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

<span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">p</font> being   <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>  holds <br/> ( (  for <font color="Olive" title="b2">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="b2">H</font> is <font color="Olive" title="b1">p</font> <a href="group_10.html#V2" title="GROUP_10:attr.2">-group</a>  holds <br/> ex <font color="Olive" title="b3">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="b3">P</font> <a href="group_10.html#R3" title="GROUP_10:pred.3">is_Sylow_p-subgroup_of_prime</a> <font color="Olive" title="b1">p</font> &amp; <font color="Olive" title="b2">H</font> is    <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Olive" title="b3">P</font> ) ) &amp; (  for <font color="Olive" title="b2">P1</font>, <font color="Olive" title="b3">P2</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="b2">P1</font> <a href="group_10.html#R3" title="GROUP_10:pred.3">is_Sylow_p-subgroup_of_prime</a> <font color="Olive" title="b1">p</font> &amp; <font color="Olive" title="b3">P2</font> <a href="group_10.html#R3" title="GROUP_10:pred.3">is_Sylow_p-subgroup_of_prime</a> <font color="Olive" title="b1">p</font> holds <br/><font color="Olive" title="b2">P1</font>,<font color="Olive" title="b3">P2</font> <a href="group_3.html#R5" title="GROUP_3:pred.5">are_conjugated</a>  ) )</span><br/><span class="kw">let </span><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  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> is <font color="Maroon" title="c2">p</font> <a href="group_10.html#V2" title="GROUP_10:attr.2">-group</a>  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#R3" title="GROUP_10:pred.3">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> ) ) &amp; (  for <font color="Olive" title="b1">P1</font>, <font color="Olive" title="b2">P2</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">P1</font> <a href="group_10.html#R3" title="GROUP_10:pred.3">is_Sylow_p-subgroup_of_prime</a> <font color="Maroon" title="c2">p</font> &amp; <font color="Olive" title="b2">P2</font> <a href="group_10.html#R3" title="GROUP_10:pred.3">is_Sylow_p-subgroup_of_prime</a> <font color="Maroon" title="c2">p</font> holds <br/><font color="Olive" title="b1">P1</font>,<font color="Olive" title="b2">P2</font> <a href="group_3.html#R5" title="GROUP_3:pred.5">are_conjugated</a>  ) )</span><br/>



<span class="kw">reconsider </span><font color="Maroon" title="c3">p9</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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></span>;<br/>
<a NAME="E2:49"/><span class="lab"><font color="Green" title="E34">A1</font></span>: 
 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> is <font color="Maroon" title="c2">p</font> <a href="group_10.html#V2" title="GROUP_10:attr.2">-group</a>  &amp; <font color="Olive" title="b2">P</font> <a href="group_10.html#R3" title="GROUP_10:pred.3">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#K8" title="GROUP_2:func.8">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#K8" title="GROUP_2:func.8">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="49_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">H</font> is <font color="Maroon" title="c2">p</font> <a href="group_10.html#V2" title="GROUP_10:attr.2">-group</a>  &amp; <font color="Maroon" title="c5">P</font> <a href="group_10.html#R3" title="GROUP_10:pred.3">is_Sylow_p-subgroup_of_prime</a> <font color="Maroon" title="c2">p</font> implies  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  <a href="group_2.html#K8" title="GROUP_2:func.8">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#K8" title="GROUP_2:func.8">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="Olive" title="b1">g</font></span>)</span> )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c6">H9</font> = <font color="Maroon" title="c4">H</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">E</font> =  <a href="group_2.html#K15" title="GROUP_2:func.15">Left_Cosets</a> <font color="Maroon" title="c5">P</font>;<br/>
<span class="kw">set </span><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/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">E</font> =  <a href="group_2.html#K15" title="GROUP_2:func.15">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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="group_2.html#T135" title="GROUP_2:th.135">GROUP_2:135</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">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="c4">H</font>,<font color="Maroon" title="c9">E</font> ;<br/>
<span class="kw">assume </span><a NAME="E3:49_1"/>
<font color="Maroon" title="c4">H</font> is <font color="Maroon" title="c2">p</font> <a href="group_10.html#V2" title="GROUP_10:attr.2">-group</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c5">P</font> <a href="group_10.html#R3" title="GROUP_10:pred.3">is_Sylow_p-subgroup_of_prime</a> <font color="Maroon" title="c2">p</font> or  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  <a href="group_2.html#K8" title="GROUP_2:func.8">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#K8" title="GROUP_2:func.8">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="Olive" title="b1">g</font></span>)</span> )</span><br/>

<a NAME="E4:49_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A2</font></span>: 
<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="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> <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="c9">E</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_10.html#T9" target="_self" title="GROUP_10:th.9">Th11</a></span>;<br/>
<span class="kw">assume </span><a NAME="E5:49_1"/>
<font color="Maroon" title="c5">P</font> <a href="group_10.html#R3" title="GROUP_10:pred.3">is_Sylow_p-subgroup_of_prime</a> <font color="Maroon" title="c2">p</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  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  <a href="group_2.html#K8" title="GROUP_2:func.8">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#K8" title="GROUP_2:func.8">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="Olive" title="b1">g</font></span>)</span></span><br/>

<a NAME="E6:49_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A3</font></span>: 
 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#K18" title="GROUP_2:func.18">index</a> <font color="Maroon" title="c5">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_10.html#D18" target="_self" title="GROUP_10:def.18">Def19</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:49_1_1"/>
<span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c9">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="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:49_1_1"/><span class="kw">then </span>
(  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="c9">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="nat_1.html#K1" title="NAT_1:func.1">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">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="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#D2" title="NAT_D:def.2">NAT_D:def 2</a></span>;<br/><a NAME="E3:49_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">p9</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="c9">E</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">NAT_D:def 3</a></span>;<br/><span class="kw">hence </span><a NAME="E4:49_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:49_1"><span class="lab"><font color="Green" title="E36">A3</font></span></a>, <a class="ref" href="group_2.html#D18" title="GROUP_2:def.18">GROUP_2:def 18</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E8:49_1"/><span class="kw">then </span>
 <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="c10">T</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:49_1"><span class="lab"><font color="Green" title="E35">A2</font></span></a>, <a class="ref" href="nat_d.html#T26" title="NAT_D:th.26">NAT_D:26</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E10:49_1"/><span class="lab"><font color="Green" title="E37">A4</font></span>: 
<font color="Maroon" title="c11">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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#T27" title="CARD_1:th.27">CARD_1:27</a>, <a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></span>;<br/>
<a NAME="E11:49_1"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">x9</font> where <font color="Olive" title="b1">x9</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c9">E</font> : <font color="Olive" title="b1">x9</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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:49_1"><span class="lab"><font color="Green" title="E37">A4</font></span></a>, <a class="ref" href="group_10.html#D13" target="_self" title="GROUP_10:def.13">Def13</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">x9</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c9">E</font><span class="kw"> such that </span><br/><a NAME="E13:49_1"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">x9</font>
 <span class="kw">and </span><br/><a NAME="E14:49_1"/><span class="lab"><font color="Green" title="E38">A5</font></span>: 
<font color="Maroon" title="c12">x9</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="E15:49_1"/>
<font color="Maroon" title="c12">x9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_2.html#K15" title="GROUP_2:func.15">Left_Cosets</a> <font color="Maroon" title="c5">P</font>
 
;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">g9</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font><span class="kw"> such that </span><br/><a NAME="E17:49_1"/><span class="lab"><font color="Green" title="E39">A6</font></span>: 
<font color="Maroon" title="c12">x9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">g9</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c5">P</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_2.html#D15" title="GROUP_2:def.15">GROUP_2:def 15</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c14">g</font> = <font color="Maroon" title="c13">g9</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> ;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c13">g9</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> 
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="group_2.html#K8" title="GROUP_2:func.8">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#K8" title="GROUP_2:func.8">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="c13">g9</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a></span>)</span></span>)</span></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c15">P1</font> = <font color="Maroon" title="c12">x9</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="group_2.html#K15" title="GROUP_2:func.15">Left_Cosets</a> <font color="Maroon" title="c5">P</font> ;<br/><span class="kw">let </span><font color="Maroon" title="c16">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c16">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c4">H</font> implies <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="c13">g9</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c5">P</font></span>)</span> <a href="group_2.html#K5" title="GROUP_2:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">g9</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a></span>)</span> )</span><br/><span class="kw">assume </span><a NAME="E2:49_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#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c4">H</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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="c13">g9</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c5">P</font></span>)</span> <a href="group_2.html#K5" title="GROUP_2:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">g9</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a></span>)</span></span><br/><span class="kw">then </span><span class="kw">reconsider </span><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="c4">H</font> ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c18">h9</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/><span class="kw">consider </span><font color="Maroon" title="c19">P2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="group_2.html#K15" title="GROUP_2:func.15">Left_Cosets</a> <font color="Maroon" title="c5">P</font>, <font color="Maroon" title="c20">A1</font>, <font color="Maroon" title="c21">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="c22">g99</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font><span class="kw"> such that </span><br/><a NAME="E6:49_1_2"/><span class="lab"><font color="Green" title="E40">A7</font></span>: 
( <font color="Maroon" title="c19">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">h9</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="c15">P1</font> &amp; <font color="Maroon" title="c21">A2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">g99</font> <a href="group_2.html#K4" title="GROUP_2:func.4">*</a> <font color="Maroon" title="c20">A1</font> &amp; <font color="Maroon" title="c20">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">P1</font> &amp; <font color="Maroon" title="c21">A2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">P2</font> )
 <span class="kw">and </span><br/><a NAME="E7:49_1_2"/><span class="lab"><font color="Green" title="E41">A8</font></span>: 
<font color="Maroon" title="c22">g99</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">h9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_10.html#D8" target="_self" title="GROUP_10:def.8">Def8</a></span>;<br/><a NAME="E8:49_1_2"/>
<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">h9</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">h9</font>,<font color="Maroon" title="c5">P</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_10.html#D9" target="_self" title="GROUP_10:def.9">Def9</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A9</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c13">g9</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c5">P</font></span>)</span> <a href="group_2.html#K5" title="GROUP_2:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">g9</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c22">g99</font> <a href="group_2.html#K4" title="GROUP_2:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">g9</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c5">P</font></span>)</span></span>)</span> <a href="group_2.html#K5" title="GROUP_2:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">g9</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:49_1"><span class="lab"><font color="Green" title="E38">A5</font></span></a>, <a class="txt" href="#E17:49_1"><span class="lab"><font color="Green" title="E39">A6</font></span></a>, <a class="txt" href="#E6:49_1_2"><span class="lab"><font color="Green" title="E40">A7</font></span></a>, <a class="ref" href="group_10.html#D12" target="_self" title="GROUP_10:def.12">Def12</a></span>
<br/>.= 
<font color="Maroon" title="c22">g99</font> <a href="group_2.html#K4" title="GROUP_2:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">g9</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c5">P</font></span>)</span> <a href="group_2.html#K5" title="GROUP_2:func.5">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">g9</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_2.html#T33" title="GROUP_2:th.33">GROUP_2:33</a></span>
;<br/><a NAME="E10:49_1_2"/>
<font color="Maroon" title="c22">g99</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">g9</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c5">P</font></span>)</span> <a href="group_2.html#K5" title="GROUP_2:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">g9</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_2_2"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:49_1_2_2"/>
 <a href="group_1.html#K1" title="GROUP_1:func.1">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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_2.html#T46" title="GROUP_2:th.46">GROUP_2:46</a></span>;<br/>
<a NAME="E2:49_1_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A10</font></span>: 
 <a href="group_1.html#K1" title="GROUP_1:func.1">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#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c5">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></span>;<br/>
<a NAME="E3:49_1_2_2"/>
<font color="Maroon" title="c13">g9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">g9</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">G</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">GROUP_1:def 4</a></span>;<br/>
<a NAME="E4:49_1_2_2"/><span class="kw">then </span>
( <font color="Maroon" title="c13">g9</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">g9</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">G</font> &amp; <font color="Maroon" title="c13">g9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">g9</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c5">P</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:49_1_2_2"><span class="lab"><font color="Green" title="E43">A10</font></span></a>, <a class="ref" href="group_1.html#D5" title="GROUP_1:def.5">GROUP_1:def 5</a>, <a class="ref" href="group_2.html#T27" title="GROUP_2:th.27">GROUP_2:27</a></span>;<br/>
<a NAME="E5:49_1_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A11</font></span>: 
( <font color="Maroon" title="c22">g99</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">g99</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">G</font></span>)</span> &amp;  <a href="group_1.html#K1" title="GROUP_1:func.1">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="c13">g9</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c5">P</font></span>)</span> <a href="group_2.html#K5" title="GROUP_2:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">g9</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">GROUP_1:def 4</a>, <a class="ref" href="group_2.html#T28" title="GROUP_2:th.28">GROUP_2:28</a></span>;<br/>
<span class="kw">assume </span><a NAME="E6:49_1_2_2"/>
 not <font color="Maroon" title="c22">g99</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">g9</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c5">P</font></span>)</span> <a href="group_2.html#K5" title="GROUP_2:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">g9</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">hence </span><a NAME="E7:49_1_2_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:49_1_2"><span class="lab"><font color="Green" title="E42">A9</font></span></a>, <a class="txt" href="#E5:49_1_2_2"><span class="lab"><font color="Green" title="E44">A11</font></span></a>, <a class="ref" href="group_2.html#T27" title="GROUP_2:th.27">GROUP_2:27</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E11:49_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="c13">g9</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c5">P</font></span>)</span> <a href="group_2.html#K5" title="GROUP_2:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">g9</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:49_1_2"><span class="lab"><font color="Green" title="E41">A8</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E19:49_1"/><span class="kw">then </span>
 <a href="group_2.html#K8" title="GROUP_2:func.8">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="c13">g9</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a></span>)</span> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a></span>)</span> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c5">P</font></span>)</span> <a href="group_2.html#K5" title="GROUP_2:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">g9</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E20:49_1"/>
 <a href="group_2.html#K8" title="GROUP_2:func.8">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#K8" title="GROUP_2:func.8">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="c13">g9</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_3.html#T59" title="GROUP_3:th.59">GROUP_3:59</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">thus </span><a NAME="E3:49"/>
 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> is <font color="Maroon" title="c2">p</font> <a href="group_10.html#V2" title="GROUP_10:attr.2">-group</a>  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#R3" title="GROUP_10:pred.3">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> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">P1</font>, <font color="Olive" title="b2">P2</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">P1</font> <a href="group_10.html#R3" title="GROUP_10:pred.3">is_Sylow_p-subgroup_of_prime</a> <font color="Maroon" title="c2">p</font> &amp; <font color="Olive" title="b2">P2</font> <a href="group_10.html#R3" title="GROUP_10:pred.3">is_Sylow_p-subgroup_of_prime</a> <font color="Maroon" title="c2">p</font> holds <br/><font color="Olive" title="b1">P1</font>,<font color="Olive" title="b2">P2</font> <a href="group_3.html#R5" title="GROUP_3:pred.5">are_conjugated</a> </span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">H</font> is <font color="Maroon" title="c2">p</font> <a href="group_10.html#V2" title="GROUP_10:attr.2">-group</a>  implies  ex <font color="Olive" title="b1">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="b1">P</font> <a href="group_10.html#R3" title="GROUP_10:pred.3">is_Sylow_p-subgroup_of_prime</a> <font color="Maroon" title="c2">p</font> &amp; <font color="Maroon" title="c4">H</font> is    <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Olive" title="b1">P</font> ) )</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c5">P9</font> being   <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a>   <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font><span class="kw"> such that </span><br/><a NAME="E2:49_2"/><span class="lab"><font color="Green" title="E35">A12</font></span>: 
<font color="Maroon" title="c5">P9</font> <a href="group_10.html#R3" title="GROUP_10:pred.3">is_Sylow_p-subgroup_of_prime</a> <font color="Maroon" title="c2">p</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_10.html#T10" target="_self" title="GROUP_10:th.10">Th12</a></span>;<br/>
<span class="kw">assume </span><a NAME="E3:49_2"/>
<font color="Maroon" title="c4">H</font> is <font color="Maroon" title="c2">p</font> <a href="group_10.html#V2" title="GROUP_10:attr.2">-group</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">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="b1">P</font> <a href="group_10.html#R3" title="GROUP_10:pred.3">is_Sylow_p-subgroup_of_prime</a> <font color="Maroon" title="c2">p</font> &amp; <font color="Maroon" title="c4">H</font> is    <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Olive" title="b1">P</font> )</span><br/>

<span class="kw">then </span><span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E5:49_2"/><span class="lab"><font color="Green" title="E36">A13</font></span>: 
 <a href="group_2.html#K8" title="GROUP_2:func.8">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#K8" title="GROUP_2:func.8">carr</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">P9</font> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Maroon" title="c6">g</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:49"><span class="lab"><font color="Green" title="E34">A1</font></span></a>, <a class="txt" href="#E2:49_2"><span class="lab"><font color="Green" title="E35">A12</font></span></a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">P</font> = <font color="Maroon" title="c5">P9</font> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Maroon" title="c6">g</font>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c5">P9</font> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Maroon" title="c6">g</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">P9</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#R3" title="GROUP_10:pred.3">is_Sylow_p-subgroup_of_prime</a> <font color="Maroon" title="c2">p</font> &amp; <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">P9</font> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Maroon" title="c6">g</font> )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c8">H2</font> = <font color="Maroon" title="c5">P9</font> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Maroon" title="c6">g</font>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">H2</font> = <font color="Maroon" title="c5">P9</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> ;<br/>
<a NAME="E7:49_2"/><span class="lab"><font color="Green" title="E37">A14</font></span>: 
 <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c5">P9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">P9</font> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Maroon" title="c6">g</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_3.html#T66" title="GROUP_3:th.66">GROUP_3:66</a></span>;<br/>
<a NAME="E8:49_2"/>
<font color="Maroon" title="c5">P9</font> is <font color="Maroon" title="c2">p</font> <a href="group_10.html#V2" title="GROUP_10:attr.2">-group</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:49_2"><span class="lab"><font color="Green" title="E35">A12</font></span></a>, <a class="ref" href="group_10.html#D18" target="_self" title="GROUP_10:def.18">Def19</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">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><span class="kw"> such that </span><br/><a NAME="E10:49_2"/><span class="lab"><font color="Green" title="E38">A15</font></span>: 
<font color="Maroon" title="c5">P9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">H1</font>
 <span class="kw">and </span><br/><a NAME="E11:49_2"/><span class="lab"><font color="Green" title="E39">A16</font></span>: 
<font color="Maroon" title="c10">H1</font> is <font color="Maroon" title="c2">p</font> <a href="group_10.html#V2" title="GROUP_10:attr.2">-group</a> 
 ;<br/>
<a NAME="E12:49_2"/>
 ex <font color="Olive" title="b1">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>  st  <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c10">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="Olive" title="b1">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:49_2"><span class="lab"><font color="Green" title="E39">A16</font></span></a>, <a class="ref" href="group_10.html#D17" target="_self" title="GROUP_10:def.17">Def17</a></span>;<br/>
<a NAME="E13:49_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A17</font></span>: 
<font color="Maroon" title="c9">H2</font> is <font color="Maroon" title="c2">p</font> <a href="group_10.html#V2" title="GROUP_10:attr.2">-group</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:49_2"><span class="lab"><font color="Green" title="E38">A15</font></span></a>, <a class="txt" href="#E7:49_2"><span class="lab"><font color="Green" title="E37">A14</font></span></a>, <a class="ref" href="group_10.html#D17" target="_self" title="GROUP_10:def.17">Def17</a></span>;<br/>
<a NAME="E14:49_2"/><span class="lab"><font color="Green" title="E41">A18</font></span>: 
 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#K18" title="GROUP_2:func.18">index</a> <font color="Maroon" title="c5">P9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:49_2"><span class="lab"><font color="Green" title="E35">A12</font></span></a>, <a class="ref" href="group_10.html#D18" target="_self" title="GROUP_10:def.18">Def19</a></span>;<br/>
<span class="p1">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">P9</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="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><a href="group_2.html#K18" title="GROUP_2:func.18">index</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">P9</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#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c1">G</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_2.html#T147" title="GROUP_2:th.147">GROUP_2:147</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">P9</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="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><a href="group_2.html#K18" title="GROUP_2:func.18">index</a> <font color="Maroon" title="c5">P9</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:49_2"><span class="lab"><font color="Green" title="E37">A14</font></span></a>, <a class="ref" href="group_2.html#T147" title="GROUP_2:th.147">GROUP_2:147</a></span>
;<br/>
<a NAME="E16:49_2"/><span class="kw">then </span>
 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#K18" title="GROUP_2:func.18">index</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">P9</font> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Maroon" title="c6">g</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:49_2"><span class="lab"><font color="Green" title="E41">A18</font></span></a>, <a class="ref" href="xcmplx_1.html#T5" title="XCMPLX_1:th.5">XCMPLX_1:5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E17:49_2"/>
<font color="Maroon" title="c5">P9</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#R3" title="GROUP_10:pred.3">is_Sylow_p-subgroup_of_prime</a> <font color="Maroon" title="c2">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:49_2"><span class="lab"><font color="Green" title="E40">A17</font></span></a>, <a class="ref" href="group_10.html#D18" target="_self" title="GROUP_10:def.18">Def19</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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">P9</font> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Maroon" title="c6">g</font></span><br/>

<span class="kw">thus </span><a NAME="E18:49_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">P9</font> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Maroon" title="c6">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:49_2"><span class="lab"><font color="Green" title="E36">A13</font></span></a>, <a class="ref" href="group_2.html#T57" title="GROUP_2:th.57">GROUP_2:57</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">P1</font> <a href="group_10.html#R3" title="GROUP_10:pred.3">is_Sylow_p-subgroup_of_prime</a> <font color="Maroon" title="c2">p</font> &amp; <font color="Maroon" title="c5">P2</font> <a href="group_10.html#R3" title="GROUP_10:pred.3">is_Sylow_p-subgroup_of_prime</a> <font color="Maroon" title="c2">p</font> implies <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>  )</span><br/>

<span class="kw">assume </span><a NAME="E4:49"/><span class="lab"><font color="Green" title="E35">A19</font></span>: 
<font color="Maroon" title="c4">P1</font> <a href="group_10.html#R3" title="GROUP_10:pred.3">is_Sylow_p-subgroup_of_prime</a> <font color="Maroon" title="c2">p</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c5">P2</font> <a href="group_10.html#R3" title="GROUP_10:pred.3">is_Sylow_p-subgroup_of_prime</a> <font color="Maroon" title="c2">p</font> or <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>  )</span><br/>

<a NAME="E5:49"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A20</font></span>: 
<font color="Maroon" title="c4">P1</font> is <font color="Maroon" title="c2">p</font> <a href="group_10.html#V2" title="GROUP_10:attr.2">-group</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_10.html#D18" target="_self" title="GROUP_10:def.18">Def19</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E7:49"/><span class="lab"><font color="Green" title="E37">A21</font></span>: 
<font color="Maroon" title="c4">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">H1</font>
 <span class="kw">and </span><br/><a NAME="E8:49"/><span class="lab"><font color="Green" title="E38">A22</font></span>: 
<font color="Maroon" title="c6">H1</font> is <font color="Maroon" title="c2">p</font> <a href="group_10.html#V2" title="GROUP_10:attr.2">-group</a> 
 ;<br/>
<a NAME="E9:49"/><span class="lab"><font color="Green" title="E39">A23</font></span>: 
 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#K18" title="GROUP_2:func.18">index</a> <font color="Maroon" title="c4">P1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:49"><span class="lab"><font color="Green" title="E35">A19</font></span></a>, <a class="ref" href="group_10.html#D18" target="_self" title="GROUP_10:def.18">Def19</a></span>;<br/>
<span class="kw">consider </span><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> <span class="kw"> such that </span><br/><a NAME="E11:49"/><span class="lab"><font color="Green" title="E40">A24</font></span>: 
 <a href="group_1.html#K7" title="GROUP_1:func.7">card</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:49"><span class="lab"><font color="Green" title="E38">A22</font></span></a>, <a class="ref" href="group_10.html#D17" target="_self" title="GROUP_10:def.17">Def17</a></span>;<br/>
<span class="kw">assume </span><a NAME="E12:49"/><span class="lab"><font color="Green" title="E41">A25</font></span>: 
<font color="Maroon" title="c5">P2</font> <a href="group_10.html#R3" title="GROUP_10:pred.3">is_Sylow_p-subgroup_of_prime</a> <font color="Maroon" title="c2">p</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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> </span><br/>

<a NAME="E13:49"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A26</font></span>: 
 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#K18" title="GROUP_2:func.18">index</a> <font color="Maroon" title="c5">P2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_10.html#D18" target="_self" title="GROUP_10:def.18">Def19</a></span>;<br/>
<a NAME="E14:49"/>
<font color="Maroon" title="c5">P2</font> is <font color="Maroon" title="c2">p</font> <a href="group_10.html#V2" title="GROUP_10:attr.2">-group</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:49"><span class="lab"><font color="Green" title="E41">A25</font></span></a>, <a class="ref" href="group_10.html#D18" target="_self" title="GROUP_10:def.18">Def19</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">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><span class="kw"> such that </span><br/><a NAME="E16:49"/><span class="lab"><font color="Green" title="E43">A27</font></span>: 
<font color="Maroon" title="c5">P2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">H2</font>
 <span class="kw">and </span><br/><a NAME="E17:49"/><span class="lab"><font color="Green" title="E44">A28</font></span>: 
<font color="Maroon" title="c8">H2</font> is <font color="Maroon" title="c2">p</font> <a href="group_10.html#V2" title="GROUP_10:attr.2">-group</a> 
 ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c9">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> <span class="kw"> such that </span><br/><a NAME="E19:49"/><span class="lab"><font color="Green" title="E45">A29</font></span>: 
 <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c8">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="c9">r2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:49"><span class="lab"><font color="Green" title="E44">A28</font></span></a>, <a class="ref" href="group_10.html#D17" target="_self" title="GROUP_10:def.17">Def17</a></span>;<br/>
<a NAME="E20:49"/><span class="lab"><font color="Green" title="E46">A30</font></span>: 
 <a href="group_1.html#K7" title="GROUP_1:func.7">card</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#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c5">P2</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><a href="group_2.html#K18" title="GROUP_2:func.18">index</a> <font color="Maroon" title="c5">P2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_2.html#T147" title="GROUP_2:th.147">GROUP_2:147</a></span>;<br/>
<a NAME="E21:49"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A31</font></span>: 
<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#K18" title="GROUP_2:func.18">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="c9">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#K18" title="GROUP_2:func.18">index</a> <font color="Maroon" title="c5">P2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:49"><span class="lab"><font color="Green" title="E37">A21</font></span></a>, <a class="txt" href="#E11:49"><span class="lab"><font color="Green" title="E40">A24</font></span></a>, <a class="txt" href="#E16:49"><span class="lab"><font color="Green" title="E43">A27</font></span></a>, <a class="txt" href="#E19:49"><span class="lab"><font color="Green" title="E45">A29</font></span></a>, <a class="ref" href="group_2.html#T147" title="GROUP_2:th.147">GROUP_2:147</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_3"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:49_3"/><span class="lab"><font color="Green" title="E48">A32</font></span>: 
<font color="Maroon" title="c7">r1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">r2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_3_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:49_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="c9">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="c9">r2</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:49_3"><span class="lab"><font color="Green" title="E48">A32</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_3_1_1"><span class="kw">suppose </span></a><a NAME="E1:49_3_1_1"/><span class="lab"><font color="Green" title="E49">A33</font></span>: 
<font color="Maroon" title="c7">r1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c9">r2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c10">r</font> = <font color="Maroon" title="c7">r1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c9">r2</font>;<br/><a NAME="E2:49_3_1_1"/><span class="lab"><font color="Green" title="E50">A34</font></span>: 
<font color="Maroon" title="c7">r1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c9">r2</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c9">r2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c9">r2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:49_3_1_1"><span class="lab"><font color="Green" title="E49">A33</font></span></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/><a NAME="E3:49_3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A35</font></span>: 
<font color="Maroon" title="c7">r1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c9">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="c9">r2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D2" title="XREAL_0:def.2">XREAL_0:def 2</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E5:49_3_1_1"/><span class="lab"><font color="Green" title="E52">A36</font></span>: 
<font color="Maroon" title="c7">r1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c9">r2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:49_3_1_1"><span class="lab"><font color="Green" title="E50">A34</font></span></a>, <a class="ref" href="nat_1.html#T6" title="NAT_1:th.6">NAT_1:6</a></span>;<br/><a NAME="E6:49_3_1_1"/>
<font color="Maroon" title="c7">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">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="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c9">r2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:49_3_1_1"><span class="lab"><font color="Green" title="E51">A35</font></span></a></span>;<br/><a NAME="E7:49_3_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">p9</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">p9</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c9">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">p9</font> <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">r1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c9">r2</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T8" title="NEWTON:th.8">NEWTON:8</a></span>;<br/><a NAME="E8:49_3_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">p9</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c9">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">p9</font> <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">r1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c9">r2</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><a href="group_2.html#K18" title="GROUP_2:func.18">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">p9</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c9">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#K18" title="GROUP_2:func.18">index</a> <font color="Maroon" title="c5">P2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:49"><span class="lab"><font color="Green" title="E47">A31</font></span></a></span>;<br/><a NAME="E9:49_3_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">p9</font> <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">r1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c9">r2</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><a href="group_2.html#K18" title="GROUP_2:func.18">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#K18" title="GROUP_2:func.18">index</a> <font color="Maroon" title="c5">P2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_1.html#T5" title="XCMPLX_1:th.5">XCMPLX_1:5</a></span>;<br/><a NAME="E10:49_3_1_1"/><span class="kw">then </span>
<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="c11">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#K18" title="GROUP_2:func.18">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#K18" title="GROUP_2:func.18">index</a> <font color="Maroon" title="c5">P2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:49_3_1_1"><span class="lab"><font color="Green" title="E52">A36</font></span></a>, <a class="ref" href="newton.html#T6" title="NEWTON:th.6">NEWTON:6</a></span>;<br/><a NAME="E11:49_3_1_1"/><span class="kw">then </span>
<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="c11">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#K18" title="GROUP_2:func.18">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#K18" title="GROUP_2:func.18">index</a> <font color="Maroon" title="c5">P2</font>
 ;<br/><span class="kw">hence </span><a NAME="E12:49_3_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:49"><span class="lab"><font color="Green" title="E42">A26</font></span></a>, <a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">NAT_D:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_3_1_2"><span class="kw">suppose </span></a><a NAME="E1:49_3_1_2"/><span class="lab"><font color="Green" title="E49">A37</font></span>: 
<font color="Maroon" title="c7">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">r2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c10">r</font> = <font color="Maroon" title="c9">r2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c7">r1</font>;<br/><a NAME="E2:49_3_1_2"/><span class="lab"><font color="Green" title="E50">A38</font></span>: 
<font color="Maroon" title="c9">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:49_3_1_2"><span class="lab"><font color="Green" title="E49">A37</font></span></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/><a NAME="E3:49_3_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A39</font></span>: 
<font color="Maroon" title="c9">r2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c7">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">r2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c7">r1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D2" title="XREAL_0:def.2">XREAL_0:def 2</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E5:49_3_1_2"/><span class="lab"><font color="Green" title="E52">A40</font></span>: 
<font color="Maroon" title="c9">r2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c7">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:49_3_1_2"><span class="lab"><font color="Green" title="E50">A38</font></span></a>, <a class="ref" href="nat_1.html#T6" title="NAT_1:th.6">NAT_1:6</a></span>;<br/><a NAME="E6:49_3_1_2"/>
<font color="Maroon" title="c9">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="c9">r2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c7">r1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:49_3_1_2"><span class="lab"><font color="Green" title="E51">A39</font></span></a></span>;<br/><a NAME="E7:49_3_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">p9</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c9">r2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p9</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">p9</font> <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">r2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c7">r1</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T8" title="NEWTON:th.8">NEWTON:8</a></span>;<br/><a NAME="E8:49_3_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">p9</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">p9</font> <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">r2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c7">r1</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><a href="group_2.html#K18" title="GROUP_2:func.18">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">p9</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#K18" title="GROUP_2:func.18">index</a> <font color="Maroon" title="c4">P1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:49"><span class="lab"><font color="Green" title="E37">A21</font></span></a>, <a class="txt" href="#E11:49"><span class="lab"><font color="Green" title="E40">A24</font></span></a>, <a class="txt" href="#E16:49"><span class="lab"><font color="Green" title="E43">A27</font></span></a>, <a class="txt" href="#E19:49"><span class="lab"><font color="Green" title="E45">A29</font></span></a>, <a class="txt" href="#E20:49"><span class="lab"><font color="Green" title="E46">A30</font></span></a>, <a class="ref" href="group_2.html#T147" title="GROUP_2:th.147">GROUP_2:147</a></span>;<br/><a NAME="E9:49_3_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">p9</font> <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">r2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c7">r1</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><a href="group_2.html#K18" title="GROUP_2:func.18">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#K18" title="GROUP_2:func.18">index</a> <font color="Maroon" title="c4">P1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_1.html#T5" title="XCMPLX_1:th.5">XCMPLX_1:5</a></span>;<br/><a NAME="E10:49_3_1_2"/><span class="kw">then </span>
<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="c11">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#K18" title="GROUP_2:func.18">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#K18" title="GROUP_2:func.18">index</a> <font color="Maroon" title="c4">P1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:49_3_1_2"><span class="lab"><font color="Green" title="E52">A40</font></span></a>, <a class="ref" href="newton.html#T6" title="NEWTON:th.6">NEWTON:6</a></span>;<br/><a NAME="E11:49_3_1_2"/><span class="kw">then </span>
<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="c11">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#K18" title="GROUP_2:func.18">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#K18" title="GROUP_2:func.18">index</a> <font color="Maroon" title="c4">P1</font>
 ;<br/><span class="kw">hence </span><a NAME="E12:49_3_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:49"><span class="lab"><font color="Green" title="E39">A23</font></span></a>, <a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">NAT_D:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<a NAME="E23:49"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A41</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c4">P1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c5">P2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:49"><span class="lab"><font color="Green" title="E37">A21</font></span></a>, <a class="txt" href="#E11:49"><span class="lab"><font color="Green" title="E40">A24</font></span></a>, <a class="txt" href="#E16:49"><span class="lab"><font color="Green" title="E43">A27</font></span></a>, <a class="txt" href="#E19:49"><span class="lab"><font color="Green" title="E45">A29</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c10">g</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font><span class="kw"> such that </span><br/><a NAME="E25:49"/><span class="lab"><font color="Green" title="E49">A42</font></span>: 
 <a href="group_2.html#K8" title="GROUP_2:func.8">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#K8" title="GROUP_2:func.8">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="c10">g</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:49"><span class="lab"><font color="Green" title="E34">A1</font></span></a>, <a class="txt" href="#E5:49"><span class="lab"><font color="Green" title="E36">A20</font></span></a>, <a class="txt" href="#E12:49"><span class="lab"><font color="Green" title="E41">A25</font></span></a></span>;<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#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c5">P2</font></span>)</span> = 
 <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c5">P2</font>

<br/>.= 
 <a href="group_1.html#K7" title="GROUP_1:func.7">card</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="c10">g</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_3.html#T66" title="GROUP_3:th.66">GROUP_3:66</a></span>
<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#K8" title="GROUP_2:func.8">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="c10">g</font></span>)</span></span>)</span>

;<br/>
<a NAME="E27:49"/><span class="kw">then </span>
 <a href="algstr_0.html#G3" title="ALGSTR_0:aggr.3">multMagma</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">P1</font>, the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c4">P1</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#G3" title="ALGSTR_0:aggr.3">multMagma</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <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="c10">g</font></span>)</span>, the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</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="c10">g</font></span>)</span> #)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:49"><span class="lab"><font color="Green" title="E49">A42</font></span></a>, <a class="txt" href="#E23:49"><span class="lab"><font color="Green" title="E48">A41</font></span></a>, <a class="ref" href="card_fin.html#T1" title="CARD_FIN:th.1">CARD_FIN:1</a>, <a class="ref" href="group_2.html#T59" title="GROUP_2:th.59">GROUP_2:59</a></span>;<br/>
<span class="kw">hence </span><a NAME="E28:49"/>
<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> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_3.html#D11" title="GROUP_3:def.11">GROUP_3:def 11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
