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

<b>let </b><font color="Maroon" title="c1">G</font> be   <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">G</font> is <a href="struct_0.html#V8" title="STRUCT_0:attr.8">finite</a> implies for <font color="Olive" title="b1">C</font> being   <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font><br/> for <font color="Olive" title="b2">A</font>, <font color="Olive" title="b3">B</font> being   <a href="group_6.html#M1" title="GROUP_6:mode.1">Subgroup</a> of <font color="Olive" title="b1">C</font>  st <font color="Olive" title="b1">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">A</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Olive" title="b3">B</font> holds <br/>for <font color="Olive" title="b4">D</font> being   <a href="group_6.html#M1" title="GROUP_6:mode.1">Subgroup</a> of <font color="Olive" title="b2">A</font>  st <font color="Olive" title="b4">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">A</font> <a href="group_2.html#K9" title="GROUP_2:func.9">/\</a> <font color="Olive" title="b3">B</font> holds <br/>for <font color="Olive" title="b5">E</font> being   <a href="group_6.html#M1" title="GROUP_6:mode.1">Subgroup</a> of <font color="Olive" title="b3">B</font>  st <font color="Olive" title="b5">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">A</font> <a href="group_2.html#K9" title="GROUP_2:func.9">/\</a> <font color="Olive" title="b3">B</font> holds <br/>for <font color="Olive" title="b6">F</font> being   <a href="group_6.html#M1" title="GROUP_6:mode.1">Subgroup</a> of <font color="Olive" title="b1">C</font>  st <font color="Olive" title="b6">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">A</font> <a href="group_2.html#K9" title="GROUP_2:func.9">/\</a> <font color="Olive" title="b3">B</font> &amp;  <a href="group_2.html#K14" title="GROUP_2:func.14">Left_Cosets</a> <font color="Olive" title="b3">B</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp;  <a href="group_2.html#K14" title="GROUP_2:func.14">Left_Cosets</a> <font color="Olive" title="b2">A</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp;  <a href="group_8.html#NK2" title="GROUP_8:NK.2">index</a> <font color="Olive" title="b1">C</font>,<font color="Olive" title="b2">A</font>, <a href="group_8.html#NK2" title="GROUP_8:NK.2">index</a> <font color="Olive" title="b1">C</font>,<font color="Olive" title="b3">B</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a>  holds <br/>(  <a href="group_8.html#NK2" title="GROUP_8:NK.2">index</a> <font color="Olive" title="b1">C</font>,<font color="Olive" title="b3">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_8.html#NK2" title="GROUP_8:NK.2">index</a> <font color="Olive" title="b2">A</font>,<font color="Olive" title="b4">D</font> &amp;  <a href="group_8.html#NK2" title="GROUP_8:NK.2">index</a> <font color="Olive" title="b1">C</font>,<font color="Olive" title="b2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_8.html#NK2" title="GROUP_8:NK.2">index</a> <font color="Olive" title="b3">B</font>,<font color="Olive" title="b5">E</font> ) )</span><br/>

<b>assume </b><a NAME="E1:21"/><i><font color="Green" title="E11">A1</font></i>: 
<font color="Maroon" title="c1">G</font> is <a href="struct_0.html#V8" title="STRUCT_0:attr.8">finite</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">C</font> being   <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font><br/> for <font color="Olive" title="b2">A</font>, <font color="Olive" title="b3">B</font> being   <a href="group_6.html#M1" title="GROUP_6:mode.1">Subgroup</a> of <font color="Olive" title="b1">C</font>  st <font color="Olive" title="b1">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">A</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Olive" title="b3">B</font> holds <br/>for <font color="Olive" title="b4">D</font> being   <a href="group_6.html#M1" title="GROUP_6:mode.1">Subgroup</a> of <font color="Olive" title="b2">A</font>  st <font color="Olive" title="b4">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">A</font> <a href="group_2.html#K9" title="GROUP_2:func.9">/\</a> <font color="Olive" title="b3">B</font> holds <br/>for <font color="Olive" title="b5">E</font> being   <a href="group_6.html#M1" title="GROUP_6:mode.1">Subgroup</a> of <font color="Olive" title="b3">B</font>  st <font color="Olive" title="b5">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">A</font> <a href="group_2.html#K9" title="GROUP_2:func.9">/\</a> <font color="Olive" title="b3">B</font> holds <br/>for <font color="Olive" title="b6">F</font> being   <a href="group_6.html#M1" title="GROUP_6:mode.1">Subgroup</a> of <font color="Olive" title="b1">C</font>  st <font color="Olive" title="b6">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">A</font> <a href="group_2.html#K9" title="GROUP_2:func.9">/\</a> <font color="Olive" title="b3">B</font> &amp;  <a href="group_2.html#K14" title="GROUP_2:func.14">Left_Cosets</a> <font color="Olive" title="b3">B</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp;  <a href="group_2.html#K14" title="GROUP_2:func.14">Left_Cosets</a> <font color="Olive" title="b2">A</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp;  <a href="group_8.html#NK2" title="GROUP_8:NK.2">index</a> <font color="Olive" title="b1">C</font>,<font color="Olive" title="b2">A</font>, <a href="group_8.html#NK2" title="GROUP_8:NK.2">index</a> <font color="Olive" title="b1">C</font>,<font color="Olive" title="b3">B</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a>  holds <br/>(  <a href="group_8.html#NK2" title="GROUP_8:NK.2">index</a> <font color="Olive" title="b1">C</font>,<font color="Olive" title="b3">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_8.html#NK2" title="GROUP_8:NK.2">index</a> <font color="Olive" title="b2">A</font>,<font color="Olive" title="b4">D</font> &amp;  <a href="group_8.html#NK2" title="GROUP_8:NK.2">index</a> <font color="Olive" title="b1">C</font>,<font color="Olive" title="b2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_8.html#NK2" title="GROUP_8:NK.2">index</a> <font color="Olive" title="b3">B</font>,<font color="Olive" title="b5">E</font> )</span><br/>

<b>let </b><font color="Maroon" title="c2">C</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</font> being   <a href="group_6.html#M1" title="GROUP_6:mode.1">Subgroup</a> of <font color="Maroon" title="c2">C</font>  st <font color="Maroon" title="c2">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">A</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Olive" title="b2">B</font> holds <br/>for <font color="Olive" title="b3">D</font> being   <a href="group_6.html#M1" title="GROUP_6:mode.1">Subgroup</a> of <font color="Olive" title="b1">A</font>  st <font color="Olive" title="b3">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">A</font> <a href="group_2.html#K9" title="GROUP_2:func.9">/\</a> <font color="Olive" title="b2">B</font> holds <br/>for <font color="Olive" title="b4">E</font> being   <a href="group_6.html#M1" title="GROUP_6:mode.1">Subgroup</a> of <font color="Olive" title="b2">B</font>  st <font color="Olive" title="b4">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">A</font> <a href="group_2.html#K9" title="GROUP_2:func.9">/\</a> <font color="Olive" title="b2">B</font> holds <br/>for <font color="Olive" title="b5">F</font> being   <a href="group_6.html#M1" title="GROUP_6:mode.1">Subgroup</a> of <font color="Maroon" title="c2">C</font>  st <font color="Olive" title="b5">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">A</font> <a href="group_2.html#K9" title="GROUP_2:func.9">/\</a> <font color="Olive" title="b2">B</font> &amp;  <a href="group_2.html#K14" title="GROUP_2:func.14">Left_Cosets</a> <font color="Olive" title="b2">B</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp;  <a href="group_2.html#K14" title="GROUP_2:func.14">Left_Cosets</a> <font color="Olive" title="b1">A</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp;  <a href="group_8.html#NK2" title="GROUP_8:NK.2">index</a> <font color="Maroon" title="c2">C</font>,<font color="Olive" title="b1">A</font>, <a href="group_8.html#NK2" title="GROUP_8:NK.2">index</a> <font color="Maroon" title="c2">C</font>,<font color="Olive" title="b2">B</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a>  holds <br/>(  <a href="group_8.html#NK2" title="GROUP_8:NK.2">index</a> <font color="Maroon" title="c2">C</font>,<font color="Olive" title="b2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_8.html#NK2" title="GROUP_8:NK.2">index</a> <font color="Olive" title="b1">A</font>,<font color="Olive" title="b3">D</font> &amp;  <a href="group_8.html#NK2" title="GROUP_8:NK.2">index</a> <font color="Maroon" title="c2">C</font>,<font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_8.html#NK2" title="GROUP_8:NK.2">index</a> <font color="Olive" title="b2">B</font>,<font color="Olive" title="b4">E</font> )</span><br/><b>let </b><font color="Maroon" title="c3">A</font>, <font color="Maroon" title="c4">B</font> be    <a href="group_6.html#M1" title="GROUP_6:mode.1">Subgroup</a> of <font color="Maroon" title="c2">C</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c4">B</font> implies for <font color="Olive" title="b1">D</font> being   <a href="group_6.html#M1" title="GROUP_6:mode.1">Subgroup</a> of <font color="Maroon" title="c3">A</font>  st <font color="Olive" title="b1">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A</font> <a href="group_2.html#K9" title="GROUP_2:func.9">/\</a> <font color="Maroon" title="c4">B</font> holds <br/>for <font color="Olive" title="b2">E</font> being   <a href="group_6.html#M1" title="GROUP_6:mode.1">Subgroup</a> of <font color="Maroon" title="c4">B</font>  st <font color="Olive" title="b2">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A</font> <a href="group_2.html#K9" title="GROUP_2:func.9">/\</a> <font color="Maroon" title="c4">B</font> holds <br/>for <font color="Olive" title="b3">F</font> being   <a href="group_6.html#M1" title="GROUP_6:mode.1">Subgroup</a> of <font color="Maroon" title="c2">C</font>  st <font color="Olive" title="b3">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A</font> <a href="group_2.html#K9" title="GROUP_2:func.9">/\</a> <font color="Maroon" title="c4">B</font> &amp;  <a href="group_2.html#K14" title="GROUP_2:func.14">Left_Cosets</a> <font color="Maroon" title="c4">B</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp;  <a href="group_2.html#K14" title="GROUP_2:func.14">Left_Cosets</a> <font color="Maroon" title="c3">A</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp;  <a href="group_8.html#NK2" title="GROUP_8:NK.2">index</a> <font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c3">A</font>, <a href="group_8.html#NK2" title="GROUP_8:NK.2">index</a> <font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c4">B</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a>  holds <br/>(  <a href="group_8.html#NK2" title="GROUP_8:NK.2">index</a> <font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c4">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_8.html#NK2" title="GROUP_8:NK.2">index</a> <font color="Maroon" title="c3">A</font>,<font color="Olive" title="b1">D</font> &amp;  <a href="group_8.html#NK2" title="GROUP_8:NK.2">index</a> <font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_8.html#NK2" title="GROUP_8:NK.2">index</a> <font color="Maroon" title="c4">B</font>,<font color="Olive" title="b2">E</font> ) )</span><br/>



<b>assume </b><a NAME="E2:21"/><i><font color="Green" title="E12">A2</font></i>: 
<font color="Maroon" title="c2">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c4">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">D</font> being   <a href="group_6.html#M1" title="GROUP_6:mode.1">Subgroup</a> of <font color="Maroon" title="c3">A</font>  st <font color="Olive" title="b1">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A</font> <a href="group_2.html#K9" title="GROUP_2:func.9">/\</a> <font color="Maroon" title="c4">B</font> holds <br/>for <font color="Olive" title="b2">E</font> being   <a href="group_6.html#M1" title="GROUP_6:mode.1">Subgroup</a> of <font color="Maroon" title="c4">B</font>  st <font color="Olive" title="b2">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A</font> <a href="group_2.html#K9" title="GROUP_2:func.9">/\</a> <font color="Maroon" title="c4">B</font> holds <br/>for <font color="Olive" title="b3">F</font> being   <a href="group_6.html#M1" title="GROUP_6:mode.1">Subgroup</a> of <font color="Maroon" title="c2">C</font>  st <font color="Olive" title="b3">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A</font> <a href="group_2.html#K9" title="GROUP_2:func.9">/\</a> <font color="Maroon" title="c4">B</font> &amp;  <a href="group_2.html#K14" title="GROUP_2:func.14">Left_Cosets</a> <font color="Maroon" title="c4">B</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp;  <a href="group_2.html#K14" title="GROUP_2:func.14">Left_Cosets</a> <font color="Maroon" title="c3">A</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp;  <a href="group_8.html#NK2" title="GROUP_8:NK.2">index</a> <font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c3">A</font>, <a href="group_8.html#NK2" title="GROUP_8:NK.2">index</a> <font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c4">B</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a>  holds <br/>(  <a href="group_8.html#NK2" title="GROUP_8:NK.2">index</a> <font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c4">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_8.html#NK2" title="GROUP_8:NK.2">index</a> <font color="Maroon" title="c3">A</font>,<font color="Olive" title="b1">D</font> &amp;  <a href="group_8.html#NK2" title="GROUP_8:NK.2">index</a> <font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_8.html#NK2" title="GROUP_8:NK.2">index</a> <font color="Maroon" title="c4">B</font>,<font color="Olive" title="b2">E</font> )</span><br/>

<b>let </b><font color="Maroon" title="c5">D</font> be    <a href="group_6.html#M1" title="GROUP_6:mode.1">Subgroup</a> of <font color="Maroon" title="c3">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A</font> <a href="group_2.html#K9" title="GROUP_2:func.9">/\</a> <font color="Maroon" title="c4">B</font> implies for <font color="Olive" title="b1">E</font> being   <a href="group_6.html#M1" title="GROUP_6:mode.1">Subgroup</a> of <font color="Maroon" title="c4">B</font>  st <font color="Olive" title="b1">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A</font> <a href="group_2.html#K9" title="GROUP_2:func.9">/\</a> <font color="Maroon" title="c4">B</font> holds <br/>for <font color="Olive" title="b2">F</font> being   <a href="group_6.html#M1" title="GROUP_6:mode.1">Subgroup</a> of <font color="Maroon" title="c2">C</font>  st <font color="Olive" title="b2">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A</font> <a href="group_2.html#K9" title="GROUP_2:func.9">/\</a> <font color="Maroon" title="c4">B</font> &amp;  <a href="group_2.html#K14" title="GROUP_2:func.14">Left_Cosets</a> <font color="Maroon" title="c4">B</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp;  <a href="group_2.html#K14" title="GROUP_2:func.14">Left_Cosets</a> <font color="Maroon" title="c3">A</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp;  <a href="group_8.html#NK2" title="GROUP_8:NK.2">index</a> <font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c3">A</font>, <a href="group_8.html#NK2" title="GROUP_8:NK.2">index</a> <font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c4">B</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a>  holds <br/>(  <a href="group_8.html#NK2" title="GROUP_8:NK.2">index</a> <font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c4">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_8.html#NK2" title="GROUP_8:NK.2">index</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c5">D</font> &amp;  <a href="group_8.html#NK2" title="GROUP_8:NK.2">index</a> <font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_8.html#NK2" title="GROUP_8:NK.2">index</a> <font color="Maroon" title="c4">B</font>,<font color="Olive" title="b1">E</font> ) )</span><br/>

<b>assume </b><a NAME="E3:21"/><i><font color="Green" title="E13">A3</font></i>: 
<font color="Maroon" title="c5">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A</font> <a href="group_2.html#K9" title="GROUP_2:func.9">/\</a> <font color="Maroon" title="c4">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">E</font> being   <a href="group_6.html#M1" title="GROUP_6:mode.1">Subgroup</a> of <font color="Maroon" title="c4">B</font>  st <font color="Olive" title="b1">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A</font> <a href="group_2.html#K9" title="GROUP_2:func.9">/\</a> <font color="Maroon" title="c4">B</font> holds <br/>for <font color="Olive" title="b2">F</font> being   <a href="group_6.html#M1" title="GROUP_6:mode.1">Subgroup</a> of <font color="Maroon" title="c2">C</font>  st <font color="Olive" title="b2">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A</font> <a href="group_2.html#K9" title="GROUP_2:func.9">/\</a> <font color="Maroon" title="c4">B</font> &amp;  <a href="group_2.html#K14" title="GROUP_2:func.14">Left_Cosets</a> <font color="Maroon" title="c4">B</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp;  <a href="group_2.html#K14" title="GROUP_2:func.14">Left_Cosets</a> <font color="Maroon" title="c3">A</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp;  <a href="group_8.html#NK2" title="GROUP_8:NK.2">index</a> <font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c3">A</font>, <a href="group_8.html#NK2" title="GROUP_8:NK.2">index</a> <font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c4">B</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a>  holds <br/>(  <a href="group_8.html#NK2" title="GROUP_8:NK.2">index</a> <font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c4">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_8.html#NK2" title="GROUP_8:NK.2">index</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c5">D</font> &amp;  <a href="group_8.html#NK2" title="GROUP_8:NK.2">index</a> <font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_8.html#NK2" title="GROUP_8:NK.2">index</a> <font color="Maroon" title="c4">B</font>,<font color="Olive" title="b1">E</font> )</span><br/>

<b>let </b><font color="Maroon" title="c6">E</font> be    <a href="group_6.html#M1" title="GROUP_6:mode.1">Subgroup</a> of <font color="Maroon" title="c4">B</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c6">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A</font> <a href="group_2.html#K9" title="GROUP_2:func.9">/\</a> <font color="Maroon" title="c4">B</font> implies for <font color="Olive" title="b1">F</font> being   <a href="group_6.html#M1" title="GROUP_6:mode.1">Subgroup</a> of <font color="Maroon" title="c2">C</font>  st <font color="Olive" title="b1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A</font> <a href="group_2.html#K9" title="GROUP_2:func.9">/\</a> <font color="Maroon" title="c4">B</font> &amp;  <a href="group_2.html#K14" title="GROUP_2:func.14">Left_Cosets</a> <font color="Maroon" title="c4">B</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp;  <a href="group_2.html#K14" title="GROUP_2:func.14">Left_Cosets</a> <font color="Maroon" title="c3">A</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp;  <a href="group_8.html#NK2" title="GROUP_8:NK.2">index</a> <font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c3">A</font>, <a href="group_8.html#NK2" title="GROUP_8:NK.2">index</a> <font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c4">B</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a>  holds <br/>(  <a href="group_8.html#NK2" title="GROUP_8:NK.2">index</a> <font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c4">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_8.html#NK2" title="GROUP_8:NK.2">index</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c5">D</font> &amp;  <a href="group_8.html#NK2" title="GROUP_8:NK.2">index</a> <font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_8.html#NK2" title="GROUP_8:NK.2">index</a> <font color="Maroon" title="c4">B</font>,<font color="Maroon" title="c6">E</font> ) )</span><br/>

<b>assume </b><a NAME="E4:21"/><i><font color="Green" title="E14">A4</font></i>: 
<font color="Maroon" title="c6">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A</font> <a href="group_2.html#K9" title="GROUP_2:func.9">/\</a> <font color="Maroon" title="c4">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">F</font> being   <a href="group_6.html#M1" title="GROUP_6:mode.1">Subgroup</a> of <font color="Maroon" title="c2">C</font>  st <font color="Olive" title="b1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A</font> <a href="group_2.html#K9" title="GROUP_2:func.9">/\</a> <font color="Maroon" title="c4">B</font> &amp;  <a href="group_2.html#K14" title="GROUP_2:func.14">Left_Cosets</a> <font color="Maroon" title="c4">B</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp;  <a href="group_2.html#K14" title="GROUP_2:func.14">Left_Cosets</a> <font color="Maroon" title="c3">A</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp;  <a href="group_8.html#NK2" title="GROUP_8:NK.2">index</a> <font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c3">A</font>, <a href="group_8.html#NK2" title="GROUP_8:NK.2">index</a> <font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c4">B</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a>  holds <br/>(  <a href="group_8.html#NK2" title="GROUP_8:NK.2">index</a> <font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c4">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_8.html#NK2" title="GROUP_8:NK.2">index</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c5">D</font> &amp;  <a href="group_8.html#NK2" title="GROUP_8:NK.2">index</a> <font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_8.html#NK2" title="GROUP_8:NK.2">index</a> <font color="Maroon" title="c4">B</font>,<font color="Maroon" title="c6">E</font> )</span><br/>

<b>let </b><font color="Maroon" title="c7">F</font> be    <a href="group_6.html#M1" title="GROUP_6:mode.1">Subgroup</a> of <font color="Maroon" title="c2">C</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c7">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A</font> <a href="group_2.html#K9" title="GROUP_2:func.9">/\</a> <font color="Maroon" title="c4">B</font> &amp;  <a href="group_2.html#K14" title="GROUP_2:func.14">Left_Cosets</a> <font color="Maroon" title="c4">B</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp;  <a href="group_2.html#K14" title="GROUP_2:func.14">Left_Cosets</a> <font color="Maroon" title="c3">A</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp;  <a href="group_8.html#NK2" title="GROUP_8:NK.2">index</a> <font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c3">A</font>, <a href="group_8.html#NK2" title="GROUP_8:NK.2">index</a> <font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c4">B</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a>  implies (  <a href="group_8.html#NK2" title="GROUP_8:NK.2">index</a> <font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c4">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_8.html#NK2" title="GROUP_8:NK.2">index</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c5">D</font> &amp;  <a href="group_8.html#NK2" title="GROUP_8:NK.2">index</a> <font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_8.html#NK2" title="GROUP_8:NK.2">index</a> <font color="Maroon" title="c4">B</font>,<font color="Maroon" title="c6">E</font> ) )</span><br/>

<b>assume </b><a NAME="E5:21"/><i><font color="Green" title="E15">A5</font></i>: 
<font color="Maroon" title="c7">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A</font> <a href="group_2.html#K9" title="GROUP_2:func.9">/\</a> <font color="Maroon" title="c4">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not  <a href="group_2.html#K14" title="GROUP_2:func.14">Left_Cosets</a> <font color="Maroon" title="c4">B</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> or  not  <a href="group_2.html#K14" title="GROUP_2:func.14">Left_Cosets</a> <font color="Maroon" title="c3">A</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> or  not  <a href="group_8.html#NK2" title="GROUP_8:NK.2">index</a> <font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c3">A</font>, <a href="group_8.html#NK2" title="GROUP_8:NK.2">index</a> <font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c4">B</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a>  or (  <a href="group_8.html#NK2" title="GROUP_8:NK.2">index</a> <font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c4">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_8.html#NK2" title="GROUP_8:NK.2">index</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c5">D</font> &amp;  <a href="group_8.html#NK2" title="GROUP_8:NK.2">index</a> <font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_8.html#NK2" title="GROUP_8:NK.2">index</a> <font color="Maroon" title="c4">B</font>,<font color="Maroon" title="c6">E</font> ) )</span><br/>

<b>assume </b><a NAME="E6:21"/><i><font color="Green" title="E16">A6</font></i>: 
(  <a href="group_2.html#K14" title="GROUP_2:func.14">Left_Cosets</a> <font color="Maroon" title="c4">B</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp;  <a href="group_2.html#K14" title="GROUP_2:func.14">Left_Cosets</a> <font color="Maroon" title="c3">A</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp;  <a href="group_2.html#K17" title="GROUP_2:func.17">index</a> <font color="Maroon" title="c3">A</font>, <a href="group_2.html#K17" title="GROUP_2:func.17">index</a> <font color="Maroon" title="c4">B</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="group_8.html#NK2" title="GROUP_8:NK.2">index</a> <font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c4">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_8.html#NK2" title="GROUP_8:NK.2">index</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c5">D</font> &amp;  <a href="group_8.html#NK2" title="GROUP_8:NK.2">index</a> <font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_8.html#NK2" title="GROUP_8:NK.2">index</a> <font color="Maroon" title="c4">B</font>,<font color="Maroon" title="c6">E</font> )</span><br/>

<a NAME="E7:21"/><i><font color="Green" title="E17">A7</font></i>: 
( <font color="Maroon" title="c2">C</font> is <a href="struct_0.html#V8" title="STRUCT_0:attr.8">finite</a> &amp; <font color="Maroon" title="c7">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">E</font> &amp; <font color="Maroon" title="c7">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D</font> )
 
<b>by </b><i><a class="txt" href="group_8.html#E1:21"><i><font color="Green" title="E11">A1</font></i></a>, <a class="txt" href="group_8.html#E3:21"><i><font color="Green" title="E13">A3</font></i></a>, <a class="txt" href="group_8.html#E4:21"><i><font color="Green" title="E14">A4</font></i></a>, <a class="txt" href="group_8.html#E5:21"><i><font color="Green" title="E15">A5</font></i></a>, <a class="ref" href="group_2.html#T48" title="GROUP_2:th.48">GROUP_2:48</a></i>;<br/>
<a NAME="E8:21"/><b>then </b>
 <a href="group_2.html#K17" title="GROUP_2:func.17">index</a> <font color="Maroon" title="c7">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="group_2.html#K17" title="GROUP_2:func.17">index</a> <font color="Maroon" title="c6">E</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#K17" title="GROUP_2:func.17">index</a> <font color="Maroon" title="c4">B</font></span>)</span>
 
<b>by </b><i><a class="ref" href="group_2.html#T179" title="GROUP_2:th.179">GROUP_2:179</a></i>;<br/>
<a NAME="E9:21"/><b>then </b><i><font color="Green" title="E18">A8</font></i>: 
<span class="p1">(<span class="default"><a href="group_2.html#K17" title="GROUP_2:func.17">index</a> <font color="Maroon" title="c6">E</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#K17" title="GROUP_2:func.17">index</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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">D</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#K17" title="GROUP_2:func.17">index</a> <font color="Maroon" title="c3">A</font></span>)</span>
 
<b>by </b><i><a class="txt" href="group_8.html#E7:21"><i><font color="Green" title="E17">A7</font></i></a>, <a class="ref" href="group_2.html#T179" title="GROUP_2:th.179">GROUP_2:179</a></i>;<br/>
<a NAME="E10:21"/><b>then </b>
 <a href="group_2.html#K17" title="GROUP_2:func.17">index</a> <font color="Maroon" title="c4">B</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <span class="p1">(<span class="default"><a href="group_2.html#K17" title="GROUP_2:func.17">index</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0: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">D</font></span>)</span>
 
<b>by </b><i><a class="ref" href="int_1.html#D9" title="INT_1:def.9">INT_1:def 9</a></i>;<br/>
<a NAME="E11:21"/><b>then </b><i><font color="Green" title="E19">A9</font></i>: 
 <a href="group_2.html#K17" title="GROUP_2:func.17">index</a> <font color="Maroon" title="c4">B</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a>  <a href="group_2.html#K17" title="GROUP_2:func.17">index</a> <font color="Maroon" title="c5">D</font>
 
<b>by </b><i><a class="txt" href="group_8.html#E6:21"><i><font color="Green" title="E16">A6</font></i></a>, <a class="ref" href="int_2.html#T40" title="INT_2:th.40">INT_2:40</a></i>;<br/>
<a NAME="E12:21"/><i><font color="Green" title="E20">A10</font></i>: 
 <a href="group_2.html#K17" title="GROUP_2:func.17">index</a> <font color="Maroon" title="c4">B</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a>  <a href="group_2.html#K17" title="GROUP_2:func.17">index</a> <font color="Maroon" title="c5">D</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1"><b>proof </b></a><div class="add">

<a NAME="E1:21_1"/>
 ex <font color="Olive" title="b1">n</font> being   <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>  st  <a href="group_2.html#K17" title="GROUP_2:func.17">index</a> <font color="Maroon" title="c5">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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">B</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Olive" title="b1">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon" title="c8">i</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><b> such that </b><br/><a NAME="E2:21_1_1"/><i><font color="Green" title="E21">A11</font></i>: 
 <a href="group_2.html#K17" title="GROUP_2:func.17">index</a> <font color="Maroon" title="c5">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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">B</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c8">i</font>
 <b>by </b><i><a class="txt" href="group_8.html#E11:21"><i><font color="Green" title="E19">A9</font></i></a>, <a class="ref" href="int_1.html#D9" title="INT_1:def.9">INT_1:def 9</a></i>;<br/>
<a NAME="E3:21_1_1"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:21_1_1_1"/><i><font color="Green" title="E22">A12</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c8">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:21_1_1_1"/>
<font color="Maroon" title="c3">A</font> is <a href="struct_0.html#V8" title="STRUCT_0:attr.8">finite</a>
 
<b>by </b><i><a class="txt" href="group_8.html#E1:21"><i><font color="Green" title="E11">A1</font></i></a>, <a class="ref" href="group_2.html#T48" title="GROUP_2:th.48">GROUP_2:48</a></i>;<br/>
<b>hence </b><a NAME="E3:21_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="group_8.html#E2:21_1_1"><i><font color="Green" title="E21">A11</font></i></a>, <a class="txt" href="group_8.html#E1:21_1_1_1"><i><font color="Green" title="E22">A12</font></i></a>, <a class="ref" href="group_8.html#T20" target="_self" title="GROUP_8:th.20">Th20</a>, <a class="ref" href="xreal_1.html#T133" title="XREAL_1:th.133">XREAL_1:133</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c9">i</font> = <font color="Maroon" title="c8">i</font> as    <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="int_1.html#T16" title="INT_1:th.16">INT_1:16</a></i>;<br/>
<b>take </b><font color="Maroon" title="c10">n</font> = <font color="Maroon" title="c9">i</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="group_2.html#K17" title="GROUP_2:func.17">index</a> <font color="Maroon" title="c5">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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">B</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c10">n</font></span><br/>

<b>thus </b><a NAME="E5:21_1_1"/>
 <a href="group_2.html#K17" title="GROUP_2:func.17">index</a> <font color="Maroon" title="c5">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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">B</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c10">n</font>
 <b>by </b><i><a class="txt" href="group_8.html#E2:21_1_1"><i><font color="Green" title="E21">A11</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:21_1"/>
 <a href="group_2.html#K17" title="GROUP_2:func.17">index</a> <font color="Maroon" title="c4">B</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a>  <a href="group_2.html#K17" title="GROUP_2:func.17">index</a> <font color="Maroon" title="c5">D</font>
 <b>by </b><i><a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">NAT_D:def 3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E13:21"/><i><font color="Green" title="E21">A13</font></i>: 
 <a href="group_8.html#NK2" title="GROUP_8:NK.2">index</a> <font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c4">B</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="group_8.html#NK2" title="GROUP_8:NK.2">index</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c5">D</font>
 
<b>by </b><i><a class="txt" href="group_8.html#E2:21"><i><font color="Green" title="E12">A2</font></i></a>, <a class="txt" href="group_8.html#E3:21"><i><font color="Green" title="E13">A3</font></i></a>, <a class="txt" href="group_8.html#E7:21"><i><font color="Green" title="E17">A7</font></i></a>, <a class="ref" href="group_8.html#T19" target="_self" title="GROUP_8:th.19">Th19</a></i>;<br/>
<a NAME="E14:21"/><i><font color="Green" title="E22">A14</font></i>: 
 <a href="group_2.html#K17" title="GROUP_2:func.17">index</a> <font color="Maroon" title="c4">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_2.html#K17" title="GROUP_2:func.17">index</a> <font color="Maroon" title="c5">D</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:21_2"/><i><font color="Green" title="E23">A15</font></i>: 
 <a href="group_2.html#K17" title="GROUP_2:func.17">index</a> <font color="Maroon" title="c4">B</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="group_2.html#K17" title="GROUP_2:func.17">index</a> <font color="Maroon" title="c5">D</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:21_2"/>
<font color="Maroon" title="c3">A</font> is <a href="struct_0.html#V8" title="STRUCT_0:attr.8">finite</a>
 
<b>by </b><i><a class="txt" href="group_8.html#E1:21"><i><font color="Green" title="E11">A1</font></i></a>, <a class="ref" href="group_2.html#T48" title="GROUP_2:th.48">GROUP_2:48</a></i>;<br/>
<a NAME="E3:21_2"/><b>then </b>
 <a href="group_2.html#K17" title="GROUP_2:func.17">index</a> <font color="Maroon" title="c5">D</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="group_8.html#T20" target="_self" title="GROUP_8:th.20">Th20</a></i>;<br/>
<a NAME="E4:21_2"/><b>then </b>
 <a href="group_2.html#K17" title="GROUP_2:func.17">index</a> <font color="Maroon" title="c4">B</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="group_2.html#K17" title="GROUP_2:func.17">index</a> <font color="Maroon" title="c5">D</font>
 
<b>by </b><i><a class="txt" href="group_8.html#E12:21"><i><font color="Green" title="E20">A10</font></i></a>, <a class="ref" href="nat_d.html#T7" title="NAT_D:th.7">NAT_D:7</a></i>;<br/>
<b>hence </b><a NAME="E5:21_2"/>
contradiction
 <b>by </b><i><a class="txt" href="group_8.html#E13:21"><i><font color="Green" title="E21">A13</font></i></a>, <a class="txt" href="group_8.html#E1:21_2"><i><font color="Green" title="E23">A15</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E15:21"/>
 <a href="group_2.html#K17" title="GROUP_2:func.17">index</a> <font color="Maroon" title="c3">A</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</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">B</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0: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="c6">E</font></span>)</span>
 
<b>by </b><i><a class="txt" href="group_8.html#E9:21"><i><font color="Green" title="E18">A8</font></i></a>, <a class="ref" href="int_1.html#D9" title="INT_1:def.9">INT_1:def 9</a></i>;<br/>
<a NAME="E16:21"/><b>then </b><i><font color="Green" title="E23">A16</font></i>: 
 <a href="group_2.html#K17" title="GROUP_2:func.17">index</a> <font color="Maroon" title="c3">A</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a>  <a href="group_2.html#K17" title="GROUP_2:func.17">index</a> <font color="Maroon" title="c6">E</font>
 
<b>by </b><i><a class="txt" href="group_8.html#E6:21"><i><font color="Green" title="E16">A6</font></i></a>, <a class="ref" href="int_2.html#T40" title="INT_2:th.40">INT_2:40</a></i>;<br/>
<a NAME="E17:21"/><i><font color="Green" title="E24">A17</font></i>: 
 <a href="group_2.html#K17" title="GROUP_2:func.17">index</a> <font color="Maroon" title="c3">A</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a>  <a href="group_2.html#K17" title="GROUP_2:func.17">index</a> <font color="Maroon" title="c6">E</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_3"><b>proof </b></a><div class="add">

<a NAME="E1:21_3"/>
 ex <font color="Olive" title="b1">n</font> being   <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>  st  <a href="group_2.html#K17" title="GROUP_2:func.17">index</a> <font color="Maroon" title="c6">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="group_2.html#K17" title="GROUP_2:func.17">index</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Olive" title="b1">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_3_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon" title="c8">i</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><b> such that </b><br/><a NAME="E2:21_3_1"/><i><font color="Green" title="E25">A18</font></i>: 
 <a href="group_2.html#K17" title="GROUP_2:func.17">index</a> <font color="Maroon" title="c6">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="group_2.html#K17" title="GROUP_2:func.17">index</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c8">i</font>
 <b>by </b><i><a class="txt" href="group_8.html#E16:21"><i><font color="Green" title="E23">A16</font></i></a>, <a class="ref" href="int_1.html#D9" title="INT_1:def.9">INT_1:def 9</a></i>;<br/>
<a NAME="E3:21_3_1"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_3_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:21_3_1_1"/><i><font color="Green" title="E26">A19</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c8">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:21_3_1_1"/>
<font color="Maroon" title="c4">B</font> is <a href="struct_0.html#V8" title="STRUCT_0:attr.8">finite</a>
 
<b>by </b><i><a class="txt" href="group_8.html#E1:21"><i><font color="Green" title="E11">A1</font></i></a>, <a class="ref" href="group_2.html#T48" title="GROUP_2:th.48">GROUP_2:48</a></i>;<br/>
<b>hence </b><a NAME="E3:21_3_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="group_8.html#E2:21_3_1"><i><font color="Green" title="E25">A18</font></i></a>, <a class="txt" href="group_8.html#E1:21_3_1_1"><i><font color="Green" title="E26">A19</font></i></a>, <a class="ref" href="group_8.html#T20" target="_self" title="GROUP_8:th.20">Th20</a>, <a class="ref" href="xreal_1.html#T133" title="XREAL_1:th.133">XREAL_1:133</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c9">i</font> = <font color="Maroon" title="c8">i</font> as    <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="int_1.html#T16" title="INT_1:th.16">INT_1:16</a></i>;<br/>
<b>take </b><font color="Maroon" title="c10">n</font> = <font color="Maroon" title="c9">i</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="group_2.html#K17" title="GROUP_2:func.17">index</a> <font color="Maroon" title="c6">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="group_2.html#K17" title="GROUP_2:func.17">index</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c10">n</font></span><br/>

<b>thus </b><a NAME="E5:21_3_1"/>
 <a href="group_2.html#K17" title="GROUP_2:func.17">index</a> <font color="Maroon" title="c6">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="group_2.html#K17" title="GROUP_2:func.17">index</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c10">n</font>
 <b>by </b><i><a class="txt" href="group_8.html#E2:21_3_1"><i><font color="Green" title="E25">A18</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:21_3"/>
 <a href="group_2.html#K17" title="GROUP_2:func.17">index</a> <font color="Maroon" title="c3">A</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a>  <a href="group_2.html#K17" title="GROUP_2:func.17">index</a> <font color="Maroon" title="c6">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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E18:21"/><i><font color="Green" title="E25">A20</font></i>: 
 <a href="group_2.html#K17" title="GROUP_2:func.17">index</a> <font color="Maroon" title="c3">A</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="group_2.html#K17" title="GROUP_2:func.17">index</a> <font color="Maroon" title="c6">E</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_4"><b>proof </b></a><div class="add">

<a NAME="E1:21_4"/>
<font color="Maroon" title="c2">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">B</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c3">A</font>
 
<b>by </b><i><a class="txt" href="group_8.html#E2:21"><i><font color="Green" title="E12">A2</font></i></a>, <a class="ref" href="group_4.html#T74" title="GROUP_4:th.74">GROUP_4:74</a></i>;<br/>
<b>hence </b><a NAME="E2:21_4"/>
 <a href="group_2.html#K17" title="GROUP_2:func.17">index</a> <font color="Maroon" title="c3">A</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="group_2.html#K17" title="GROUP_2:func.17">index</a> <font color="Maroon" title="c6">E</font>
 <b>by </b><i><a class="txt" href="group_8.html#E4:21"><i><font color="Green" title="E14">A4</font></i></a>, <a class="txt" href="group_8.html#E7:21"><i><font color="Green" title="E17">A7</font></i></a>, <a class="ref" href="group_8.html#T19" target="_self" title="GROUP_8:th.19">Th19</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E19:21"/>
 <a href="group_2.html#K17" title="GROUP_2:func.17">index</a> <font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_2.html#K17" title="GROUP_2:func.17">index</a> <font color="Maroon" title="c6">E</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_5"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:21_5"/><i><font color="Green" title="E26">A21</font></i>: 
 <a href="group_2.html#K17" title="GROUP_2:func.17">index</a> <font color="Maroon" title="c3">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="group_2.html#K17" title="GROUP_2:func.17">index</a> <font color="Maroon" title="c6">E</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:21_5"/>
<font color="Maroon" title="c4">B</font> is <a href="struct_0.html#V8" title="STRUCT_0:attr.8">finite</a>
 
<b>by </b><i><a class="txt" href="group_8.html#E1:21"><i><font color="Green" title="E11">A1</font></i></a>, <a class="ref" href="group_2.html#T48" title="GROUP_2:th.48">GROUP_2:48</a></i>;<br/>
<a NAME="E3:21_5"/><b>then </b>
 <a href="group_2.html#K17" title="GROUP_2:func.17">index</a> <font color="Maroon" title="c6">E</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="group_8.html#T20" target="_self" title="GROUP_8:th.20">Th20</a></i>;<br/>
<a NAME="E4:21_5"/><b>then </b>
 <a href="group_2.html#K17" title="GROUP_2:func.17">index</a> <font color="Maroon" title="c3">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="group_2.html#K17" title="GROUP_2:func.17">index</a> <font color="Maroon" title="c6">E</font>
 
<b>by </b><i><a class="txt" href="group_8.html#E17:21"><i><font color="Green" title="E24">A17</font></i></a>, <a class="ref" href="nat_d.html#T7" title="NAT_D:th.7">NAT_D:7</a></i>;<br/>
<b>hence </b><a NAME="E5:21_5"/>
contradiction
 <b>by </b><i><a class="txt" href="group_8.html#E18:21"><i><font color="Green" title="E25">A20</font></i></a>, <a class="txt" href="group_8.html#E1:21_5"><i><font color="Green" title="E26">A21</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E20:21"/>
(  <a href="group_8.html#NK2" title="GROUP_8:NK.2">index</a> <font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c4">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_8.html#NK2" title="GROUP_8:NK.2">index</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c5">D</font> &amp;  <a href="group_8.html#NK2" title="GROUP_8:NK.2">index</a> <font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_8.html#NK2" title="GROUP_8:NK.2">index</a> <font color="Maroon" title="c4">B</font>,<font color="Maroon" title="c6">E</font> )
 <b>by </b><i><a class="txt" href="group_8.html#E14:21"><i><font color="Green" title="E22">A14</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
