<?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>reconsider </b><font color="Maroon" title="c3">p'</font> = <font color="Maroon" title="c2">p</font> as  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">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="E2:48"/><i><font color="Green" title="E37">A1</font></i>: 
for <font color="Olive" title="b1">H</font>, <font color="Olive" title="b2">P</font> being   <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font>  st <font color="Olive" title="b1">H</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; <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> holds <br/> ex <font color="Olive" title="b3">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  <a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <font color="Olive" title="b1">H</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="Olive" title="b2">P</font> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Olive" title="b3">g</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">H</font>, <font color="Maroon" title="c5">P</font> be    <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font>;<br/>

<b>assume </b><a NAME="E1:48_1"/><i><font color="Green" title="E38">A2</font></i>: 
<font color="Maroon" title="c4">H</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>
 ;<br/>

<b>assume </b><a NAME="E2:48_1"/><i><font color="Green" title="E39">A3</font></i>: 
<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>
 ;<br/>

<b>set </b><font color="Maroon" title="c6">H'</font> = <font color="Maroon" title="c4">H</font>;<br/>
<b>set </b><font color="Maroon" title="c7">E</font> =  <a href="group_2.html#K14" title="GROUP_2:func.14">Left_Cosets</a> <font color="Maroon" title="c5">P</font>;<br/>
<b>set </b><font color="Maroon" title="c8">T</font> =  <a href="group_10.html#K9" title="GROUP_10:func.9">the_left_operation_of</a> <font color="Maroon" title="c4">H</font>,<font color="Maroon" title="c5">P</font>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">H'</font> = <font color="Maroon" title="c4">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/>
<a NAME="E4:48_1"/><i><font color="Green" title="E40">A4</font></i>: 
 ex <font color="Olive" title="b1">I</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="c4">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">I</font> &amp; <font color="Olive" title="b1">I</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#E1:48_1"><i><font color="Green" title="E38">A2</font></i></a>, <a class="ref" href="group_10.html#D18" target="_self" title="GROUP_10:def.18">Def18</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">E</font> =  <a href="group_2.html#K14" title="GROUP_2:func.14">Left_Cosets</a> <font color="Maroon" title="c5">P</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <b>by </b><i><a class="ref" href="group_2.html#T164" title="GROUP_2:th.164">GROUP_2:164</a>, <a class="ref" href="group_2.html#T165" title="GROUP_2:th.165">GROUP_2:165</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">T</font> =  <a href="group_10.html#K9" title="GROUP_10:func.9">the_left_operation_of</a> <font color="Maroon" title="c4">H</font>,<font color="Maroon" title="c5">P</font> as   <a href="group_10.html#NM1" title="GROUP_10:NM.1">LeftOperation</a> of <font color="Maroon" title="c9">H'</font>,<font color="Maroon" title="c10">E</font> ;<br/>
<a NAME="E7:48_1"/><i><font color="Green" title="E41">A5</font></i>: 
<span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">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="c11">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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c10">E</font></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#E4:48_1"><i><font color="Green" title="E40">A4</font></i></a>, <a class="ref" href="group_10.html#T11" target="_self" title="GROUP_10:th.11">Th11</a></i>;<br/>
<a NAME="E8:48_1"/><i><font color="Green" title="E42">A6</font></i>: 
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#E2:48_1"><i><font color="Green" title="E39">A3</font></i></a>, <a class="ref" href="group_10.html#D19" target="_self" title="GROUP_10:def.19">Def19</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:48_1_1"/><i><font color="Green" title="E43">A7</font></i>: 
<span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c10">E</font></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>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 ;<br/><a NAME="E2:48_1_1"/>
(  ex <font color="Olive" title="b1">t</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>(  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c10">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b1">t</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  &amp;  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">p</font> ) or (  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  &amp; <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  ) )
 <b>by </b><i><a class="txt" href="group_10.html#E1:48_1_1"><i><font color="Green" title="E43">A7</font></i></a>, <a class="ref" href="nat_d.html#D2" title="NAT_D:def.2">NAT_D:def 2</a></i>;<br/><a NAME="E3:48_1_1"/><b>then </b>
<font color="Maroon" title="c3">p'</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c10">E</font>
 <b>by </b><i><a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">NAT_D:def 3</a></i>;<br/><b>hence </b><a NAME="E4:48_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="group_10.html#E8:48_1"><i><font color="Green" title="E42">A6</font></i></a>, <a class="ref" href="group_2.html#D18" title="GROUP_2:def.18">GROUP_2:def 18</a></i>;<br/></div><b>end;</b></div>
<a NAME="E10:48_1"/><b>then </b>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="group_10.html#K12" title="GROUP_10:func.12">the_fixed_points_of</a> <font color="Maroon" title="c11">T</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 
<b>by </b><i><a class="txt" href="group_10.html#E7:48_1"><i><font color="Green" title="E41">A5</font></i></a>, <a class="ref" href="nat_d.html#T26" title="NAT_D:th.26">NAT_D:26</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c12">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E12:48_1"/><i><font color="Green" title="E43">A8</font></i>: 
<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="c11">T</font>
 <b>by </b><i><a class="ref" href="card_1.html#T78" title="CARD_1:th.78">CARD_1:78</a>, <a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/>
<a NAME="E13:48_1"/>
<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#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c10">E</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="c11">T</font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="group_10.html#E12:48_1"><i><font color="Green" title="E43">A8</font></i></a>, <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">x'</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c10">E</font><b> such that </b><br/><a NAME="E15:48_1"/><i><font color="Green" title="E44">A9</font></i>: 
( <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">x'</font> &amp; <font color="Maroon" title="c13">x'</font> <a href="group_10.html#R1" title="GROUP_10:pred.1">is_fixed_under</a> <font color="Maroon" title="c11">T</font> )
 ;<br/>
<a NAME="E16:48_1"/>
<font color="Maroon" title="c13">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_2.html#K14" title="GROUP_2:func.14">Left_Cosets</a> <font color="Maroon" title="c5">P</font>
 
;<br/>
<b>then consider </b><font color="Maroon" title="c14">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="E18:48_1"/><i><font color="Green" title="E45">A10</font></i>: 
<font color="Maroon" title="c13">x'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">g'</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="ref" href="group_2.html#D15" title="GROUP_2:def.15">GROUP_2:def 15</a></i>;<br/>
<b>set </b><font color="Maroon" title="c15">g</font> = <font color="Maroon" title="c14">g'</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> ;<br/>
<b>take </b>
<font color="Maroon" title="c14">g'</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> 
;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c16">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:48_1_2"/>
<font color="Maroon" title="c16">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="c4">H</font>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c17">h</font> = <font color="Maroon" title="c16">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c9">H'</font> ;<br/><b>reconsider </b><font color="Maroon" title="c18">h'</font> = <font color="Maroon" title="c17">h</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">H</font> ;<br/><a NAME="E4:48_1_2"/><i><font color="Green" title="E46">A11</font></i>: 
<span class="p1">(<span class="default"><a href="group_10.html#K9" title="GROUP_10:func.9">the_left_operation_of</a> <font color="Maroon" title="c4">H</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">h'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_10.html#K8" title="GROUP_10:func.8">the_left_translation_of</a> <font color="Maroon" title="c18">h'</font>,<font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="ref" href="group_10.html#D9" target="_self" title="GROUP_10:def.9">Def9</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c19">P1</font> = <font color="Maroon" title="c13">x'</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="group_2.html#K14" title="GROUP_2:func.14">Left_Cosets</a> <font color="Maroon" title="c5">P</font> ;<br/><b>consider </b><font color="Maroon" title="c20">P2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="group_2.html#K14" title="GROUP_2:func.14">Left_Cosets</a> <font color="Maroon" title="c5">P</font>, <font color="Maroon" title="c21">A1</font>, <font color="Maroon" title="c22">A2</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">G</font>, <font color="Maroon" title="c23">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="E7:48_1_2"/><i><font color="Green" title="E47">A12</font></i>: 
( <font color="Maroon" title="c20">P2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="group_10.html#K8" title="GROUP_10:func.8">the_left_translation_of</a> <font color="Maroon" title="c18">h'</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">P1</font> &amp; <font color="Maroon" title="c22">A2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">g''</font> <a href="group_2.html#K3" title="GROUP_2:func.3">*</a> <font color="Maroon" title="c21">A1</font> &amp; <font color="Maroon" title="c21">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">P1</font> &amp; <font color="Maroon" title="c22">A2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">P2</font> &amp; <font color="Maroon" title="c23">g''</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">h'</font> )
 <b>by </b><i><a class="ref" href="group_10.html#D8" target="_self" title="GROUP_10:def.8">Def8</a></i>;<br/><i><font color="Green" title="E48">A13</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c14">g'</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c5">P</font></span>)</span> <a href="group_2.html#K4" title="GROUP_2:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">g'</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c23">g''</font> <a href="group_2.html#K3" title="GROUP_2:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">g'</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c5">P</font></span>)</span></span>)</span> <a href="group_2.html#K4" title="GROUP_2:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">g'</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span>
<b>by </b><i><a class="txt" href="group_10.html#E15:48_1"><i><font color="Green" title="E44">A9</font></i></a>, <a class="txt" href="group_10.html#E18:48_1"><i><font color="Green" title="E45">A10</font></i></a>, <a class="txt" href="group_10.html#E4:48_1_2"><i><font color="Green" title="E46">A11</font></i></a>, <a class="txt" href="group_10.html#E7:48_1_2"><i><font color="Green" title="E47">A12</font></i></a>, <a class="ref" href="group_10.html#D12" target="_self" title="GROUP_10:def.12">Def12</a></i>
<br/>.= 
<font color="Maroon" title="c23">g''</font> <a href="group_2.html#K3" title="GROUP_2:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">g'</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c5">P</font></span>)</span> <a href="group_2.html#K4" title="GROUP_2:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">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_2.html#T39" title="GROUP_2:th.39">GROUP_2:39</a></i>
;<br/><a NAME="E9:48_1_2"/>
<font color="Maroon" title="c23">g''</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">g'</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c5">P</font></span>)</span> <a href="group_2.html#K4" title="GROUP_2:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">g'</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_2_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:48_1_2_2"/><i><font color="Green" title="E49">A14</font></i>: 
not <font color="Maroon" title="c23">g''</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">g'</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c5">P</font></span>)</span> <a href="group_2.html#K4" title="GROUP_2:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">g'</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span>
 ;<br/>

<a NAME="E2:48_1_2_2"/><i><font color="Green" title="E50">A15</font></i>: 
<font color="Maroon" title="c23">g''</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">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#D5" title="GROUP_1:def.5">GROUP_1:def 5</a></i>;<br/>
<a NAME="E3:48_1_2_2"/><i><font color="Green" title="E51">A16</font></i>: 
<font color="Maroon" title="c14">g'</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">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>  <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="E4:48_1_2_2"/>
 <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c1">G</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c5">P</font>
 
<b>by </b><i><a class="ref" href="group_2.html#T55" title="GROUP_2:th.55">GROUP_2:55</a></i>;<br/>
<a NAME="E5:48_1_2_2"/><b>then </b><i><font color="Green" title="E52">A17</font></i>: 
 <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c1">G</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="c5">P</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/>
<a NAME="E6:48_1_2_2"/>
<font color="Maroon" title="c14">g'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">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#D5" title="GROUP_1:def.5">GROUP_1:def 5</a></i>;<br/>
<a NAME="E7:48_1_2_2"/><b>then </b>
<font color="Maroon" title="c14">g'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">g'</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c5">P</font>
 
<b>by </b><i><a class="txt" href="group_10.html#E5:48_1_2_2"><i><font color="Green" title="E52">A17</font></i></a>, <a class="ref" href="group_2.html#T33" title="GROUP_2:th.33">GROUP_2:33</a></i>;<br/>
<a NAME="E8:48_1_2_2"/><b>then </b>
 <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c1">G</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">g'</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c5">P</font></span>)</span> <a href="group_2.html#K4" title="GROUP_2:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">g'</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="group_10.html#E3:48_1_2_2"><i><font color="Green" title="E51">A16</font></i></a>, <a class="ref" href="group_2.html#T34" title="GROUP_2:th.34">GROUP_2:34</a></i>;<br/>
<b>hence </b><a NAME="E9:48_1_2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="group_10.html#E8:48_1_2"><i><font color="Green" title="E48">A13</font></i></a>, <a class="txt" href="group_10.html#E1:48_1_2_2"><i><font color="Green" title="E49">A14</font></i></a>, <a class="txt" href="group_10.html#E2:48_1_2_2"><i><font color="Green" title="E50">A15</font></i></a>, <a class="ref" href="group_2.html#T33" title="GROUP_2:th.33">GROUP_2:33</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E10:48_1_2"/>
<font color="Maroon" title="c16">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">g'</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c5">P</font></span>)</span> <a href="group_2.html#K4" title="GROUP_2:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">g'</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span>
 <b>by </b><i><a class="txt" href="group_10.html#E7:48_1_2"><i><font color="Green" title="E47">A12</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E20:48_1"/><b>then </b><i><font color="Green" title="E46">A18</font></i>: 
 <a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <font color="Maroon" title="c4">H</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">g'</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c5">P</font></span>)</span> <a href="group_2.html#K4" title="GROUP_2:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">g'</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<a NAME="E21:48_1"/>
 <a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <font color="Maroon" title="c4">H</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c14">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> </span>)</span> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c5">P</font></span>)</span> <a href="group_2.html#K4" title="GROUP_2:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">g'</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="group_10.html#E20:48_1"><i><font color="Green" title="E46">A18</font></i></a>, <a class="ref" href="group_1.html#T19" title="GROUP_1:th.19">GROUP_1:19</a></i>;<br/>
<b>hence </b><a NAME="E22:48_1"/>
 <a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <font color="Maroon" title="c4">H</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="c5">P</font> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">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_3.html#T71" title="GROUP_3:th.71">GROUP_3:71</a></i>;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E3:48"/>
for <font color="Olive" title="b1">H</font> being   <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font>  st <font color="Olive" title="b1">H</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> holds <br/> ex <font color="Olive" title="b2">P</font> being   <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="group_10.html#R5" title="GROUP_10:pred.5">is_Sylow_p-subgroup_of_prime</a> <font color="Maroon" title="c2">p</font> &amp; <font color="Olive" title="b1">H</font> is    <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Olive" title="b2">P</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">H</font> be    <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font>;<br/>

<b>assume </b><a NAME="E1:48_2"/><i><font color="Green" title="E38">A19</font></i>: 
<font color="Maroon" title="c4">H</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>
 ;<br/>

<b>consider </b><font color="Maroon" title="c5">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="E3:48_2"/><i><font color="Green" title="E39">A20</font></i>: 
<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#T12" target="_self" title="GROUP_10:th.12">Th12</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c6">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="E5:48_2"/><i><font color="Green" title="E40">A21</font></i>: 
 <a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <font color="Maroon" title="c4">H</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="c5">P'</font> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Maroon" title="c6">g</font></span>)</span>
 <b>by </b><i><a class="txt" href="group_10.html#E2:48"><i><font color="Green" title="E37">A1</font></i></a>, <a class="txt" href="group_10.html#E1:48_2"><i><font color="Green" title="E38">A19</font></i></a>, <a class="txt" href="group_10.html#E3:48_2"><i><font color="Green" title="E39">A20</font></i></a></i>;<br/>
<b>set </b><font color="Maroon" title="c7">P</font> = <font color="Maroon" title="c5">P'</font> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Maroon" title="c6">g</font>;<br/>
<b>take </b>
<font color="Maroon" title="c5">P'</font> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Maroon" title="c6">g</font>
;<br/>

<a NAME="E6:48_2"/><i><font color="Green" title="E41">A22</font></i>: 
( <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#E3:48_2"><i><font color="Green" title="E39">A20</font></i></a>, <a class="ref" href="group_10.html#D19" target="_self" title="GROUP_10:def.19">Def19</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">H1</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="E8:48_2"/><i><font color="Green" title="E42">A23</font></i>: 
( <font color="Maroon" title="c5">P'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">H1</font> &amp; <font color="Maroon" title="c8">H1</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#E6:48_2"><i><font color="Green" title="E41">A22</font></i></a>, <a class="ref" href="group_10.html#D18" target="_self" title="GROUP_10:def.18">Def18</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c9">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="E10:48_2"/><i><font color="Green" title="E43">A24</font></i>: 
 <a href="group_1.html#K9" title="GROUP_1:func.9">ord</a> <font color="Maroon" title="c8">H1</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="c9">r</font>
 <b>by </b><i><a class="txt" href="group_10.html#E8:48_2"><i><font color="Green" title="E42">A23</font></i></a>, <a class="ref" href="group_10.html#D17" target="_self" title="GROUP_10:def.17">Def17</a></i>;<br/>
<b>set </b><font color="Maroon" title="c10">H2</font> = <font color="Maroon" title="c5">P'</font> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Maroon" title="c6">g</font>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">H2</font> = <font color="Maroon" title="c5">P'</font> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Maroon" title="c6">g</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/>
<a NAME="E12:48_2"/>
<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="E13:48_2"/><b>then </b><i><font color="Green" title="E44">A25</font></i>: 
 <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="group_1.html#K9" title="GROUP_1:func.9">ord</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">P'</font> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Maroon" title="c6">g</font></span>)</span>
 
<b>by </b><i><a class="ref" href="group_3.html#T79" title="GROUP_3:th.79">GROUP_3:79</a></i>;<br/>
<a NAME="E14:48_2"/><b>then </b>
<font color="Maroon" title="c11">H2</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#E8:48_2"><i><font color="Green" title="E42">A23</font></i></a>, <a class="txt" href="group_10.html#E10:48_2"><i><font color="Green" title="E43">A24</font></i></a>, <a class="ref" href="group_10.html#D17" target="_self" title="GROUP_10:def.17">Def17</a></i>;<br/>
<a NAME="E15:48_2"/><b>then </b><i><font color="Green" title="E45">A26</font></i>: 
<font color="Maroon" title="c5">P'</font> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Maroon" title="c6">g</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>
 
<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="E16:48_2"/>
<font color="Maroon" title="c5">P'</font> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Maroon" title="c6">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="E17:48_2"/><b>then </b><i><font color="Green" title="E46">A27</font></i>: 
 <a href="group_1.html#K9" title="GROUP_1:func.9">ord</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">P'</font> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Maroon" title="c6">g</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 
<b>by </b><i><a class="ref" href="group_1.html#T90" title="GROUP_1:th.90">GROUP_1:90</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"><font color="Maroon" title="c5">P'</font> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Maroon" title="c6">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"><font color="Maroon" title="c5">P'</font> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Maroon" title="c6">g</font></span>)</span></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> <span class="p2">(<span class="default"><font color="Maroon" title="c5">P'</font> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Maroon" title="c6">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> <font color="Maroon" title="c5">P'</font></span>)</span>
<b>by </b><i><a class="txt" href="group_10.html#E13:48_2"><i><font color="Green" title="E44">A25</font></i></a>, <a class="ref" href="group_2.html#T177" title="GROUP_2:th.177">GROUP_2:177</a></i>
;<br/>
<a NAME="E19:48_2"/><b>then </b>
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> <span class="p1">(<span class="default"><font color="Maroon" title="c5">P'</font> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Maroon" title="c6">g</font></span>)</span>
 
<b>by </b><i><a class="txt" href="group_10.html#E6:48_2"><i><font color="Green" title="E41">A22</font></i></a>, <a class="txt" href="group_10.html#E17:48_2"><i><font color="Green" title="E46">A27</font></i></a>, <a class="ref" href="xcmplx_1.html#T5" title="XCMPLX_1:th.5">XCMPLX_1:5</a></i>;<br/>
<b>hence </b><a NAME="E20:48_2"/>
<font color="Maroon" title="c5">P'</font> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Maroon" title="c6">g</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#E15:48_2"><i><font color="Green" title="E45">A26</font></i></a>, <a class="ref" href="group_10.html#D19" target="_self" title="GROUP_10:def.19">Def19</a></i>;<br/>

<b>thus </b><a NAME="E21:48_2"/>
<font color="Maroon" title="c4">H</font> is    <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c5">P'</font> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Maroon" title="c6">g</font>
 <b>by </b><i><a class="txt" href="group_10.html#E5:48_2"><i><font color="Green" title="E40">A21</font></i></a>, <a class="ref" href="group_2.html#T66" title="GROUP_2:th.66">GROUP_2:66</a></i>;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon" title="c4">P1</font>, <font color="Maroon" title="c5">P2</font> be    <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font>;<br/>

<b>assume </b><a NAME="E4:48"/>
<font color="Maroon" title="c4">P1</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/>

<a NAME="E5:48"/><b>then </b><i><font color="Green" title="E38">A28</font></i>: 
( <font color="Maroon" title="c4">P1</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="c4">P1</font> )
 
<b>by </b><i><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="c6">H1</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="E7:48"/><i><font color="Green" title="E39">A29</font></i>: 
( <font color="Maroon" title="c4">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">H1</font> &amp; <font color="Maroon" title="c6">H1</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>consider </b><font color="Maroon" title="c7">r1</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="E9:48"/><i><font color="Green" title="E40">A30</font></i>: 
 <a href="group_1.html#K9" title="GROUP_1:func.9">ord</a> <font color="Maroon" title="c6">H1</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="c7">r1</font>
 <b>by </b><i><a class="txt" href="group_10.html#E7:48"><i><font color="Green" title="E39">A29</font></i></a>, <a class="ref" href="group_10.html#D17" target="_self" title="GROUP_10:def.17">Def17</a></i>;<br/>
<b>assume </b><a NAME="E10:48"/><i><font color="Green" title="E41">A31</font></i>: 
<font color="Maroon" title="c5">P2</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>then consider </b><font color="Maroon" title="c8">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="E12:48"/><i><font color="Green" title="E42">A32</font></i>: 
 <a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <font color="Maroon" title="c4">P1</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="c5">P2</font> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Maroon" title="c8">g</font></span>)</span>
 <b>by </b><i><a class="txt" href="group_10.html#E2:48"><i><font color="Green" title="E37">A1</font></i></a>, <a class="txt" href="group_10.html#E5:48"><i><font color="Green" title="E38">A28</font></i></a></i>;<br/>
<a NAME="E13:48"/><i><font color="Green" title="E43">A33</font></i>: 
<font color="Maroon" title="c5">P2</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="E14:48"/><i><font color="Green" title="E44">A34</font></i>: 
( <font color="Maroon" title="c5">P2</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">P2</font> )
 
<b>by </b><i><a class="txt" href="group_10.html#E10:48"><i><font color="Green" title="E41">A31</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="c9">H2</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="E16:48"/><i><font color="Green" title="E45">A35</font></i>: 
( <font color="Maroon" title="c5">P2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">H2</font> &amp; <font color="Maroon" title="c9">H2</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>consider </b><font color="Maroon" title="c10">r2</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="E18:48"/><i><font color="Green" title="E46">A36</font></i>: 
 <a href="group_1.html#K9" title="GROUP_1:func.9">ord</a> <font color="Maroon" title="c9">H2</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">r2</font>
 <b>by </b><i><a class="txt" href="group_10.html#E16:48"><i><font color="Green" title="E45">A35</font></i></a>, <a class="ref" href="group_10.html#D17" target="_self" title="GROUP_10:def.17">Def17</a></i>;<br/>
<a NAME="E19:48"/><i><font color="Green" title="E47">A37</font></i>: 
 <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="c5">P2</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">P2</font></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/>
<a NAME="E20:48"/><i><font color="Green" title="E48">A38</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c7">r1</font></span>)</span> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> <span class="p1">(<span class="default"><a href="group_2.html#K17" title="GROUP_2:func.17">index</a> <font color="Maroon" title="c4">P1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c10">r2</font></span>)</span> <a href="nat_1.html#K3" title="NAT_1:func.3">*</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">P2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="group_10.html#E7:48"><i><font color="Green" title="E39">A29</font></i></a>, <a class="txt" href="group_10.html#E9:48"><i><font color="Green" title="E40">A30</font></i></a>, <a class="txt" href="group_10.html#E16:48"><i><font color="Green" title="E45">A35</font></i></a>, <a class="txt" href="group_10.html#E18:48"><i><font color="Green" title="E46">A36</font></i></a>, <a class="txt" href="group_10.html#E19:48"><i><font color="Green" title="E47">A37</font></i></a>, <a class="ref" href="group_2.html#T177" title="GROUP_2:th.177">GROUP_2:177</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:48_3"/><i><font color="Green" title="E49">A39</font></i>: 
<font color="Maroon" title="c7">r1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">r2</font>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:48_3_1"/>
( <font color="Maroon" title="c7">r1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c10">r2</font> or <font color="Maroon" title="c7">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">r2</font> )
 </span><b>by </b><i><a class="txt" href="group_10.html#E1:48_3"><i><font color="Green" title="E49">A39</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_3_1_1"><b>suppose </b></a><a NAME="E1:48_3_1_1"/>
<font color="Maroon" title="c7">r1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c10">r2</font>
 ;<br/><div class="add"><a NAME="E2:48_3_1_1"/><b>then </b><i><font color="Green" title="E50">A40</font></i>: 
<font color="Maroon" title="c7">r1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c10">r2</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c10">r2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c10">r2</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><b>set </b><font color="Maroon" title="c11">r</font> = <font color="Maroon" title="c7">r1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c10">r2</font>;<br/><a NAME="E3:48_3_1_1"/><i><font color="Green" title="E51">A41</font></i>: 
<font color="Maroon" title="c7">r1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c10">r2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">r1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c10">r2</font>
 <b>by </b><i><a class="txt" href="group_10.html#E2:48_3_1_1"><i><font color="Green" title="E50">A40</font></i></a>, <a class="ref" href="binarith.html#D3" title="BINARITH:def.3">BINARITH:def 3</a></i>;<br/><a NAME="E4:48_3_1_1"/><b>then </b>
<font color="Maroon" title="c7">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">r2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">r1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c10">r2</font></span>)</span>
 ;<br/><a NAME="E5:48_3_1_1"/><b>then </b>
<font color="Maroon" title="c3">p'</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c7">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p'</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c10">r2</font></span>)</span> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p'</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">r1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c10">r2</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="newton.html#T13" title="NEWTON:th.13">NEWTON:13</a></i>;<br/><a NAME="E6:48_3_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">p'</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c10">r2</font></span>)</span> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">p'</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">r1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c10">r2</font></span>)</span></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p2">(<span class="default"><a href="group_2.html#K17" title="GROUP_2:func.17">index</a> <font color="Maroon" title="c4">P1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p'</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c10">r2</font></span>)</span> <a href="nat_1.html#K3" title="NAT_1:func.3">*</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">P2</font></span>)</span>
 <b>by </b><i><a class="txt" href="group_10.html#E20:48"><i><font color="Green" title="E48">A38</font></i></a></i>;<br/><a NAME="E7:48_3_1_1"/><b>then </b><i><font color="Green" title="E52">A42</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">p'</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">r1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c10">r2</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> <font color="Maroon" title="c4">P1</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="c5">P2</font>
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T5" title="XCMPLX_1:th.5">XCMPLX_1:5</a></i>;<br/><b>consider </b><font color="Maroon" title="c12">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E9:48_3_1_1"/><i><font color="Green" title="E53">A43</font></i>: 
<font color="Maroon" title="c7">r1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c10">r2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="group_10.html#E2:48_3_1_1"><i><font color="Green" title="E50">A40</font></i></a>, <a class="txt" href="group_10.html#E3:48_3_1_1"><i><font color="Green" title="E51">A41</font></i></a>, <a class="ref" href="nat_1.html#T6" title="NAT_1:th.6">NAT_1:6</a></i>;<br/><a NAME="E10:48_3_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c12">k</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> <span class="p1">(<span class="default"><a href="group_2.html#K17" title="GROUP_2:func.17">index</a> <font color="Maroon" title="c4">P1</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="c5">P2</font>
 <b>by </b><i><a class="txt" href="group_10.html#E7:48_3_1_1"><i><font color="Green" title="E52">A42</font></i></a>, <a class="txt" href="group_10.html#E9:48_3_1_1"><i><font color="Green" title="E53">A43</font></i></a>, <a class="ref" href="newton.html#T11" title="NEWTON:th.11">NEWTON:11</a></i>;<br/><a NAME="E11:48_3_1_1"/><b>then </b>
<font color="Maroon" title="c2">p</font> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c12">k</font></span>)</span> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> <span class="p2">(<span class="default"><a href="group_2.html#K17" title="GROUP_2:func.17">index</a> <font color="Maroon" title="c4">P1</font></span>)</span></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="c5">P2</font>
 ;<br/><b>hence </b><a NAME="E12:48_3_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="group_10.html#E14:48"><i><font color="Green" title="E44">A34</font></i></a>, <a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">NAT_D:def 3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_3_1_2"><b>suppose </b></a><a NAME="E1:48_3_1_2"/>
<font color="Maroon" title="c7">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">r2</font>
 ;<br/><div class="add"><a NAME="E2:48_3_1_2"/><b>then </b><i><font color="Green" title="E50">A44</font></i>: 
<font color="Maroon" title="c10">r2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c7">r1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c7">r1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c7">r1</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><b>set </b><font color="Maroon" title="c11">r</font> = <font color="Maroon" title="c10">r2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c7">r1</font>;<br/><a NAME="E3:48_3_1_2"/><i><font color="Green" title="E51">A45</font></i>: 
<font color="Maroon" title="c10">r2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c7">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">r2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c7">r1</font>
 <b>by </b><i><a class="txt" href="group_10.html#E2:48_3_1_2"><i><font color="Green" title="E50">A44</font></i></a>, <a class="ref" href="binarith.html#D3" title="BINARITH:def.3">BINARITH:def 3</a></i>;<br/><a NAME="E4:48_3_1_2"/><b>then </b>
<font color="Maroon" title="c10">r2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">r1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">r2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c7">r1</font></span>)</span>
 ;<br/><a NAME="E5:48_3_1_2"/><b>then </b>
<font color="Maroon" title="c3">p'</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c10">r2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p'</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c7">r1</font></span>)</span> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p'</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">r2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c7">r1</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="newton.html#T13" title="NEWTON:th.13">NEWTON:13</a></i>;<br/><a NAME="E6:48_3_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">p'</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c7">r1</font></span>)</span> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">p'</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">r2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c7">r1</font></span>)</span></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p2">(<span class="default"><a href="group_2.html#K17" title="GROUP_2:func.17">index</a> <font color="Maroon" title="c5">P2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p'</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c7">r1</font></span>)</span> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> <span class="p1">(<span class="default"><a href="group_2.html#K17" title="GROUP_2:func.17">index</a> <font color="Maroon" title="c4">P1</font></span>)</span>
 <b>by </b><i><a class="txt" href="group_10.html#E7:48"><i><font color="Green" title="E39">A29</font></i></a>, <a class="txt" href="group_10.html#E9:48"><i><font color="Green" title="E40">A30</font></i></a>, <a class="txt" href="group_10.html#E16:48"><i><font color="Green" title="E45">A35</font></i></a>, <a class="txt" href="group_10.html#E18:48"><i><font color="Green" title="E46">A36</font></i></a>, <a class="txt" href="group_10.html#E19:48"><i><font color="Green" title="E47">A37</font></i></a>, <a class="ref" href="group_2.html#T177" title="GROUP_2:th.177">GROUP_2:177</a></i>;<br/><a NAME="E7:48_3_1_2"/><b>then </b><i><font color="Green" title="E52">A46</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">p'</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">r2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c7">r1</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> <font color="Maroon" title="c5">P2</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="c4">P1</font>
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T5" title="XCMPLX_1:th.5">XCMPLX_1:5</a></i>;<br/><b>consider </b><font color="Maroon" title="c12">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E9:48_3_1_2"/><i><font color="Green" title="E53">A47</font></i>: 
<font color="Maroon" title="c10">r2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c7">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="group_10.html#E2:48_3_1_2"><i><font color="Green" title="E50">A44</font></i></a>, <a class="txt" href="group_10.html#E3:48_3_1_2"><i><font color="Green" title="E51">A45</font></i></a>, <a class="ref" href="nat_1.html#T6" title="NAT_1:th.6">NAT_1:6</a></i>;<br/><a NAME="E10:48_3_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c12">k</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="nat_1.html#K3" title="NAT_1:func.3">*</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">P2</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="c4">P1</font>
 <b>by </b><i><a class="txt" href="group_10.html#E7:48_3_1_2"><i><font color="Green" title="E52">A46</font></i></a>, <a class="txt" href="group_10.html#E9:48_3_1_2"><i><font color="Green" title="E53">A47</font></i></a>, <a class="ref" href="newton.html#T11" title="NEWTON:th.11">NEWTON:11</a></i>;<br/><a NAME="E11:48_3_1_2"/><b>then </b>
<font color="Maroon" title="c2">p</font> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c12">k</font></span>)</span> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> <span class="p2">(<span class="default"><a href="group_2.html#K17" title="GROUP_2:func.17">index</a> <font color="Maroon" title="c5">P2</font></span>)</span></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="c4">P1</font>
 ;<br/><b>hence </b><a NAME="E12:48_3_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="group_10.html#E5:48"><i><font color="Green" title="E38">A28</font></i></a>, <a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">NAT_D:def 3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>then </b><i><font color="Green" title="E49">A48</font></i>:  <a href="card_1.html#K5" title="CARD_1:func.5">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="c4">P1</font></span>)</span> = 
 <a href="group_1.html#K9" title="GROUP_1:func.9">ord</a> <font color="Maroon" title="c5">P2</font>
<b>by </b><i><a class="txt" href="group_10.html#E7:48"><i><font color="Green" title="E39">A29</font></i></a>, <a class="txt" href="group_10.html#E9:48"><i><font color="Green" title="E40">A30</font></i></a>, <a class="txt" href="group_10.html#E16:48"><i><font color="Green" title="E45">A35</font></i></a>, <a class="txt" href="group_10.html#E18:48"><i><font color="Green" title="E46">A36</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="card_1.html#K5" title="CARD_1:func.5">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="c5">P2</font></span>)</span>
<b>by </b><i><a class="txt" href="group_10.html#E13:48"><i><font color="Green" title="E43">A33</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="E23:48"/><i><font color="Green" title="E50">A49</font></i>: 
<font color="Maroon" title="c5">P2</font> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Maroon" title="c8">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 href="card_1.html#K5" title="CARD_1:func.5">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="c5">P2</font></span>)</span> = 
 <a href="group_1.html#K9" title="GROUP_1:func.9">ord</a> <font color="Maroon" title="c5">P2</font>
<b>by </b><i><a class="txt" href="group_10.html#E13:48"><i><font color="Green" title="E43">A33</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="c5">P2</font> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Maroon" title="c8">g</font></span>)</span>
<b>by </b><i><a class="txt" href="group_10.html#E13:48"><i><font color="Green" title="E43">A33</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#K5" title="CARD_1:func.5">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="c5">P2</font> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Maroon" title="c8">g</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="group_10.html#E23:48"><i><font color="Green" title="E50">A49</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="E25:48"/><b>then </b>
 <a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <font color="Maroon" title="c4">P1</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="c5">P2</font> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Maroon" title="c8">g</font></span>)</span>
 
<b>by </b><i><a class="txt" href="group_10.html#E12:48"><i><font color="Green" title="E42">A32</font></i></a>, <a class="txt" href="group_10.html#E22:48"><i><font color="Green" title="E49">A48</font></i></a>, <a class="ref" href="card_fin.html#T1" title="CARD_FIN:th.1">CARD_FIN:1</a></i>;<br/>
<a NAME="E26:48"/><b>then </b>
 <a href="group_1.html#G1" title="GROUP_1:aggr.1">HGrStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">P1</font>,the <a href="group_1.html#U1" title="GROUP_1:sel.1">mult</a> of <font color="Maroon" title="c4">P1</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#G1" title="GROUP_1:aggr.1">HGrStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c5">P2</font> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Maroon" title="c8">g</font></span>)</span>,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="c5">P2</font> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Maroon" title="c8">g</font></span>)</span> #)
 
<b>by </b><i><a class="ref" href="group_2.html#T68" title="GROUP_2:th.68">GROUP_2:68</a></i>;<br/>
<b>hence </b><a NAME="E27:48"/>
<font color="Maroon" title="c4">P1</font>,<font color="Maroon" title="c5">P2</font> <a href="group_3.html#R5" title="GROUP_3:pred.5">are_conjugated</a> 
 <b>by </b><i><a class="ref" href="group_3.html#D11" title="GROUP_3:def.11">GROUP_3:def 11</a></i>;<br/>


</div>
