<?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">N1</font> be  <a href="group_3.html#V1">normal</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font>;<br/><b>let </b><font color="Maroon">N2</font> be  <a href="group_3.html#V1">normal</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font>;<br/>





<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font>;<br/><b>assume </b><a NAME="E1:74_1"/>
<font color="Maroon">a</font> <a href="rlvect_1.html#R1">in</a> <span class="p1"><a href="group_5.html#K8" target="_self">[.</a><span class="default"><font color="Maroon">N1</font>,<font color="Maroon">N2</font></span><a href="group_5.html#K8" target="_self">.]</a></span>
 ;<br/><b>then consider </b><font color="Maroon">F</font> being    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font>, <font color="Maroon">I</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K4">INT</a> <b> such that </b><br/><a NAME="E3:74_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">I</font>
 <b>and </b><br/><a NAME="E4:74_1"/><i><font color="Green">E58</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a>  <a href="group_5.html#K5" target="_self">commutators</a> <font color="Maroon">N1</font>,<font color="Maroon">N2</font>
 <b>and </b><br/><a NAME="E5:74_1"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="group_4.html#K3">Product</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="group_4.html#K4">|^</a> <font color="Maroon">I</font></span>)</span>
 <b>by </b><i/>;<br/><b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font> = <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="group_1.html#K3">"</a> ;<br/><b>consider </b><font color="Maroon">F1</font> being    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E7:74_1"/><i><font color="Green">E63</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">F1</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font>
 <b>and </b><br/><a NAME="E8:74_1"/><i><font color="Green">E64</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span> holds <br/><font color="Maroon">F1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">k</font>)
 <b>from </b><i><a class="ref" href="finseq_2.html#S1">FINSEQ_2:sch 1</a>();<br/></i><b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K4">INT</a>  =  <a href="group_4.html#K2">@</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p2">(<span class="default"><a href="group_4.html#K2">@</a> <span class="p3">(<span class="default"><font color="Maroon">I</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span></span>)</span></span>)</span>;<br/><b>consider </b><font color="Maroon">I1</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K4">INT</a> <b> such that </b><br/><a NAME="E10:74_1"/><i><font color="Green">E65</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">I1</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font>
 <b>and </b><br/><a NAME="E11:74_1"/><i><font color="Green">E66</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span> holds <br/><font color="Maroon">I1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">k</font>)
 <b>from </b><i><a class="ref" href="finseq_2.html#S1">FINSEQ_2:sch 1</a>();<br/></i><a NAME="E12:74_1"/><i><font color="Green">E74</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E13:74_1"/><i><font color="Green">E90</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F1</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/><a NAME="E14:74_1"/><i><font color="Green">E105</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">I1</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/><b>set </b><font color="Maroon">J</font> = <font color="Maroon">F1</font> <a href="group_4.html#K4">|^</a> <font color="Maroon">I1</font>;<br/><a NAME="E15:74_1"/><i><font color="Green">E106</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="group_4.html#K4">|^</a> <font color="Maroon">I1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="group_4.html#K4">|^</a> <font color="Maroon">I</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> )
 <b>by </b><i>, <a class="ref" href="group_4.html#D4">GROUP_4:def 4</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:74_1_1"/><i><font color="Green">E107</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span>
 ;<br/><a NAME="E2:74_1_1"/><b>then </b><i><font color="Green">E108</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E3:74_1_1"/>
( <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="group_4.html#K4">|^</a> <font color="Maroon">I1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font></span>)</span> <a href="group_1.html#K6">|^</a> <span class="p1">(<span class="default"><a href="group_4.html#K2">@</a> <span class="p2">(<span class="default"><font color="Maroon">I1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font></span>)</span></span>)</span> &amp; <font color="Maroon">F1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> &amp; <font color="Maroon">F1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font></span>)</span> <a href="group_1.html#K3">"</a>  &amp; <font color="Maroon">I1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font> &amp;  <a href="group_4.html#K2">@</a> <span class="p1">(<span class="default"><font color="Maroon">I1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">I1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font> &amp; <font color="Maroon">I1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="group_4.html#K2">@</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p2">(<span class="default"><a href="group_4.html#K2">@</a> <span class="p3">(<span class="default"><font color="Maroon">I</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> &amp;  <a href="group_4.html#K2">@</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p2">(<span class="default"><a href="group_4.html#K2">@</a> <span class="p3">(<span class="default"><font color="Maroon">I</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xcmplx_0.html#K4">-</a> <span class="p1">(<span class="default"><a href="group_4.html#K2">@</a> <span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font></span>)</span></span>)</span> )
 <b>by </b><i>, , , , , , <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a>, <a class="ref" href="group_4.html#D2">GROUP_4:def 2</a>, <a class="ref" href="group_4.html#D4">GROUP_4:def 4</a></i>;<br/><b>then </b><span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="group_4.html#K4">|^</a> <font color="Maroon">I1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font></span>)</span> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K6">|^</a> <span class="p2">(<span class="default"><a href="group_4.html#K2">@</a> <span class="p3">(<span class="default"><font color="Maroon">I</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K3">"</a> 
<b>by </b><i><a class="ref" href="group_1.html#T70">GROUP_1:70</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font></span>)</span> <a href="group_1.html#K6">|^</a> <span class="p3">(<span class="default"><a href="group_4.html#K2">@</a> <span class="p4">(<span class="default"><font color="Maroon">I</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K3">"</a> 
<b>by </b><i><a class="ref" href="group_1.html#T72">GROUP_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font></span>)</span> <a href="group_1.html#K6">|^</a> <span class="p1">(<span class="default"><a href="group_4.html#K2">@</a> <span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#T19">GROUP_1:19</a></i>
;<br/><b>hence </b><a NAME="E5:74_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="group_4.html#K4">|^</a> <font color="Maroon">I</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="group_4.html#K4">|^</a> <font color="Maroon">I1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
 <b>by </b><i>, <a class="ref" href="group_4.html#D4">GROUP_4:def 4</a></i>;<br/></div><b>end;</b></div><a NAME="E17:74_1"/><b>then </b><i><font color="Green">E111</font></i>: 
<font color="Maroon">F1</font> <a href="group_4.html#K4">|^</a> <font color="Maroon">I1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="group_4.html#K4">|^</a> <font color="Maroon">I</font>
 <b>by </b><i>, <a class="ref" href="finseq_2.html#T10">FINSEQ_2:10</a></i>;<br/><a NAME="E18:74_1"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">F1</font> <a href="tarski.html#R1">c=</a>  <a href="group_5.html#K5" target="_self">commutators</a> <font color="Maroon">N2</font>,<font color="Maroon">N1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:74_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F1</font>
 ;<br/>

<b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:74_1_2"/><i><font color="Green">E112</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F1</font>
 <b>and </b><br/><a NAME="E4:74_1_2"/><i><font color="Green">E113</font></i>: 
<font color="Maroon">F1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">y</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/>
<a NAME="E6:74_1_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font>
 
<b>by </b><i>, , <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E7:74_1_2"/><b>then </b>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon">b</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font>, <font color="Maroon">c</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E9:74_1_2"/><i><font color="Green">E114</font></i>: 
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="group_5.html#K2" target="_self">[.</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">c</font></span><a href="group_5.html#K2" target="_self">.]</a></span>
 <b>and </b><br/><a NAME="E10:74_1_2"/><i><font color="Green">E115</font></i>: 
( <font color="Maroon">b</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">N1</font> &amp; <font color="Maroon">c</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">N2</font> )
 <b>by </b><i>, </i>;<br/>
<a NAME="E11:74_1_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font>
 
<b>by </b><i>, , <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E12:74_1_2"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">y</font></span>)</span> <a href="group_1.html#K3">"</a>  &amp; <font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> )
 
<b>by </b><i>, , , <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<a NAME="E13:74_1_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="group_5.html#K2" target="_self">[.</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="group_5.html#K2" target="_self">.]</a></span>
 
<b>by </b><i><a class="ref" href="group_5.html#T1" target="_self">Th1</a>, </i>;<br/>
<b>hence </b><a NAME="E14:74_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="group_5.html#K5" target="_self">commutators</a> <font color="Maroon">N2</font>,<font color="Maroon">N1</font>
 <b>by </b><i><a class="ref" href="group_5.html#T3" target="_self">Th3</a>, </i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E19:74_1"/>
<font color="Maroon">a</font> <a href="rlvect_1.html#R1">in</a> <span class="p1"><a href="group_5.html#K8" target="_self">[.</a><span class="default"><font color="Maroon">N2</font>,<font color="Maroon">N1</font></span><a href="group_5.html#K8" target="_self">.]</a></span>
 <b>by </b><i>, , , , </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:74"/>
<span class="p1"><a href="group_5.html#K8" target="_self">[.</a><span class="default"><font color="Maroon">N1</font>,<font color="Maroon">N2</font></span><a href="group_5.html#K8" target="_self">.]</a></span> is    <a href="group_2.html#M1">Subgroup</a> of <span class="p1"><a href="group_5.html#K8" target="_self">[.</a><span class="default"><font color="Maroon">N2</font>,<font color="Maroon">N1</font></span><a href="group_5.html#K8" target="_self">.]</a></span>
 <b>by </b><i><a class="ref" href="group_2.html#T67">GROUP_2:67</a></i>;<br/>


</div>
