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

<b>let </b><font color="Maroon">G</font> be   <a href="group_1.html#NM1">Group</a>;<br/><b>let </b><font color="Maroon">H</font> be    <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:138"/><i><font color="Green">E38</font></i>: 
 <a href="group_2.html#K14">Left_Cosets</a> <font color="Maroon">H</font> is <a href="finset_1.html#V1">finite</a>
 <b>and </b><br/><a NAME="E2:138"/><i><font color="Green">E39</font></i>: 
 <a href="group_2.html#K17">index</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> 2
 ;<br/>

<b>consider </b><font color="Maroon">B</font> being  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:138"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="group_2.html#K14">Left_Cosets</a> <font color="Maroon">H</font> &amp;  <a href="group_2.html#K17">index</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">B</font> )
 <b>by </b><i><a class="ref" href="group_3.html#T61" target="_self">Th61</a>, <a class="ref" href="group_2.html#T176">GROUP_2:176</a></i>;<br/>
<b>consider </b><font color="Maroon">C</font> being  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:138"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">C</font> <a href="hidden.html#R1">=</a>  <a href="group_2.html#K15">Right_Cosets</a> <font color="Maroon">H</font> &amp;  <a href="group_2.html#K17">index</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">C</font> )
 <b>by </b><i><a class="ref" href="group_3.html#T61" target="_self">Th61</a>, <a class="ref" href="group_2.html#T176">GROUP_2:176</a></i>;<br/>
<b>consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E8:138"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font>
 <b>and </b><br/><a NAME="E9:138"/><i><font color="Green">E51</font></i>: 
 <a href="group_2.html#K14">Left_Cosets</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K2">}</a></span>
 <b>by </b><i><a class="ref" href="group_3.html#T63" target="_self">Th63</a>, <a class="ref" href="group_3.html#T65" target="_self">Th65</a>, <a class="ref" href="card_2.html#T79">CARD_2:79</a></i>;<br/>
<b>consider </b><font color="Maroon">z1</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">z2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E11:138"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">z1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">z2</font>
 <b>and </b><br/><a NAME="E12:138"/><i><font color="Green">E53</font></i>: 
 <a href="group_2.html#K15">Right_Cosets</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">z1</font>,<font color="Maroon">z2</font></span><a href="tarski.html#K2">}</a></span>
 <b>by </b><i><a class="ref" href="group_3.html#T63" target="_self">Th63</a>, <a class="ref" href="group_3.html#D6" target="_self">Def6</a>, <a class="ref" href="card_2.html#T79">CARD_2:79</a></i>;<br/>
<a NAME="E13:138"/><i><font color="Green">E54</font></i>: 
(  <a href="group_2.html#K7">carr</a> <font color="Maroon">H</font> <a href="hidden.html#R2">in</a>  <a href="group_2.html#K14">Left_Cosets</a> <font color="Maroon">H</font> &amp;  <a href="group_2.html#K7">carr</a> <font color="Maroon">H</font> <a href="hidden.html#R2">in</a>  <a href="group_2.html#K15">Right_Cosets</a> <font color="Maroon">H</font> )
 
<b>by </b><i><a class="ref" href="group_2.html#T165">GROUP_2:165</a></i>;<br/>
<a NAME="E14:138"/><b>then </b>
( <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">x</font>,<span class="p2">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">H</font></span>)</span></span><a href="tarski.html#K2">}</a></span> or <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">H</font></span>)</span>,<font color="Maroon">y</font></span><a href="tarski.html#K2">}</a></span> )
 
<b>by </b><i><a class="ref" href="group_3.html#T71" target="_self">Th71</a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<b>then consider </b><font color="Maroon">z3</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E16:138"/><i><font color="Green">E55</font></i>: 
<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">H</font></span>)</span>,<font color="Maroon">z3</font></span><a href="tarski.html#K2">}</a></span>
 ;<br/>
<a NAME="E17:138"/>
( <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">z1</font>,<font color="Maroon">z2</font></span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">z1</font>,<span class="p2">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">H</font></span>)</span></span><a href="tarski.html#K2">}</a></span> or <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">z1</font>,<font color="Maroon">z2</font></span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">H</font></span>)</span>,<font color="Maroon">z2</font></span><a href="tarski.html#K2">}</a></span> )
 
<b>by </b><i><a class="ref" href="group_3.html#T73" target="_self">Th73</a>, <a class="ref" href="group_3.html#T74" target="_self">Th74</a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<b>then consider </b><font color="Maroon">z4</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E19:138"/><i><font color="Green">E56</font></i>: 
<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">z1</font>,<font color="Maroon">z2</font></span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">H</font></span>)</span>,<font color="Maroon">z4</font></span><a href="tarski.html#K2">}</a></span>
 ;<br/>
<a NAME="E20:138"/><i><font color="Green">E59</font></i>: 
(  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="group_2.html#K14">Left_Cosets</a> <font color="Maroon">H</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font> &amp;  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="group_2.html#K15">Right_Cosets</a> <font color="Maroon">H</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font> )
 
<b>by </b><i><a class="ref" href="group_2.html#T167">GROUP_2:167</a></i>;<br/>
<a NAME="E21:138"/><i><font color="Green">E60</font></i>: 
 <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="group_2.html#K14">Left_Cosets</a> <font color="Maroon">H</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">H</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">z3</font>
 
<b>by </b><i><a class="ref" href="group_3.html#T71" target="_self">Th71</a>, <a class="ref" href="group_3.html#T76" target="_self">Th76</a>, <a class="ref" href="zfmisc_1.html#T93">ZFMISC_1:93</a></i>;<br/>
<a NAME="E22:138"/>
 <a href="group_2.html#K7">carr</a> <font color="Maroon">H</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">z3</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="138_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:138_1"/><i><font color="Green">E108</font></i>: 
not  <a href="group_2.html#K7">carr</a> <font color="Maroon">H</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">z3</font>
 ;<br/>

<a NAME="E2:138_1"/>
<font color="Maroon">z3</font> <a href="hidden.html#R2">in</a>  <a href="group_2.html#K14">Left_Cosets</a> <font color="Maroon">H</font>
 
<b>by </b><i><a class="ref" href="group_3.html#T71" target="_self">Th71</a>, <a class="ref" href="group_3.html#T76" target="_self">Th76</a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<a NAME="E3:138_1"/><b>then </b><i><font color="Green">E109</font></i>: 
 ex <font color="Olive">a</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font> st <font color="Maroon">z3</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font> <a href="group_2.html#K12">*</a> <font color="Maroon">H</font>
 
<b>by </b><i><a class="ref" href="group_2.html#D15">GROUP_2:def 15</a></i>;<br/>
<a NAME="E4:138_1"/>
 <a href="group_2.html#K7">carr</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">G</font></span>)</span> <a href="group_2.html#K12">*</a> <font color="Maroon">H</font>
 
<b>by </b><i><a class="ref" href="group_2.html#T132">GROUP_2:132</a></i>;<br/>
<a NAME="E5:138_1"/><b>then </b>
 <a href="group_2.html#K7">carr</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z3</font>
 
<b>by </b><i><a class="ref" href="group_3.html#T80" target="_self">Th80</a>, <a class="ref" href="group_3.html#T82" target="_self">Th82</a>, <a class="ref" href="group_2.html#T138">GROUP_2:138</a></i>;<br/>
<a NAME="E6:138_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">H</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="group_3.html#T76" target="_self">Th76</a>, <a class="ref" href="enumset1.html#T69">ENUMSET1:69</a></i>;<br/>
<a NAME="E7:138_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="group_2.html#K7">carr</a> <font color="Maroon">H</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="group_2.html#K7">carr</a> <font color="Maroon">H</font> )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T8">ZFMISC_1:8</a></i>;<br/>
<b>hence </b><a NAME="E8:138_1"/>
contradiction
 <b>by </b><i><a class="ref" href="group_3.html#T70" target="_self">Th70</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E23:138"/><b>then </b><i><font color="Green">E110</font></i>: 
<font color="Maroon">z3</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">H</font></span>)</span>
 
<b>by </b><i><a class="ref" href="group_3.html#T78" target="_self">Th78</a>, <a class="ref" href="group_3.html#T79" target="_self">Th79</a>, <a class="ref" href="xboole_1.html#T88">XBOOLE_1:88</a></i>;<br/>
<a NAME="E24:138"/><i><font color="Green">E111</font></i>: 
 <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="group_2.html#K15">Right_Cosets</a> <font color="Maroon">H</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">H</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">z4</font>
 
<b>by </b><i><a class="ref" href="group_3.html#T73" target="_self">Th73</a>, <a class="ref" href="group_3.html#T77" target="_self">Th77</a>, <a class="ref" href="zfmisc_1.html#T93">ZFMISC_1:93</a></i>;<br/>
<a NAME="E25:138"/><i><font color="Green">E112</font></i>: 
 <a href="group_2.html#K7">carr</a> <font color="Maroon">H</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">z4</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="138_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:138_2"/><i><font color="Green">E113</font></i>: 
not  <a href="group_2.html#K7">carr</a> <font color="Maroon">H</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">z4</font>
 ;<br/>

<a NAME="E2:138_2"/>
<font color="Maroon">z4</font> <a href="hidden.html#R2">in</a>  <a href="group_2.html#K15">Right_Cosets</a> <font color="Maroon">H</font>
 
<b>by </b><i><a class="ref" href="group_3.html#T73" target="_self">Th73</a>, <a class="ref" href="group_3.html#T77" target="_self">Th77</a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<a NAME="E3:138_2"/><b>then </b><i><font color="Green">E114</font></i>: 
 ex <font color="Olive">a</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font> st <font color="Maroon">z4</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H</font> <a href="group_2.html#K13">*</a> <font color="Olive">a</font>
 
<b>by </b><i><a class="ref" href="group_2.html#D16">GROUP_2:def 16</a></i>;<br/>
<a NAME="E4:138_2"/>
 <a href="group_2.html#K7">carr</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H</font> <a href="group_2.html#K13">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">G</font></span>)</span>
 
<b>by </b><i><a class="ref" href="group_2.html#T132">GROUP_2:132</a></i>;<br/>
<a NAME="E5:138_2"/><b>then </b>
 <a href="group_2.html#K7">carr</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z4</font>
 
<b>by </b><i>, , <a class="ref" href="group_2.html#T144">GROUP_2:144</a></i>;<br/>
<a NAME="E6:138_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">z1</font>,<font color="Maroon">z2</font></span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">H</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="group_3.html#T77" target="_self">Th77</a>, <a class="ref" href="enumset1.html#T69">ENUMSET1:69</a></i>;<br/>
<a NAME="E7:138_2"/><b>then </b>
( <font color="Maroon">z1</font> <a href="hidden.html#R1">=</a>  <a href="group_2.html#K7">carr</a> <font color="Maroon">H</font> &amp; <font color="Maroon">z2</font> <a href="hidden.html#R1">=</a>  <a href="group_2.html#K7">carr</a> <font color="Maroon">H</font> )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T8">ZFMISC_1:8</a></i>;<br/>
<b>hence </b><a NAME="E8:138_2"/>
contradiction
 <b>by </b><i><a class="ref" href="group_3.html#T72" target="_self">Th72</a></i>;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="138_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="138_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="138_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:138_3_1_1"/>
( <font color="Maroon">c</font> <a href="group_2.html#K12">*</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a>  <a href="group_2.html#K7">carr</a> <font color="Maroon">H</font> or <font color="Maroon">c</font> <a href="group_2.html#K12">*</a> <font color="Maroon">H</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="group_2.html#K7">carr</a> <font color="Maroon">H</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="138_3_1_1_1"><b>suppose </b></a><a NAME="E1:138_3_1_1_1"/><i><font color="Green">E115</font></i>: 
<font color="Maroon">c</font> <a href="group_2.html#K12">*</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a>  <a href="group_2.html#K7">carr</a> <font color="Maroon">H</font>
 ;<br/><div class="add"><a NAME="E2:138_3_1_1_1"/><b>then </b>
<font color="Maroon">c</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">H</font>
 <b>by </b><i><a class="ref" href="group_2.html#T136">GROUP_2:136</a></i>;<br/><b>hence </b><a NAME="E3:138_3_1_1_1"/>
<font color="Maroon">c</font> <a href="group_2.html#K12">*</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H</font> <a href="group_2.html#K13">*</a> <font color="Maroon">c</font>
 <b>by </b><i>, <a class="ref" href="group_2.html#T142">GROUP_2:142</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="138_3_1_1_2"><b>suppose </b></a><a NAME="E1:138_3_1_1_2"/><i><font color="Green">E116</font></i>: 
<font color="Maroon">c</font> <a href="group_2.html#K12">*</a> <font color="Maroon">H</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="group_2.html#K7">carr</a> <font color="Maroon">H</font>
 ;<br/><div class="add"><a NAME="E2:138_3_1_1_2"/><b>then </b>
not <font color="Maroon">c</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">H</font>
 <b>by </b><i><a class="ref" href="group_2.html#T136">GROUP_2:136</a></i>;<br/><a NAME="E3:138_3_1_1_2"/><b>then </b><i><font color="Green">E117</font></i>: 
<font color="Maroon">H</font> <a href="group_2.html#K13">*</a> <font color="Maroon">c</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="group_2.html#K7">carr</a> <font color="Maroon">H</font>
 <b>by </b><i><a class="ref" href="group_2.html#T142">GROUP_2:142</a></i>;<br/><a NAME="E4:138_3_1_1_2"/>
( <font color="Maroon">c</font> <a href="group_2.html#K12">*</a> <font color="Maroon">H</font> <a href="hidden.html#R2">in</a>  <a href="group_2.html#K14">Left_Cosets</a> <font color="Maroon">H</font> &amp; <font color="Maroon">H</font> <a href="group_2.html#K13">*</a> <font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="group_2.html#K15">Right_Cosets</a> <font color="Maroon">H</font> )
 <b>by </b><i><a class="ref" href="group_2.html#D15">GROUP_2:def 15</a>, <a class="ref" href="group_2.html#D16">GROUP_2:def 16</a></i>;<br/><a NAME="E5:138_3_1_1_2"/><b>then </b>
( <font color="Maroon">c</font> <a href="group_2.html#K12">*</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z3</font> &amp; <font color="Maroon">H</font> <a href="group_2.html#K13">*</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z4</font> )
 <b>by </b><i><a class="ref" href="group_3.html#T71" target="_self">Th71</a>, <a class="ref" href="group_3.html#T73" target="_self">Th73</a>, <a class="ref" href="group_3.html#T76" target="_self">Th76</a>, <a class="ref" href="group_3.html#T77" target="_self">Th77</a>, , , <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><b>hence </b><a NAME="E6:138_3_1_1_2"/>
<font color="Maroon">c</font> <a href="group_2.html#K12">*</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H</font> <a href="group_2.html#K13">*</a> <font color="Maroon">c</font>
 <b>by </b><i><a class="ref" href="group_3.html#T78" target="_self">Th78</a>, <a class="ref" href="group_3.html#T80" target="_self">Th80</a>, <a class="ref" href="group_3.html#T82" target="_self">Th82</a>, <a class="ref" href="group_3.html#T84" target="_self">Th84</a>, <a class="ref" href="xboole_1.html#T88">XBOOLE_1:88</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E2:138_3"/>
<font color="Maroon">c</font> <a href="group_2.html#K12">*</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H</font> <a href="group_2.html#K13">*</a> <font color="Maroon">c</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E27:138"/>
<font color="Maroon">H</font> is  <a href="group_3.html#V1" target="_self">normal</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="group_3.html#T51" target="_self">Th51</a></i>;<br/>


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