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

<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="E2:50_1_1"/><i><font color="Green" title="E37">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">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="E3:50_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> &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="c5">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="c6">H'</font> = <font color="Maroon" title="c5">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="E6:50_1_1"/>
( <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="txt" href="group_10.html#E2:50_1_1"><i><font color="Green" title="E37">A1</font></i></a>, <a class="ref" href="group_10.html#D19" target="_self" title="GROUP_10:def.19">Def19</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c7">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="E8:50_1_1"/><i><font color="Green" title="E38">A2</font></i>: 
( <font color="Maroon" title="c3">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">H</font> &amp; <font color="Maroon" title="c7">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/>
<a NAME="E9:50_1_1"/><i><font color="Green" title="E39">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#E8:50_1_1"><i><font color="Green" title="E38">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="E10:50_1_1"/>
<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="E11:50_1_1"/><b>then </b><i><font color="Green" title="E40">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="E12:50_1_1"/>
<font color="Maroon" title="c5">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="E13:50_1_1"/><b>then </b><i><font color="Green" title="E41">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="c5">P'</font> &amp;  <a href="group_1.html#K9" title="GROUP_1:func.9">ord</a> <font color="Maroon" title="c5">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="c5">P'</font></span>)</span>
<b>by </b><i><a class="txt" href="group_10.html#E11:50_1_1"><i><font color="Green" title="E40">A4</font></i></a>, <a class="txt" href="group_10.html#E13:50_1_1"><i><font color="Green" title="E41">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="E15:50_1_1"/><b>then </b><i><font color="Green" title="E42">A6</font></i>: 
 <a href="group_2.html#K17" title="GROUP_2:func.17">index</a> <font color="Maroon" title="c5">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#E9:50_1_1"><i><font color="Green" title="E39">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="E16:50_1_1"/>
 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="c5">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="50_1_1_2"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon" title="c8">H''</font> =  <a href="group_2.html#K6" title="GROUP_2:func.6">(Omega).</a> <font color="Maroon" title="c7">H</font>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">H''</font> =  <a href="group_2.html#K6" title="GROUP_2:func.6">(Omega).</a> <font color="Maroon" title="c7">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="c9">H''</font>
;<br/>

<b>thus </b><a NAME="E2:50_1_1_2"/>
<font color="Maroon" title="c5">P'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">H''</font>
 <b>by </b><i><a class="txt" href="group_10.html#E8:50_1_1"><i><font color="Green" title="E38">A2</font></i></a></i>;<br/>

<a NAME="E3:50_1_1_2"/><i><font color="Green" title="E43">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="c7">H</font> &amp;  <a href="group_1.html#K9" title="GROUP_1:func.9">ord</a> <font color="Maroon" title="c7">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="E4:50_1_1_2"/><i><font color="Green" title="E44">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="c9">H''</font> &amp;  <a href="group_1.html#K9" title="GROUP_1:func.9">ord</a> <font color="Maroon" title="c9">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="c10">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="E6:50_1_1_2"/><i><font color="Green" title="E45">A9</font></i>: 
 <a href="group_1.html#K9" title="GROUP_1:func.9">ord</a> <font color="Maroon" title="c7">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="c10">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#E8:50_1_1"><i><font color="Green" title="E38">A2</font></i></a></i>;<br/>
<b>thus </b><a NAME="E7:50_1_1_2"/>
<font color="Maroon" title="c9">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#E6:50_1_1_2"><i><font color="Green" title="E45">A9</font></i></a>, <a class="txt" href="group_10.html#E3:50_1_1_2"><i><font color="Green" title="E43">A7</font></i></a>, <a class="txt" href="group_10.html#E4:50_1_1_2"><i><font color="Green" title="E44">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="E17:50_1_1"/><b>then </b>
( <font color="Maroon" title="c5">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="c5">P'</font> )
 
<b>by </b><i><a class="txt" href="group_10.html#E2:50_1_1"><i><font color="Green" title="E37">A1</font></i></a>, <a class="ref" href="group_10.html#D19" target="_self" title="GROUP_10:def.19">Def19</a>, <a class="txt" href="group_10.html#E15:50_1_1"><i><font color="Green" title="E42">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="E18:50_1_1"/><b>then </b>
( <font color="Maroon" title="c6">H'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P'</font> &amp; <font color="Maroon" title="c5">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="E19:50_1_1"/><b>then </b><i><font color="Green" title="E43">A10</font></i>: 
<font color="Maroon" title="c6">H'</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/>
<a NAME="E20:50_1_1"/>
 <a href="group_3.html#K1" title="GROUP_3:func.1">Subgroups</a> <font color="Maroon" title="c1">G</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 
<b>by </b><i><a class="ref" href="group_3.html#T19" title="GROUP_3:th.19">GROUP_3:19</a></i>;<br/>
<b>hence </b><a NAME="E21:50_1_1"/>
( not  <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> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp;  <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> 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#E19:50_1_1"><i><font color="Green" title="E43">A10</font></i></a>, <a class="ref" href="finset_1.html#T13" title="FINSET_1:th.13">FINSET_1:13</a></i>;<br/>


</div>
