<?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"> 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  <a href="group_2.html#K14" title="GROUP_2:func.14">Left_Cosets</a> <font color="Olive" title="b1">H</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="Olive" title="b1">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 holds <br/><font color="Olive" title="b1">H</font> is  <a href="group_3.html#V1" title="GROUP_3:attr.1">normal</a>  <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font></span><br/><b>let </b><font color="Maroon" title="c2">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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="group_2.html#K14" title="GROUP_2:func.14">Left_Cosets</a> <font color="Maroon" title="c2">H</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="c2">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 implies <font color="Maroon" title="c2">H</font> is  <a href="group_3.html#V1" title="GROUP_3:attr.1">normal</a>  <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font> )</span><br/>



<b>assume </b><b>that </b><br/><a NAME="E1:137"/><i><font color="Green" title="E87">A1</font></i>: 
 <a href="group_2.html#K14" title="GROUP_2:func.14">Left_Cosets</a> <font color="Maroon" title="c2">H</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 <b>and </b><br/><a NAME="E2:137"/><i><font color="Green" title="E88">A2</font></i>: 
 <a href="group_2.html#K17" title="GROUP_2:func.17">index</a> <font color="Maroon" title="c2">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">H</font> is  <a href="group_3.html#V1" title="GROUP_3:attr.1">normal</a>  <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font></span><br/>

<b>consider </b><font color="Maroon" title="c3">B</font> being  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:137"/><i><font color="Green" title="E89">A3</font></i>: 
( <font color="Maroon" title="c3">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_2.html#K14" title="GROUP_2:func.14">Left_Cosets</a> <font color="Maroon" title="c2">H</font> &amp;  <a href="group_2.html#K17" title="GROUP_2:func.17">index</a> <font color="Maroon" title="c2">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c3">B</font> )
 <b>by </b><i><a class="txt" href="group_3.html#E1:137"><i><font color="Green" title="E87">A1</font></i></a>, <a class="ref" href="group_2.html#T176" title="GROUP_2:th.176">GROUP_2:176</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c4">C</font> being  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E6:137"/><i><font color="Green" title="E90">A4</font></i>: 
( <font color="Maroon" title="c4">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_2.html#K15" title="GROUP_2:func.15">Right_Cosets</a> <font color="Maroon" title="c2">H</font> &amp;  <a href="group_2.html#K17" title="GROUP_2:func.17">index</a> <font color="Maroon" title="c2">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c4">C</font> )
 <b>by </b><i><a class="txt" href="group_3.html#E1:137"><i><font color="Green" title="E87">A1</font></i></a>, <a class="ref" href="group_2.html#T176" title="GROUP_2:th.176">GROUP_2:176</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c5">x</font>, <font color="Maroon" title="c6">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E8:137"/><i><font color="Green" title="E91">A5</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">y</font>
 <b>and </b><br/><a NAME="E9:137"/><i><font color="Green" title="E92">A6</font></i>: 
 <a href="group_2.html#K14" title="GROUP_2:func.14">Left_Cosets</a> <font color="Maroon" title="c2">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <b>by </b><i><a class="txt" href="group_3.html#E2:137"><i><font color="Green" title="E88">A2</font></i></a>, <a class="txt" href="group_3.html#E4:137"><i><font color="Green" title="E89">A3</font></i></a>, <a class="ref" href="card_2.html#T79" title="CARD_2:th.79">CARD_2:79</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c7">z1</font>, <font color="Maroon" title="c8">z2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E11:137"/><i><font color="Green" title="E93">A7</font></i>: 
<font color="Maroon" title="c7">z1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">z2</font>
 <b>and </b><br/><a NAME="E12:137"/><i><font color="Green" title="E94">A8</font></i>: 
 <a href="group_2.html#K15" title="GROUP_2:func.15">Right_Cosets</a> <font color="Maroon" title="c2">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c7">z1</font>,<font color="Maroon" title="c8">z2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <b>by </b><i><a class="txt" href="group_3.html#E2:137"><i><font color="Green" title="E88">A2</font></i></a>, <a class="txt" href="group_3.html#E6:137"><i><font color="Green" title="E90">A4</font></i></a>, <a class="ref" href="card_2.html#T79" title="CARD_2:th.79">CARD_2:79</a></i>;<br/>
<a NAME="E13:137"/><i><font color="Green" title="E95">A9</font></i>: 
(  <a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <font color="Maroon" title="c2">H</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_2.html#K14" title="GROUP_2:func.14">Left_Cosets</a> <font color="Maroon" title="c2">H</font> &amp;  <a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <font color="Maroon" title="c2">H</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_2.html#K15" title="GROUP_2:func.15">Right_Cosets</a> <font color="Maroon" title="c2">H</font> )
 
<b>by </b><i><a class="ref" href="group_2.html#T165" title="GROUP_2:th.165">GROUP_2:165</a></i>;<br/>
<a NAME="E14:137"/><b>then </b>
( <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c5">x</font>,<span class="p2">(<span class="default"><a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <font color="Maroon" title="c2">H</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> or <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <font color="Maroon" title="c2">H</font></span>)</span>,<font color="Maroon" title="c6">y</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> )
 
<b>by </b><i><a class="txt" href="group_3.html#E9:137"><i><font color="Green" title="E92">A6</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c9">z3</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E16:137"/><i><font color="Green" title="E96">A10</font></i>: 
<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <font color="Maroon" title="c2">H</font></span>)</span>,<font color="Maroon" title="c9">z3</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 ;<br/>
<a NAME="E17:137"/>
( <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c7">z1</font>,<font color="Maroon" title="c8">z2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c7">z1</font>,<span class="p2">(<span class="default"><a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <font color="Maroon" title="c2">H</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> or <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c7">z1</font>,<font color="Maroon" title="c8">z2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <font color="Maroon" title="c2">H</font></span>)</span>,<font color="Maroon" title="c8">z2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> )
 
<b>by </b><i><a class="txt" href="group_3.html#E12:137"><i><font color="Green" title="E94">A8</font></i></a>, <a class="txt" href="group_3.html#E13:137"><i><font color="Green" title="E95">A9</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c10">z4</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E19:137"/><i><font color="Green" title="E97">A11</font></i>: 
<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c7">z1</font>,<font color="Maroon" title="c8">z2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <font color="Maroon" title="c2">H</font></span>)</span>,<font color="Maroon" title="c10">z4</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 ;<br/>
<a NAME="E20:137"/><i><font color="Green" title="E98">A12</font></i>: 
(  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="group_2.html#K14" title="GROUP_2:func.14">Left_Cosets</a> <font color="Maroon" title="c2">H</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font> &amp;  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="group_2.html#K15" title="GROUP_2:func.15">Right_Cosets</a> <font color="Maroon" title="c2">H</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font> )
 
<b>by </b><i><a class="ref" href="group_2.html#T167" title="GROUP_2:th.167">GROUP_2:167</a></i>;<br/>
<a NAME="E21:137"/><i><font color="Green" title="E99">A13</font></i>: 
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="group_2.html#K14" title="GROUP_2:func.14">Left_Cosets</a> <font color="Maroon" title="c2">H</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <font color="Maroon" title="c2">H</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c9">z3</font>
 
<b>by </b><i><a class="txt" href="group_3.html#E9:137"><i><font color="Green" title="E92">A6</font></i></a>, <a class="txt" href="group_3.html#E16:137"><i><font color="Green" title="E96">A10</font></i></a>, <a class="ref" href="zfmisc_1.html#T93" title="ZFMISC_1:th.93">ZFMISC_1:93</a></i>;<br/>
<a NAME="E22:137"/>
 <a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <font color="Maroon" title="c2">H</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c9">z3</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="137_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:137_1"/><i><font color="Green" title="E100">A14</font></i>: 
not  <a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <font color="Maroon" title="c2">H</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c9">z3</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="E2:137_1"/>
<font color="Maroon" title="c9">z3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_2.html#K14" title="GROUP_2:func.14">Left_Cosets</a> <font color="Maroon" title="c2">H</font>
 
<b>by </b><i><a class="txt" href="group_3.html#E9:137"><i><font color="Green" title="E92">A6</font></i></a>, <a class="txt" href="group_3.html#E16:137"><i><font color="Green" title="E96">A10</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/>
<a NAME="E3:137_1"/><b>then </b><i><font color="Green" title="E101">A15</font></i>: 
 ex <font color="Olive" title="b1">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> st <font color="Maroon" title="c9">z3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c2">H</font>
 
<b>by </b><i><a class="ref" href="group_2.html#D15" title="GROUP_2:def.15">GROUP_2:def 15</a></i>;<br/>
<a NAME="E4:137_1"/>
 <a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <font color="Maroon" title="c2">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c2">H</font>
 
<b>by </b><i><a class="ref" href="group_2.html#T132" title="GROUP_2:th.132">GROUP_2:132</a></i>;<br/>
<a NAME="E5:137_1"/><b>then </b>
 <a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <font color="Maroon" title="c2">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">z3</font>
 
<b>by </b><i><a class="txt" href="group_3.html#E1:137_1"><i><font color="Green" title="E100">A14</font></i></a>, <a class="txt" href="group_3.html#E3:137_1"><i><font color="Green" title="E101">A15</font></i></a>, <a class="ref" href="group_2.html#T138" title="GROUP_2:th.138">GROUP_2:138</a></i>;<br/>
<a NAME="E6:137_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <font color="Maroon" title="c2">H</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="txt" href="group_3.html#E16:137"><i><font color="Green" title="E96">A10</font></i></a>, <a class="ref" href="enumset1.html#T69" title="ENUMSET1:th.69">ENUMSET1:69</a></i>;<br/>
<a NAME="E7:137_1"/><b>then </b>
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <font color="Maroon" title="c2">H</font> &amp; <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <font color="Maroon" title="c2">H</font> )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T8" title="ZFMISC_1:th.8">ZFMISC_1:8</a></i>;<br/>
<b>hence </b><a NAME="E8:137_1"/>
contradiction
 <b>by </b><i><a class="txt" href="group_3.html#E8:137"><i><font color="Green" title="E91">A5</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>
<a NAME="E23:137"/><b>then </b><i><font color="Green" title="E100">A16</font></i>: 
<font color="Maroon" title="c9">z3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <font color="Maroon" title="c2">H</font></span>)</span>
 
<b>by </b><i><a class="txt" href="group_3.html#E20:137"><i><font color="Green" title="E98">A12</font></i></a>, <a class="txt" href="group_3.html#E21:137"><i><font color="Green" title="E99">A13</font></i></a>, <a class="ref" href="xboole_1.html#T88" title="XBOOLE_1:th.88">XBOOLE_1:88</a></i>;<br/>
<a NAME="E24:137"/><i><font color="Green" title="E101">A17</font></i>: 
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="group_2.html#K15" title="GROUP_2:func.15">Right_Cosets</a> <font color="Maroon" title="c2">H</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <font color="Maroon" title="c2">H</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c10">z4</font>
 
<b>by </b><i><a class="txt" href="group_3.html#E12:137"><i><font color="Green" title="E94">A8</font></i></a>, <a class="txt" href="group_3.html#E19:137"><i><font color="Green" title="E97">A11</font></i></a>, <a class="ref" href="zfmisc_1.html#T93" title="ZFMISC_1:th.93">ZFMISC_1:93</a></i>;<br/>
<a NAME="E25:137"/><i><font color="Green" title="E102">A18</font></i>: 
 <a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <font color="Maroon" title="c2">H</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c10">z4</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="137_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:137_2"/><i><font color="Green" title="E103">A19</font></i>: 
not  <a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <font color="Maroon" title="c2">H</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c10">z4</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="E2:137_2"/>
<font color="Maroon" title="c10">z4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_2.html#K15" title="GROUP_2:func.15">Right_Cosets</a> <font color="Maroon" title="c2">H</font>
 
<b>by </b><i><a class="txt" href="group_3.html#E12:137"><i><font color="Green" title="E94">A8</font></i></a>, <a class="txt" href="group_3.html#E19:137"><i><font color="Green" title="E97">A11</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/>
<a NAME="E3:137_2"/><b>then </b><i><font color="Green" title="E104">A20</font></i>: 
 ex <font color="Olive" title="b1">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> st <font color="Maroon" title="c10">z4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">H</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Olive" title="b1">a</font>
 
<b>by </b><i><a class="ref" href="group_2.html#D16" title="GROUP_2:def.16">GROUP_2:def 16</a></i>;<br/>
<a NAME="E4:137_2"/>
 <a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <font color="Maroon" title="c2">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">H</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</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>
 
<b>by </b><i><a class="ref" href="group_2.html#T132" title="GROUP_2:th.132">GROUP_2:132</a></i>;<br/>
<a NAME="E5:137_2"/><b>then </b>
 <a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <font color="Maroon" title="c2">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">z4</font>
 
<b>by </b><i><a class="txt" href="group_3.html#E1:137_2"><i><font color="Green" title="E103">A19</font></i></a>, <a class="txt" href="group_3.html#E3:137_2"><i><font color="Green" title="E104">A20</font></i></a>, <a class="ref" href="group_2.html#T144" title="GROUP_2:th.144">GROUP_2:144</a></i>;<br/>
<a NAME="E6:137_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c7">z1</font>,<font color="Maroon" title="c8">z2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <font color="Maroon" title="c2">H</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="txt" href="group_3.html#E19:137"><i><font color="Green" title="E97">A11</font></i></a>, <a class="ref" href="enumset1.html#T69" title="ENUMSET1:th.69">ENUMSET1:69</a></i>;<br/>
<a NAME="E7:137_2"/><b>then </b>
( <font color="Maroon" title="c7">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <font color="Maroon" title="c2">H</font> &amp; <font color="Maroon" title="c8">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <font color="Maroon" title="c2">H</font> )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T8" title="ZFMISC_1:th.8">ZFMISC_1:8</a></i>;<br/>
<b>hence </b><a NAME="E8:137_2"/>
contradiction
 <b>by </b><i><a class="txt" href="group_3.html#E11:137"><i><font color="Green" title="E93">A7</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>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="137_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c11">c</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</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"> <font color="Maroon" title="c11">c</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c2">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">H</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c11">c</font></span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="137_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="137_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:137_3_1_1"/>
( <font color="Maroon" title="c11">c</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c2">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <font color="Maroon" title="c2">H</font> or <font color="Maroon" title="c11">c</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c2">H</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <font color="Maroon" title="c2">H</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="137_3_1_1_1"><b>suppose </b></a><a NAME="E1:137_3_1_1_1"/><i><font color="Green" title="E103">A21</font></i>: 
<font color="Maroon" title="c11">c</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c2">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <font color="Maroon" title="c2">H</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c11">c</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c2">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">H</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c11">c</font></span><br/><div class="add"><a NAME="E2:137_3_1_1_1"/><b>then </b>
<font color="Maroon" title="c11">c</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">H</font>
 <b>by </b><i><a class="ref" href="group_2.html#T136" title="GROUP_2:th.136">GROUP_2:136</a></i>;<br/><b>hence </b><a NAME="E3:137_3_1_1_1"/>
<font color="Maroon" title="c11">c</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c2">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">H</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c11">c</font>
 <b>by </b><i><a class="txt" href="group_3.html#E1:137_3_1_1_1"><i><font color="Green" title="E103">A21</font></i></a>, <a class="ref" href="group_2.html#T142" title="GROUP_2:th.142">GROUP_2:142</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="137_3_1_1_2"><b>suppose </b></a><a NAME="E1:137_3_1_1_2"/><i><font color="Green" title="E103">A22</font></i>: 
<font color="Maroon" title="c11">c</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c2">H</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <font color="Maroon" title="c2">H</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c11">c</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c2">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">H</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c11">c</font></span><br/><div class="add"><a NAME="E2:137_3_1_1_2"/><b>then </b>
not <font color="Maroon" title="c11">c</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">H</font>
 <b>by </b><i><a class="ref" href="group_2.html#T136" title="GROUP_2:th.136">GROUP_2:136</a></i>;<br/><a NAME="E3:137_3_1_1_2"/><b>then </b><i><font color="Green" title="E104">A23</font></i>: 
<font color="Maroon" title="c2">H</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c11">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <font color="Maroon" title="c2">H</font>
 <b>by </b><i><a class="ref" href="group_2.html#T142" title="GROUP_2:th.142">GROUP_2:142</a></i>;<br/><a NAME="E4:137_3_1_1_2"/>
( <font color="Maroon" title="c11">c</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c2">H</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_2.html#K14" title="GROUP_2:func.14">Left_Cosets</a> <font color="Maroon" title="c2">H</font> &amp; <font color="Maroon" title="c2">H</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c11">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_2.html#K15" title="GROUP_2:func.15">Right_Cosets</a> <font color="Maroon" title="c2">H</font> )
 <b>by </b><i><a class="ref" href="group_2.html#D15" title="GROUP_2:def.15">GROUP_2:def 15</a>, <a class="ref" href="group_2.html#D16" title="GROUP_2:def.16">GROUP_2:def 16</a></i>;<br/><a NAME="E5:137_3_1_1_2"/><b>then </b>
( <font color="Maroon" title="c11">c</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c2">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">z3</font> &amp; <font color="Maroon" title="c2">H</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c11">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">z4</font> )
 <b>by </b><i><a class="txt" href="group_3.html#E9:137"><i><font color="Green" title="E92">A6</font></i></a>, <a class="txt" href="group_3.html#E12:137"><i><font color="Green" title="E94">A8</font></i></a>, <a class="txt" href="group_3.html#E16:137"><i><font color="Green" title="E96">A10</font></i></a>, <a class="txt" href="group_3.html#E19:137"><i><font color="Green" title="E97">A11</font></i></a>, <a class="txt" href="group_3.html#E1:137_3_1_1_2"><i><font color="Green" title="E103">A22</font></i></a>, <a class="txt" href="group_3.html#E3:137_3_1_1_2"><i><font color="Green" title="E104">A23</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/><b>hence </b><a NAME="E6:137_3_1_1_2"/>
<font color="Maroon" title="c11">c</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c2">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">H</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c11">c</font>
 <b>by </b><i><a class="txt" href="group_3.html#E20:137"><i><font color="Green" title="E98">A12</font></i></a>, <a class="txt" href="group_3.html#E23:137"><i><font color="Green" title="E100">A16</font></i></a>, <a class="txt" href="group_3.html#E24:137"><i><font color="Green" title="E101">A17</font></i></a>, <a class="txt" href="group_3.html#E25:137"><i><font color="Green" title="E102">A18</font></i></a>, <a class="ref" href="xboole_1.html#T88" title="XBOOLE_1:th.88">XBOOLE_1:88</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></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E2:137_3"/>
<font color="Maroon" title="c11">c</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c2">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">H</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c11">c</font>
 ; <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="E27:137"/>
<font color="Maroon" title="c2">H</font> is  <a href="group_3.html#V1" title="GROUP_3:attr.1">normal</a>  <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="ref" href="group_3.html#T140" target="_self" title="GROUP_3:th.140">Th140</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>
