10.html#R5" title="GROUP_10:pred.5">is_Sylow_p-subgroup_of_prime</a> <font color="Maroon" title="c2">p</font>
 <b>by </b><i><a class="txt" href="group_10.html#E45"><i><font color="Green" title="E39">Lm11</font></i></a>, <a class="txt" href="group_10.html#E18:54"><i><font color="Green" title="E46">A10</font></i></a></i>;<br/><a NAME="E18:54_3_1"/><i><font color="Green" title="E55">A30</font></i>: 
<font color="Maroon" title="c19">Q''</font> <a href="group_10.html#R5" title="GROUP_10:pred.5">is_Sylow_p-subgroup_of_prime</a> <font color="Maroon" title="c2">p</font>
 <b>by </b><i><a class="txt" href="group_10.html#E45"><i><font color="Green" title="E39">Lm11</font></i></a>, <a class="txt" href="group_10.html#E9:54_3_1"><i><font color="Green" title="E50">A14</font></i></a></i>;<br/><a NAME="E19:54_3_1"/>
<font color="Maroon" title="c18">P''</font>,<font color="Maroon" title="c19">Q''</font> <a href="group_3.html#R6" title="GROUP_3:pred.6">are_conjugated</a> 
 <b>by </b><i><a class="txt" href="group_10.html#E17:54_3_1"><i><font color="Green" title="E54">A29</font></i></a>, <a class="txt" href="group_10.html#E18:54_3_1"><i><font color="Green" title="E55">A30</font></i></a>, <a class="ref" href="group_10.html#T14" target="_self" title="GROUP_10:th.14">Th14</a></i>;<br/><b>then consider </b><font color="Maroon" title="c20">n</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c17">N</font><b> such that </b><br/><a NAME="E21:54_3_1"/><i><font color="Green" title="E56">A31</font></i>: 
<font color="Maroon" title="c18">P''</font> <a href="group_2.html#R1" title="GROUP_2:pred.1">=</a> <font color="Maroon" title="c19">Q''</font> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Maroon" title="c20">n</font>
 <b>by </b><i><a class="ref" href="group_3.html#D11" title="GROUP_3:def.11">GROUP_3:def 11</a></i>;<br/><a NAME="E22:54_3_1"/>
<font color="Maroon" title="c20">n</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="group_3.html#K11" title="GROUP_3:func.11">Normalizator</a> <font color="Maroon" title="c15">Q'</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>then consider </b><font color="Maroon" title="c21">n'</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E24:54_3_1"/><i><font color="Green" title="E57">A32</font></i>: 
( <font color="Maroon" title="c20">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">n'</font> &amp; <font color="Maroon" title="c15">Q'</font> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Maroon" title="c21">n'</font> <a href="group_2.html#R1" title="GROUP_2:pred.1">=</a> <font color="Maroon" title="c15">Q'</font> )
 <b>by </b><i><a class="ref" href="group_3.html#T160" title="GROUP_3:th.160">GROUP_3:160</a></i>;<br/><div><i><font color="Green" title="E58">A33</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="54_3_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c22">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:54_3_1_3_1"/>
<font color="Maroon" title="c22">y</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 <span class="p1">(<span class="default"><font color="Maroon" title="c15">Q'</font> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Maroon" title="c21">n'</font></span>)</span>
 ;<br/><a NAME="E2:54_3_1_3_1"/><b>then </b>
<font color="Maroon" title="c22">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <font color="Maroon" title="c15">Q'</font></span>)</span> <a href="group_3.html#K4" title="GROUP_3:func.4">|^</a> <font color="Maroon" title="c21">n'</font>
 <b>by </b><i><a class="ref" href="group_3.html#D6" title="GROUP_3:def.6">GROUP_3:def 6</a></i>;<br/><b>then consider </b><font color="Maroon" title="c23">q'</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E4:54_3_1_3_1"/><i><font color="Green" title="E59">A34</font></i>: 
( <font color="Maroon" title="c22">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">q'</font> <a href="group_3.html#K2" title="GROUP_3:func.2">|^</a> <font color="Maroon" title="c21">n'</font> &amp; <font color="Maroon" title="c23">q'</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="c15">Q'</font> )
 <b>by </b><i><a class="ref" href="group_3.html#T50" title="GROUP_3:th.50">GROUP_3:50</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c24">q''</font> = <font color="Maroon" title="c23">q'</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c17">N</font> <b>by </b><i><a class="txt" href="group_10.html#E4:54_3_1_3_1"><i><font color="Green" title="E59">A34</font></i></a>, <a class="txt" href="group_10.html#E12:54_3_1"><i><font color="Green" title="E52">A21</font></i></a></i>;<br/><a NAME="E6:54_3_1_3_1"/>
<font color="Maroon" title="c21">n'</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">n</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> 
 <b>by </b><i><a class="txt" href="group_10.html#E24:54_3_1"><i><font color="Green" title="E57">A32</font></i></a>, <a class="ref" href="group_2.html#T57" title="GROUP_2:th.57">GROUP_2:57</a></i>;<br/><a NAME="E7:54_3_1_3_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c21">n'</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c23">q'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">n</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c24">q''</font>
 <b>by </b><i><a class="ref" href="group_2.html#T52" title="GROUP_2:th.52">GROUP_2:52</a></i>;<br/><a NAME="E8:54_3_1_3_1"/><b>then </b><i><font color="Green" title="E60">A35</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">n'</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c23">q'</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c21">n'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">n</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c24">q''</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c20">n</font>
 <b>by </b><i><a class="txt" href="group_10.html#E24:54_3_1"><i><font color="Green" title="E57">A32</font></i></a>, <a class="ref" href="group_2.html#T52" title="GROUP_2:th.52">GROUP_2:52</a></i>;<br/><font color="Maroon" title="c23">q'</font> <a href="group_3.html#K2" title="GROUP_3:func.2">|^</a> <font color="Maroon" title="c21">n'</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">n'</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c23">q'</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c21">n'</font>
<b>by </b><i><a class="ref" href="group_3.html#D2" title="GROUP_3:def.2">GROUP_3:def 2</a></i>
<br/>.= 
<font color="Maroon" title="c24">q''</font> <a href="group_3.html#K2" title="GROUP_3:func.2">|^</a> <font color="Maroon" title="c20">n</font>
<b>by </b><i><a class="txt" href="group_10.html#E8:54_3_1_3_1"><i><font color="Green" title="E60">A35</font></i></a>, <a class="ref" href="group_3.html#D2" title="GROUP_3:def.2">GROUP_3:def 2</a></i>
;<br/><a NAME="E10:54_3_1_3_1"/><b>then </b>
<font color="Maroon" title="c22">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <font color="Maroon" title="c19">Q''</font></span>)</span> <a href="group_3.html#K4" title="GROUP_3:func.4">|^</a> <font color="Maroon" title="c20">n</font>
 <b>by </b><i><a class="txt" href="group_10.html#E4:54_3_1_3_1"><i><font color="Green" title="E59">A34</font></i></a>, <a class="ref" href="group_3.html#T50" title="GROUP_3:th.50">GROUP_3:50</a></i>;<br/><b>hence </b><a NAME="E11:54_3_1_3_1"/>
<font color="Maroon" title="c22">y</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 <span class="p1">(<span class="default"><font color="Maroon" title="c19">Q''</font> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Maroon" title="c20">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="group_3.html#D6" title="GROUP_3:def.6">GROUP_3:def 6</a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:54_3_1_3"/>
<font color="Maroon" title="c22">y</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 <span class="p1">(<span class="default"><font color="Maroon" title="c19">Q''</font> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Maroon" title="c20">n</font></span>)</span>
 ;<br/><a NAME="E3:54_3_1_3"/><b>then </b>
<font color="Maroon" title="c22">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <font color="Maroon" title="c19">Q''</font></span>)</span> <a href="group_3.html#K4" title="GROUP_3:func.4">|^</a> <font color="Maroon" title="c20">n</font>
 <b>by </b><i><a class="ref" href="group_3.html#D6" title="GROUP_3:def.6">GROUP_3:def 6</a></i>;<br/><b>then consider </b><font color="Maroon" title="c23">q''</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c17">N</font><b> such that </b><br/><a NAME="E5:54_3_1_3"/><i><font color="Green" title="E59">A36</font></i>: 
( <font color="Maroon" title="c22">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">q''</font> <a href="group_3.html#K2" title="GROUP_3:func.2">|^</a> <font color="Maroon" title="c20">n</font> &amp; <font color="Maroon" title="c23">q''</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="c19">Q''</font> )
 <b>by </b><i><a class="ref" href="group_3.html#T50" title="GROUP_3:th.50">GROUP_3:50</a></i>;<br/><a NAME="E6:54_3_1_3"/><i><font color="Green" title="E60">A37</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c19">Q''</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="c1">G</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>reconsider </b><font color="Maroon" title="c24">q'</font> = <font color="Maroon" title="c23">q''</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> <b>by </b><i><a class="txt" href="group_10.html#E5:54_3_1_3"><i><font color="Green" title="E59">A36</font></i></a>, <a class="txt" href="group_10.html#E6:54_3_1_3"><i><font color="Green" title="E60">A37</font></i></a></i>;<br/><a NAME="E8:54_3_1_3"/>
<font color="Maroon" title="c21">n'</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">n</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> 
 <b>by </b><i><a class="txt" href="group_10.html#E24:54_3_1"><i><font color="Green" title="E57">A32</font></i></a>, <a class="ref" href="group_2.html#T57" title="GROUP_2:th.57">GROUP_2:57</a></i>;<br/><a NAME="E9:54_3_1_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c21">n'</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c24">q'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">n</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c23">q''</font>
 <b>by </b><i><a class="ref" href="group_2.html#T52" title="GROUP_2:th.52">GROUP_2:52</a></i>;<br/><a NAME="E10:54_3_1_3"/><b>then </b><i><font color="Green" title="E61">A38</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">n'</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c24">q'</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c21">n'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">n</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c23">q''</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c20">n</font>
 <b>by </b><i><a class="txt" href="group_10.html#E24:54_3_1"><i><font color="Green" title="E57">A32</font></i></a>, <a class="ref" href="group_2.html#T52" title="GROUP_2:th.52">GROUP_2:52</a></i>;<br/><font color="Maroon" title="c24">q'</font> <a href="group_3.html#K2" title="GROUP_3:func.2">|^</a> <font color="Maroon" title="c21">n'</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">n'</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c24">q'</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c21">n'</font>
<b>by </b><i><a class="ref" href="group_3.html#D2" title="GROUP_3:def.2">GROUP_3:def 2</a></i>
<br/>.= 
<font color="Maroon" title="c23">q''</font> <a href="group_3.html#K2" title="GROUP_3:func.2">|^</a> <font color="Maroon" title="c20">n</font>
<b>by </b><i><a class="txt" href="group_10.html#E10:54_3_1_3"><i><font color="Green" title="E61">A38</font></i></a>, <a class="ref" href="group_3.html#D2" title="GROUP_3:def.2">GROUP_3:def 2</a></i>
;<br/><a NAME="E12:54_3_1_3"/><b>then </b>
<font color="Maroon" title="c22">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <font color="Maroon" title="c15">Q'</font></span>)</span> <a href="group_3.html#K4" title="GROUP_3:func.4">|^</a> <font color="Maroon" title="c21">n'</font>
 <b>by </b><i><a class="txt" href="group_10.html#E5:54_3_1_3"><i><font color="Green" title="E59">A36</font></i></a>, <a class="ref" href="group_3.html#T50" title="GROUP_3:th.50">GROUP_3:50</a></i>;<br/><b>hence </b><a NAME="E13:54_3_1_3"/>
<font color="Maroon" title="c22">y</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 <span class="p1">(<span class="default"><font color="Maroon" title="c15">Q'</font> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Maroon" title="c21">n'</font></span>)</span>
 <b>by </b><i><a class="ref" href="group_3.html#D6" title="GROUP_3:def.6">GROUP_3:def 6</a></i>;<br/></div><b>end;</b></div><a NAME="E26:54_3_1"/><b>then </b><i><font color="Green" title="E59">A39</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="c15">Q'</font> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Maroon" title="c21">n'</font></span>)</span> <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"><font color="Maroon" title="c19">Q''</font> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Maroon" title="c20">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/><a NAME="E27:54_3_1"/>
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="c19">Q''</font> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Maroon" title="c20">n</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 <font color="Maroon" title="c17">N</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="E28:54_3_1"/><b>then </b><i><font color="Green" title="E60">A40</font></i>: 
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c19">Q''</font> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Maroon" title="c20">n</font></span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c19">Q''</font> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Maroon" title="c20">n</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c17">N</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c17">N</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T119" title="ZFMISC_1:th.119">ZFMISC_1:119</a></i>;<br/><i><font color="Green" title="E61">A41</font></i>: the <a href="group_1.html#U1" title="GROUP_1:sel.1">mult</a> of <font color="Maroon" title="c1">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 <span class="p1">(<span class="default"><font color="Maroon" title="c19">Q''</font> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Maroon" title="c20">n</font></span>)</span> = 
<span class="p1">(<span class="default">the <a href="group_1.html#U1" title="GROUP_1:sel.1">mult</a> of <font color="Maroon" title="c1">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="c17">N</font></span>)</span> <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 <span class="p1">(<span class="default"><font color="Maroon" title="c19">Q''</font> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Maroon" title="c20">n</font></span>)</span>
<b>by </b><i><a class="txt" href="group_10.html#E28:54_3_1"><i><font color="Green" title="E60">A40</font></i></a>, <a class="ref" href="relat_1.html#T103" title="RELAT_1:th.103">RELAT_1:103</a></i>
<br/>.= 
the <a href="group_1.html#U1" title="GROUP_1:sel.1">mult</a> of <font color="Maroon" title="c17">N</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 <span class="p1">(<span class="default"><font color="Maroon" title="c19">Q''</font> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Maroon" title="c20">n</font></span>)</span>
<b>by </b><i><a class="ref" href="group_2.html#D5" title="GROUP_2:def.5">GROUP_2:def 5</a></i>
;<br/>the <a href="group_1.html#U1" title="GROUP_1:sel.1">mult</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c15">Q'</font> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Maroon" title="c21">n'</font></span>)</span> = 
the <a href="group_1.html#U1" title="GROUP_1:sel.1">mult</a> of <font color="Maroon" title="c1">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 <span class="p1">(<span class="default"><font color="Maroon" title="c19">Q''</font> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Maroon" title="c20">n</font></span>)</span>
<b>by </b><i><a class="txt" href="group_10.html#E26:54_3_1"><i><font color="Green" title="E59">A39</font></i></a>, <a class="ref" href="group_2.html#D5" title="GROUP_2:def.5">GROUP_2:def 5</a></i>
<br/>.= 
the <a href="group_1.html#U1" title="GROUP_1:sel.1">mult</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c19">Q''</font> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Maroon" title="c20">n</font></span>)</span>
<b>by </b><i><a class="txt" href="group_10.html#E29:54_3_1"><i><font color="Green" title="E61">A41</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="E31:54_3_1"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">P'</font>
 <b>by </b><i><a class="txt" href="group_10.html#E21:54_3_1"><i><font color="Green" title="E56">A31</font></i></a>, <a class="txt" href="group_10.html#E4:54_3_1"><i><font color="Green" title="E48">A12</font></i></a>, <a class="txt" href="group_10.html#E7:54_3_1"><i><font color="Green" title="E49">A13</font></i></a>, <a class="txt" href="group_10.html#E9:54_3_1"><i><font color="Green" title="E50">A14</font></i></a>, <a class="txt" href="group_10.html#E25:54_3_1"><i><font color="Green" title="E58">A33</font></i></a>, <a class="txt" href="group_10.html#E24:54_3_1"><i><font color="Green" title="E57">A32</font></i></a>, <a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:54_3"/><i><font color="Green" title="E48">A42</font></i>: 
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">P'</font>
 ;<br/>

<a NAME="E3:54_3"/>
for <font color="Olive" title="b1">h</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">P</font> holds  <font color="Maroon" title="c11">P'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">T</font> <a href="group_10.html#K1" title="GROUP_10:func.1">^</a> <font color="Olive" title="b1">h</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">P'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_3_2"><b>proof </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="c3">P</font>;<br/>

<b>reconsider </b><font color="Maroon" title="c14">P1</font> = <font color="Maroon" title="c11">P'</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="group_10.html#K15" title="GROUP_10:func.15">the_sylow_p-subgroups_of_prime</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c1">G</font> ;<br/>
<b>consider </b><font color="Maroon" title="c15">P2</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="group_10.html#K15" title="GROUP_10:func.15">the_sylow_p-subgroups_of_prime</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c1">G</font>, <font color="Maroon" title="c16">H1</font>, <font color="Maroon" title="c17">H2</font> being  <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 <font color="Maroon" title="c1">G</font>, <font color="Maroon" title="c18">g</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E3:54_3_2"/><i><font color="Green" title="E49">A43</font></i>: 
( <font color="Maroon" title="c15">P2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="group_10.html#K16" title="GROUP_10:func.16">the_left_translation_of</a> <font color="Maroon" title="c13">h</font>,<font color="Maroon" title="c2">p</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">P1</font> &amp; <font color="Maroon" title="c14">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">H1</font> &amp; <font color="Maroon" title="c15">P2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">H2</font> &amp; <font color="Maroon" title="c13">h</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">g</font> &amp; <font color="Maroon" title="c17">H2</font> <a href="group_2.html#R1" title="GROUP_2:pred.1">=</a> <font color="Maroon" title="c16">H1</font> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Maroon" title="c18">g</font> )
 <b>by </b><i><a class="ref" href="group_10.html#D21" target="_self" title="GROUP_10:def.21">Def21</a></i>;<br/>
<a NAME="E4:54_3_2"/><i><font color="Green" title="E50">A44</font></i>: 
( <font color="Maroon" title="c11">P'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">P1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c10">T</font> <a href="group_10.html#K1" title="GROUP_10:func.1">^</a> <font color="Maroon" title="c13">h</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">P'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">P2</font> )
 
<b>by </b><i><a class="ref" href="group_10.html#D22" target="_self" title="GROUP_10:def.22">Def22</a>, <a class="txt" href="group_10.html#E3:54_3_2"><i><font color="Green" title="E49">A43</font></i></a></i>;<br/>
<a NAME="E5:54_3_2"/><i><font color="Green" title="E51">A45</font></i>: 
<font color="Maroon" title="c16">H1</font> is <a href="struct_0.html#V8" title="STRUCT_0:attr.8">finite</a>
 
<b>by </b><i><a class="ref" href="group_2.html#T48" title="GROUP_2:th.48">GROUP_2:48</a></i>;<br/>
<a NAME="E6:54_3_2"/><i><font color="Green" title="E52">A46</font></i>: 
<font color="Maroon" title="c16">H1</font> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Maroon" title="c18">g</font> is <a href="struct_0.html#V8" title="STRUCT_0:attr.8">finite</a>
 
<b>by </b><i><a class="ref" href="group_2.html#T48" title="GROUP_2:th.48">GROUP_2:48</a></i>;<br/>
<a NAME="E7:54_3_2"/><i><font color="Green" title="E53">A47</font></i>: 
 <a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <font color="Maroon" title="c16">H1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">H1</font> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Maroon" title="c18">g</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_3_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:54_3_2_1"/><i><font color="Green" title="E54">A48</font></i>: 
<font color="Maroon" title="c18">g</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c16">H1</font>
 
<b>by </b><i><a class="txt" href="group_10.html#E3:54_3_2"><i><font color="Green" title="E49">A43</font></i></a>, <a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_3_2_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c19">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:54_3_2_1_1"/><i><font color="Green" title="E55">A49</font></i>: 
<font color="Maroon" title="c19">y</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="c16">H1</font>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c20">h'</font> = <font color="Maroon" title="c19">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> ;<br/><a NAME="E3:54_3_2_1_1"/><i><font color="Green" title="E56">A50</font></i>: 
<font color="Maroon" title="c20">h'</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c16">H1</font>
 <b>by </b><i><a class="txt" href="group_10.html#E1:54_3_2_1_1"><i><font color="Green" title="E55">A49</font></i></a>, <a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/><a NAME="E4:54_3_2_1_1"/>
 ex <font color="Olive" title="b1">h</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> st <br/>( <font color="Maroon" title="c19">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">h</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c18">g</font> &amp; <font color="Olive" title="b1">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">g</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c16">H1</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_3_2_1_1_1"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon" title="c21">h''</font> = <font color="Maroon" title="c20">h'</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">g</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span>;<br/>
<a NAME="E1:54_3_2_1_1_1"/>
<font color="Maroon" title="c18">g</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a>  <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c16">H1</font>
 
<b>by </b><i><a class="txt" href="group_10.html#E1:54_3_2_1"><i><font color="Green" title="E54">A48</font></i></a>, <a class="ref" href="group_2.html#T60" title="GROUP_2:th.60">GROUP_2:60</a></i>;<br/>
<a NAME="E2:54_3_2_1_1_1"/><b>then </b><i><font color="Green" title="E57">A51</font></i>: 
<font color="Maroon" title="c20">h'</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">g</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c16">H1</font>
 
<b>by </b><i><a class="txt" href="group_10.html#E3:54_3_2_1_1"><i><font color="Green" title="E56">A50</font></i></a>, <a class="ref" href="group_2.html#T59" title="GROUP_2:th.59">GROUP_2:59</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c20">h'</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">g</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span>
;<br/>

<b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c20">h'</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">g</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c18">g</font> = 
<font color="Maroon" title="c20">h'</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">g</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c18">g</font></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">GROUP_1:def 4</a></i>
<br/>.= 
<font color="Maroon" title="c20">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> <font color="Maroon" title="c1">G</font></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D6" title="GROUP_1:def.6">GROUP_1:def 6</a></i>
<br/>.= 
<font color="Maroon" title="c19">y</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/>

<a NAME="E4:54_3_2_1_1_1"/>
 ex <font color="Olive" title="b1">h</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> st <br/>( <font color="Maroon" title="c20">h'</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">g</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">g</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b1">h</font> &amp; <font color="Olive" title="b1">h</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c16">H1</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_3_2_1_1_1_2"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon" title="c22">h'''</font> = <font color="Maroon" title="c18">g</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">h'</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">g</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span></span>)</span>;<br/>
<b>take </b>
<font color="Maroon" title="c18">g</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">h'</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">g</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span></span>)</span>
;<br/>

<b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c18">g</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">g</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">h'</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c18">g</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">g</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c18">g</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">h'</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">g</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">GROUP_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">h'</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">g</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D6" title="GROUP_1:def.6">GROUP_1:def 6</a></i>
<br/>.= 
<font color="Maroon" title="c20">h'</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">g</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span>
<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>thus </b><a NAME="E2:54_3_2_1_1_1_2"/>
<font color="Maroon" title="c18">g</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">h'</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">g</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span></span>)</span> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c16">H1</font>
 <b>by </b><i><a class="txt" href="group_10.html#E1:54_3_2_1"><i><font color="Green" title="E54">A48</font></i></a>, <a class="txt" href="group_10.html#E2:54_3_2_1_1_1"><i><font color="Green" title="E57">A51</font></i></a>, <a class="ref" href="group_2.html#T59" title="GROUP_2:th.59">GROUP_2:59</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:54_3_2_1_1_1"/>
<font color="Maroon" title="c20">h'</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">g</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">g</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c16">H1</font>
 <b>by </b><i><a class="ref" href="group_2.html#T125" title="GROUP_2:th.125">GROUP_2:125</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E5:54_3_2_1_1"/>
<font color="Maroon" title="c19">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">g</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c16">H1</font></span>)</span> <a href="group_2.html#K4" title="GROUP_2:func.4">*</a> <font color="Maroon" title="c18">g</font>
 <b>by </b><i><a class="ref" href="group_2.html#T34" title="GROUP_2:th.34">GROUP_2:34</a></i>;<br/></div><b>end;</b></div>
<a NAME="E3:54_3_2_1"/><b>then </b>
 <a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <font color="Maroon" title="c16">H1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">g</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c16">H1</font></span>)</span> <a href="group_2.html#K4" title="GROUP_2:func.4">*</a> <font color="Maroon" title="c18">g</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<b>hence </b><a NAME="E4:54_3_2_1"/>
 <a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <font color="Maroon" title="c16">H1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">H1</font> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Maroon" title="c18">g</font></span>)</span>
 <b>by </b><i><a class="ref" href="group_3.html#T71" title="GROUP_3:th.71">GROUP_3:71</a></i>;<br/>


</div><b>end;</b></div>
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <font color="Maroon" title="c16">H1</font></span>)</span> = 
 <a href="group_1.html#K9" title="GROUP_1:func.9">ord</a> <font color="Maroon" title="c16">H1</font>
<b>by </b><i><a class="txt" href="group_10.html#E5:54_3_2"><i><font color="Green" title="E51">A45</font></i></a>, <a class="ref" href="group_1.html#D15" title="GROUP_1:def.15">GROUP_1:def 15</a></i>
<br/>.= 
 <a href="group_1.html#K9" title="GROUP_1:func.9">ord</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">H1</font> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Maroon" title="c18">g</font></span>)</span>
<b>by </b><i><a class="txt" href="group_10.html#E5:54_3_2"><i><font color="Green" title="E51">A45</font></i></a>, <a class="ref" href="group_3.html#T79" title="GROUP_3:th.79">GROUP_3:79</a></i>
<br/>.= 
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">H1</font> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Maroon" title="c18">g</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="group_10.html#E6:54_3_2"><i><font color="Green" title="E52">A46</font></i></a>, <a class="ref" href="group_1.html#D15" title="GROUP_1:def.15">GROUP_1:def 15</a></i>
;<br/>
<a NAME="E9:54_3_2"/><b>then </b>
 <a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <font color="Maroon" title="c16">H1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">H1</font> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Maroon" title="c18">g</font></span>)</span>
 
<b>by </b><i><a class="txt" href="group_10.html#E7:54_3_2"><i><font color="Green" title="E53">A47</font></i></a>, <a class="ref" href="card_fin.html#T1" title="CARD_FIN:th.1">CARD_FIN:1</a></i>;<br/>
<b>hence </b><a NAME="E10:54_3_2"/>
<font color="Maroon" title="c11">P'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">T</font> <a href="group_10.html#K1" title="GROUP_10:func.1">^</a> <font color="Maroon" title="c13">h</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">P'</font>
 <b>by </b><i><a class="txt" href="group_10.html#E3:54_3_2"><i><font color="Green" title="E49">A43</font></i></a>, <a class="txt" href="group_10.html#E4:54_3_2"><i><font color="Green" title="E50">A44</font></i></a>, <a class="ref" href="group_2.html#T68" title="GROUP_2:th.68">GROUP_2:68</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:54_3"/><b>then </b>
<font color="Maroon" title="c11">P'</font> <a href="group_10.html#R1" title="GROUP_10:pred.1">is_fixed_under</a> <font color="Maroon" title="c10">T</font>
 
<b>by </b><i><a class="ref" href="group_10.html#D12" target="_self" title="GROUP_10:def.12">Def12</a></i>;<br/>
<a NAME="E5:54_3"/><b>then </b>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x'</font> where <font color="Olive" title="b1">x'</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="group_10.html#K15" title="GROUP_10:func.15">the_sylow_p-subgroups_of_prime</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c1">G</font> : <font color="Olive" title="b1">x'</font> <a href="group_10.html#R1" title="GROUP_10:pred.1">is_fixed_under</a> <font color="Maroon" title="c10">T</font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="group_10.html#E2:54_3"><i><font color="Green" title="E48">A42</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:54_3"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_10.html#K12" title="GROUP_10:func.12">the_fixed_points_of</a> <font color="Maroon" title="c10">T</font>
 <b>by </b><i><a class="ref" href="group_10.html#D13" target="_self" title="GROUP_10:def.13">Def13</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E23:54"/><b>then </b><i><font color="Green" title="E48">A52</font></i>: 
 <a href="group_10.html#K12" title="GROUP_10:func.12">the_fixed_points_of</a> <font color="Maroon" title="c10">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c11">P'</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E24:54"/>
<font color="Maroon" title="c2">p</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E25:54"/><b>then </b><i><font color="Green" title="E49">A53</font></i>: 
<font color="Maroon" title="c2">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 
<b>by </b><i><a class="ref" href="nat_4.html#T12" title="NAT_4:th.12">NAT_4:12</a></i>;<br/>
<span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p2">(<span class="default"><a href="group_10.html#K15" title="GROUP_10:func.15">the_sylow_p-subgroups_of_prime</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c1">G</font></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">p</font> = 
<span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p2">(<span class="default"><a href="group_10.html#K12" title="GROUP_10:func.12">the_fixed_points_of</a> <font color="Maroon" title="c10">T</font></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">p</font>
<b>by </b><i><a class="txt" href="group_10.html#E5:54"><i><font color="Green" title="E41">A2</font></i></a>, <a class="ref" href="group_10.html#T11" target="_self" title="GROUP_10:th.11">Th11</a></i>
<br/>.= 
1 <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">p</font>
<b>by </b><i><a class="txt" href="group_10.html#E23:54"><i><font color="Green" title="E48">A52</font></i></a>, <a class="ref" href="card_1.html#T79" title="CARD_1:th.79">CARD_1:79</a></i>
;<br/>
<b>hence </b><a NAME="E27:54"/>
<span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p2">(<span class="default"><a href="group_10.html#K15" title="GROUP_10:func.15">the_sylow_p-subgroups_of_prime</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c1">G</font></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="group_10.html#E25:54"><i><font color="Green" title="E49">A53</font></i></a>, <a class="ref" href="nat_d.html#T14" title="NAT_D:th.14">NAT_D:14</a></i>;<br/>

<b>set </b><font color="Maroon" title="c12">G'</font> =  <a href="group_2.html#K6" title="GROUP_2:func.6">(Omega).</a> <font color="Maroon" title="c1">G</font>;<br/>
<b>set </b><font color="Maroon" title="c13">T'</font> =  <a href="group_10.html#K17" title="GROUP_10:func.17">the_left_operation_of</a> <span class="p1">(<span class="default"><a href="group_2.html#K6" title="GROUP_2:func.6">(Omega).</a> <font color="Maroon" title="c1">G</font></span>)</span>,<font color="Maroon" title="c2">p</font>;<br/>
<b>set </b><font color="Maroon" title="c14">K</font> =  <a href="group_10.html#K11" title="GROUP_10:func.11">the_strict_stabilizer_of</a> <font color="Maroon" title="c11">P'</font>,<span class="p1">(<span class="default"><a href="group_10.html#K17" title="GROUP_10:func.17">the_left_operation_of</a> <span class="p2">(<span class="default"><a href="group_2.html#K6" title="GROUP_2:func.6">(Omega).</a> <font color="Maroon" title="c1">G</font></span>)</span>,<font color="Maroon" title="c2">p</font></span>)</span>;<br/>
<a NAME="E28:54"/><i><font color="Green" title="E50">A54</font></i>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="group_10.html#K13" title="GROUP_10:func.13">the_orbit_of</a> <font color="Maroon" title="c11">P'</font>,<span class="p2">(<span class="default"><a href="group_10.html#K17" title="GROUP_10:func.17">the_left_operation_of</a> <span class="p3">(<span class="default"><a href="group_2.html#K6" title="GROUP_2:func.6">(Omega).</a> <font color="Maroon" title="c1">G</font></span>)</span>,<font color="Maroon" title="c2">p</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_2.html#K16" title="GROUP_2:func.16">Index</a> <span class="p1">(<span class="default"><a href="group_10.html#K11" title="GROUP_10:func.11">the_strict_stabilizer_of</a> <font color="Maroon" title="c11">P'</font>,<span class="p2">(<span class="default"><a href="group_10.html#K17" title="GROUP_10:func.17">the_left_operation_of</a> <span class="p3">(<span class="default"><a href="group_2.html#K6" title="GROUP_2:func.6">(Omega).</a> <font color="Maroon" title="c1">G</font></span>)</span>,<font color="Maroon" title="c2">p</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="group_10.html#T10" target="_self" title="GROUP_10:th.10">Th10</a></i>;<br/>
<div><i><font color="Green" title="E51">A55</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="54_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c15">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>thus </b><a NAME="E1:54_5"/>
( <font color="Maroon" title="c15">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_10.html#K13" title="GROUP_10:func.13">the_orbit_of</a> <font color="Maroon" title="c11">P'</font>,<span class="p1">(<span class="default"><a href="group_10.html#K17" title="GROUP_10:func.17">the_left_operation_of</a> <span class="p2">(<span class="default"><a href="group_2.html#K6" title="GROUP_2:func.6">(Omega).</a> <font color="Maroon" title="c1">G</font></span>)</span>,<font color="Maroon" title="c2">p</font></span>)</span> implies <font color="Maroon" title="c15">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_10.html#K15" title="GROUP_10:func.15">the_sylow_p-subgroups_of_prime</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c1">G</font> )
 ;<br/><b>assume </b><a NAME="E2:54_5"/>
<font color="Maroon" title="c15">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_10.html#K15" title="GROUP_10:func.15">the_sylow_p-subgroups_of_prime</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c1">G</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c16">Q</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="group_3.html#K1" title="GROUP_3:func.1">Subgroups</a> <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E4:54_5"/><i><font color="Green" title="E52">A56</font></i>: 
( <font color="Maroon" title="c15">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">Q</font> &amp;  ex <font color="Olive" title="b1">P</font> being <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 <font color="Maroon" title="c1">G</font> st <br/>( <font color="Olive" title="b1">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">Q</font> &amp; <font color="Olive" title="b1">P</font> <a href="group_10.html#R5" title="GROUP_10:pred.5">is_Sylow_p-subgroup_of_prime</a> <font color="Maroon" title="c2">p</font> ) )
 ;<br/><b>consider </b><font color="Maroon" title="c17">Q'</font> being  <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 <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E6:54_5"/><i><font color="Green" title="E53">A57</font></i>: 
( <font color="Maroon" title="c17">Q'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">Q</font> &amp; <font color="Maroon" title="c17">Q'</font> <a href="group_10.html#R5" title="GROUP_10:pred.5">is_Sylow_p-subgroup_of_prime</a> <font color="Maroon" title="c2">p</font> )
 <b>by </b><i><a class="txt" href="group_10.html#E4:54_5"><i><font color="Green" title="E52">A56</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c18">P''</font> = <font color="Maroon" title="c11">P'</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 <font color="Maroon" title="c1">G</font> ;<br/><a NAME="E8:54_5"/>
<font color="Maroon" title="c18">P''</font>,<font color="Maroon" title="c17">Q'</font> <a href="group_3.html#R6" title="GROUP_3:pred.6">are_conjugated</a> 
 <b>by </b><i><a class="txt" href="group_10.html#E18:54"><i><font color="Green" title="E46">A10</font></i></a>, <a class="txt" href="group_10.html#E6:54_5"><i><font color="Green" title="E53">A57</font></i></a>, <a class="ref" href="group_10.html#T14" target="_self" title="GROUP_10:th.14">Th14</a></i>;<br/><b>then consider </b><font color="Maroon" title="c19">g</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E10:54_5"/><i><font color="Green" title="E54">A58</font></i>: 
<font color="Maroon" title="c17">Q'</font> <a href="group_2.html#R1" title="GROUP_2:pred.1">=</a> <font color="Maroon" title="c18">P''</font> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Maroon" title="c19">g</font>
 <b>by </b><i><a class="ref" href="group_3.html#D11" title="GROUP_3:def.11">GROUP_3:def 11</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c20">g'</font> = <font color="Maroon" title="c19">g</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a>  as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="group_2.html#K6" title="GROUP_2:func.6">(Omega).</a> <font color="Maroon" title="c1">G</font></span>)</span> ;<br/><b>reconsider </b><font color="Maroon" title="c21">P1</font> = <font color="Maroon" title="c11">P'</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="group_10.html#K15" title="GROUP_10:func.15">the_sylow_p-subgroups_of_prime</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c1">G</font> ;<br/><b>consider </b><font color="Maroon" title="c22">P2</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="group_10.html#K15" title="GROUP_10:func.15">the_sylow_p-subgroups_of_prime</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c1">G</font>, <font color="Maroon" title="c23">H1</font>, <font color="Maroon" title="c24">H2</font> being  <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 <font color="Maroon" title="c1">G</font>, <font color="Maroon" title="c25">g'''</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E14:54_5"/><i><font color="Green" title="E55">A59</font></i>: 
( <font color="Maroon" title="c22">P2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="group_10.html#K16" title="GROUP_10:func.16">the_left_translation_of</a> <font color="Maroon" title="c20">g'</font>,<font color="Maroon" title="c2">p</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">P1</font> &amp; <font color="Maroon" title="c21">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">H1</font> &amp; <font color="Maroon" title="c22">P2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">H2</font> &amp; <font color="Maroon" title="c20">g'</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">g'''</font> &amp; <font color="Maroon" title="c24">H2</font> <a href="group_2.html#R1" title="GROUP_2:pred.1">=</a> <font color="Maroon" title="c23">H1</font> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Maroon" title="c25">g'''</font> )
 <b>by </b><i><a class="ref" href="group_10.html#D21" target="_self" title="GROUP_10:def.21">Def21</a></i>;<br/><font color="Maroon" title="c20">g'</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a>  = 
<span class="p1">(<span class="default"><font color="Maroon" title="c19">g</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> 
<b>by </b><i><a class="ref" href="group_2.html#T57" title="GROUP_2:th.57">GROUP_2:57</a></i>
<br/>.= 
<font color="Maroon" title="c19">g</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="E16:54_5"/><b>then </b><i><font color="Green" title="E56">A60</font></i>: 
<font color="Maroon" title="c17">Q'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_10.html#K17" title="GROUP_10:func.17">the_left_operation_of</a> <span class="p3">(<span class="default"><a href="group_2.html#K6" title="GROUP_2:func.6">(Omega).</a> <font color="Maroon" title="c1">G</font></span>)</span>,<font color="Maroon" title="c2">p</font></span>)</span> <a href="group_10.html#K1" title="GROUP_10:func.1">^</a> <font color="Maroon" title="c20">g'</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">P'</font>
 <b>by </b><i><a class="txt" href="group_10.html#E14:54_5"><i><font color="Green" title="E55">A59</font></i></a>, <a class="ref" href="group_10.html#D22" target="_self" title="GROUP_10:def.22">Def22</a>, <a class="txt" href="group_10.html#E10:54_5"><i><font color="Green" title="E54">A58</font></i></a></i>;<br/><a NAME="E17:54_5"/>
<font color="Maroon" title="c17">Q'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_10.html#K15" title="GROUP_10:func.15">the_sylow_p-subgroups_of_prime</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="txt" href="group_10.html#E6:54_5"><i><font color="Green" title="E53">A57</font></i></a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c26">Q''</font> = <font color="Maroon" title="c17">Q'</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="group_10.html#K15" title="GROUP_10:func.15">the_sylow_p-subgroups_of_prime</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c1">G</font> ;<br/><a NAME="E19:54_5"/>
( <font color="Maroon" title="c15">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">Q''</font> &amp; <font color="Maroon" title="c11">P'</font>,<font color="Maroon" title="c26">Q''</font> <a href="group_10.html#R2" title="GROUP_10:pred.2">are_conjugated_under</a>  <a href="group_10.html#K17" title="GROUP_10:func.17">the_left_operation_of</a> <span class="p1">(<span class="default"><a href="group_2.html#K6" title="GROUP_2:func.6">(Omega).</a> <font color="Maroon" title="c1">G</font></span>)</span>,<font color="Maroon" title="c2">p</font> )
 <b>by </b><i><a class="txt" href="group_10.html#E4:54_5"><i><font color="Green" title="E52">A56</font></i></a>, <a class="txt" href="group_10.html#E6:54_5"><i><font color="Green" title="E53">A57</font></i></a>, <a class="txt" href="group_10.html#E16:54_5"><i><font color="Green" title="E56">A60</font></i></a>, <a class="ref" href="group_10.html#D14" target="_self" title="GROUP_10:def.14">Def14</a></i>;<br/><b>hence </b><a NAME="E20:54_5"/>
<font color="Maroon" title="c15">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_10.html#K13" title="GROUP_10:func.13">the_orbit_of</a> <font color="Maroon" title="c11">P'</font>,<span class="p1">(<span class="default"><a href="group_10.html#K17" title="GROUP_10:func.17">the_left_operation_of</a> <span class="p2">(<span class="default"><a href="group_2.html#K6" title="GROUP_2:func.6">(Omega).</a> <font color="Maroon" title="c1">G</font></span>)</span>,<font color="Maroon" title="c2">p</font></span>)</span>
 ;<br/></div><b>end;</b></div>
<a NAME="E30:54"/><i><font color="Green" title="E52">A61</font></i>: 
 <a href="group_2.html#K6" title="GROUP_2:func.6">(Omega).</a> <font color="Maroon" title="c1">G</font> is <a href="struct_0.html#V8" title="STRUCT_0:attr.8">finite</a>
 
<b>by </b><i><a class="ref" href="group_2.html#T48" title="GROUP_2:th.48">GROUP_2:48</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c15">K</font> =  <a href="group_10.html#K11" title="GROUP_10:func.11">the_strict_stabilizer_of</a> <font color="Maroon" title="c11">P'</font>,<span class="p1">(<span class="default"><a href="group_10.html#K17" title="GROUP_10:func.17">the_left_operation_of</a> <span class="p2">(<span class="default"><a href="group_2.html#K6" title="GROUP_2:func.6">(Omega).</a> <font color="Maroon" title="c1">G</font></span>)</span>,<font color="Maroon" title="c2">p</font></span>)</span> as    <a href="group_6.html#M1" title="GROUP_6:mode.1">Subgroup</a> of  <a href="group_2.html#K6" title="GROUP_2:func.6">(Omega).</a> <font color="Maroon" title="c1">G</font> ;<br/>
<a NAME="E32:54"/>
 <a href="group_2.html#K14" title="GROUP_2:func.14">Left_Cosets</a> <font color="Maroon" title="c15">K</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 
<b>by </b><i><a class="txt" href="group_10.html#E30:54"><i><font color="Green" title="E52">A61</font></i></a>, <a class="ref" href="group_2.html#T164" title="GROUP_2:th.164">GROUP_2:164</a></i>;<br/>
<a NAME="E33:54"/><b>then </b><i><font color="Green" title="E53">A62</font></i>: 
 ex <font color="Olive" title="b1">B</font> being <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_2.html#K14" title="GROUP_2:func.14">Left_Cosets</a> <font color="Maroon" title="c15">K</font> &amp;  <a href="group_2.html#K17" title="GROUP_2:func.17">index</a> <font color="Maroon" title="c15">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Olive" title="b1">B</font> )
 
<b>by </b><i><a class="ref" href="group_2.html#T176" title="GROUP_2:th.176">GROUP_2:176</a></i>;<br/>
<a NAME="E34:54"/><i><font color="Green" title="E54">A63</font></i>: 
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="group_10.html#K15" title="GROUP_10:func.15">the_sylow_p-subgroups_of_prime</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c1">G</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_2.html#K17" title="GROUP_2:func.17">index</a> <font color="Maroon" title="c15">K</font>
 
<b>by </b><i><a class="txt" href="group_10.html#E33:54"><i><font color="Green" title="E53">A62</font></i></a>, <a class="txt" href="group_10.html#E28:54"><i><font color="Green" title="E50">A54</font></i></a>, <a class="txt" href="group_10.html#E29:54"><i><font color="Green" title="E51">A55</font></i></a>, <a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/>
 <a href="group_1.html#K9" title="GROUP_1:func.9">ord</a> <font color="Maroon" title="c1">G</font> = 
<span class="p1">(<span class="default"><a href="group_1.html#K9" title="GROUP_1:func.9">ord</a> <span class="p2">(<span class="default"><a href="group_2.html#K6" title="GROUP_2:func.6">(Omega).</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><a href="group_2.html#K17" title="GROUP_2:func.17">index</a> <span class="p2">(<span class="default"><a href="group_2.html#K6" title="GROUP_2:func.6">(Omega).</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_2.html#T177" title="GROUP_2:th.177">GROUP_2:177</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_1.html#K9" title="GROUP_1:func.9">ord</a> <span class="p2">(<span class="default"><a href="group_2.html#K6" title="GROUP_2:func.6">(Omega).</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> 1
<b>by </b><i><a class="ref" href="group_2.html#T180" title="GROUP_2:th.180">GROUP_2:180</a></i>
;<br/>
<a NAME="E36:54"/><b>then </b>
 <a href="group_1.html#K9" title="GROUP_1:func.9">ord</a> <font color="Maroon" title="c1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="group_1.html#K9" title="GROUP_1:func.9">ord</a> <font color="Maroon" title="c15">K</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><a href="group_2.html#K17" title="GROUP_2:func.17">index</a> <font color="Maroon" title="c15">K</font></span>)</span>
 
<b>by </b><i><a class="txt" href="group_10.html#E30:54"><i><font color="Green" title="E52">A61</font></i></a>, <a class="ref" href="group_2.html#T177" title="GROUP_2:th.177">GROUP_2:177</a></i>;<br/>
<b>hence </b><a NAME="E37:54"/>
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="group_10.html#K15" title="GROUP_10:func.15">the_sylow_p-subgroups_of_prime</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c1">G</font></span>)</span> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a>  <a href="group_1.html#K9" title="GROUP_1:func.9">ord</a> <font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="txt" href="group_10.html#E34:54"><i><font color="Green" title="E54">A63</font></i></a>, <a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">NAT_D:def 3</a></i>;<br/>


</div>
