<?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>;<br/><b>let </b><font color="Maroon" title="c2">N1</font>, <font color="Maroon" title="c3">N2</font>, <font color="Maroon" title="c4">N3</font> be  <a href="group_1.html#V1" title="GROUP_1:attr.1">strict</a> <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>;<br/>



<a NAME="E1:76"/><i><font color="Green" title="E41">A1</font></i>: 
( <span class="p1"><a href="group_5.html#K8" title="GROUP_5:func.8">[.</a><span class="default"><font color="Maroon" title="c2">N1</font>,<font color="Maroon" title="c4">N3</font></span><a href="group_5.html#K8" title="GROUP_5:func.8">.]</a></span> 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> &amp; <span class="p1"><a href="group_5.html#K8" title="GROUP_5:func.8">[.</a><span class="default"><font color="Maroon" title="c3">N2</font>,<font color="Maroon" title="c4">N3</font></span><a href="group_5.html#K8" title="GROUP_5:func.8">.]</a></span> 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_5.html#T77" target="_self" title="GROUP_5:th.77">Th77</a></i>;<br/>
<a NAME="E2:76"/>
 <a href="group_5.html#K5" title="GROUP_5:func.5">commutators</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">N1</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c3">N2</font></span>)</span>,<font color="Maroon" title="c4">N3</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="group_5.html#K8" title="GROUP_5:func.8">[.</a><span class="default"><font color="Maroon" title="c2">N1</font>,<font color="Maroon" title="c4">N3</font></span><a href="group_5.html#K8" title="GROUP_5:func.8">.]</a></span> <a href="group_4.html#K7" title="GROUP_4:func.7">*</a> <span class="p1"><a href="group_5.html#K8" title="GROUP_5:func.8">[.</a><span class="default"><font color="Maroon" title="c3">N2</font>,<font color="Maroon" title="c4">N3</font></span><a href="group_5.html#K8" title="GROUP_5:func.8">.]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:76_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_5.html#K5" title="GROUP_5:func.5">commutators</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">N1</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c3">N2</font></span>)</span>,<font color="Maroon" title="c4">N3</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c6">a</font>, <font color="Maroon" title="c7">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E3:76_1"/><i><font color="Green" title="E42">A2</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="group_5.html#K2" title="GROUP_5:func.2">[.</a><span class="default"><font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font></span><a href="group_5.html#K2" title="GROUP_5:func.2">.]</a></span>
 <b>and </b><br/><a NAME="E4:76_1"/><i><font color="Green" title="E43">A3</font></i>: 
<font color="Maroon" title="c6">a</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">N1</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c3">N2</font>
 <b>and </b><br/><a NAME="E5:76_1"/><i><font color="Green" title="E44">A4</font></i>: 
<font color="Maroon" title="c7">b</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c4">N3</font>
 <b>by </b><i><a class="ref" href="group_5.html#T58" target="_self" title="GROUP_5:th.58">Th58</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">c</font>, <font color="Maroon" title="c9">d</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E7:76_1"/><i><font color="Green" title="E45">A5</font></i>: 
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">c</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c9">d</font>
 <b>and </b><br/><a NAME="E8:76_1"/><i><font color="Green" title="E46">A6</font></i>: 
<font color="Maroon" title="c8">c</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">N1</font>
 <b>and </b><br/><a NAME="E9:76_1"/><i><font color="Green" title="E47">A7</font></i>: 
<font color="Maroon" title="c9">d</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">N2</font>
 <b>by </b><i><a class="txt" href="group_5.html#E4:76_1"><i><font color="Green" title="E43">A3</font></i></a>, <a class="ref" href="group_5.html#T7" target="_self" title="GROUP_5:th.7">Th7</a></i>;<br/>
<a NAME="E10:76_1"/><i><font color="Green" title="E48">A8</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="group_5.html#K2" title="GROUP_5:func.2">[.</a><span class="default"><font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c7">b</font></span><a href="group_5.html#K2" title="GROUP_5:func.2">.]</a></span> <a href="group_3.html#K2" title="GROUP_3:func.2">|^</a> <font color="Maroon" title="c9">d</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1"><a href="group_5.html#K2" title="GROUP_5:func.2">[.</a><span class="default"><font color="Maroon" title="c9">d</font>,<font color="Maroon" title="c7">b</font></span><a href="group_5.html#K2" title="GROUP_5:func.2">.]</a></span>
 
<b>by </b><i><a class="txt" href="group_5.html#E3:76_1"><i><font color="Green" title="E42">A2</font></i></a>, <a class="txt" href="group_5.html#E7:76_1"><i><font color="Green" title="E45">A5</font></i></a>, <a class="ref" href="group_5.html#T28" target="_self" title="GROUP_5:th.28">Th28</a></i>;<br/>
<a NAME="E11:76_1"/>
<span class="p1"><a href="group_5.html#K2" title="GROUP_5:func.2">[.</a><span class="default"><font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c7">b</font></span><a href="group_5.html#K2" title="GROUP_5:func.2">.]</a></span> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <span class="p1"><a href="group_5.html#K8" title="GROUP_5:func.8">[.</a><span class="default"><font color="Maroon" title="c2">N1</font>,<font color="Maroon" title="c4">N3</font></span><a href="group_5.html#K8" title="GROUP_5:func.8">.]</a></span>
 
<b>by </b><i><a class="txt" href="group_5.html#E5:76_1"><i><font color="Green" title="E44">A4</font></i></a>, <a class="txt" href="group_5.html#E8:76_1"><i><font color="Green" title="E46">A6</font></i></a>, <a class="ref" href="group_5.html#T74" target="_self" title="GROUP_5:th.74">Th74</a></i>;<br/>
<a NAME="E12:76_1"/><b>then </b>
<span class="p1"><a href="group_5.html#K2" title="GROUP_5:func.2">[.</a><span class="default"><font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c7">b</font></span><a href="group_5.html#K2" title="GROUP_5:func.2">.]</a></span> <a href="group_3.html#K2" title="GROUP_3:func.2">|^</a> <font color="Maroon" title="c9">d</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <span class="p1"><a href="group_5.html#K8" title="GROUP_5:func.8">[.</a><span class="default"><font color="Maroon" title="c2">N1</font>,<font color="Maroon" title="c4">N3</font></span><a href="group_5.html#K8" title="GROUP_5:func.8">.]</a></span> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Maroon" title="c9">d</font>
 
<b>by </b><i><a class="ref" href="group_3.html#T70" title="GROUP_3:th.70">GROUP_3:70</a></i>;<br/>
<a NAME="E13:76_1"/><b>then </b>
( <span class="p1"><a href="group_5.html#K2" title="GROUP_5:func.2">[.</a><span class="default"><font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c7">b</font></span><a href="group_5.html#K2" title="GROUP_5:func.2">.]</a></span> <a href="group_3.html#K2" title="GROUP_3:func.2">|^</a> <font color="Maroon" title="c9">d</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <span class="p1"><a href="group_5.html#K8" title="GROUP_5:func.8">[.</a><span class="default"><font color="Maroon" title="c2">N1</font>,<font color="Maroon" title="c4">N3</font></span><a href="group_5.html#K8" title="GROUP_5:func.8">.]</a></span> &amp; <span class="p1"><a href="group_5.html#K2" title="GROUP_5:func.2">[.</a><span class="default"><font color="Maroon" title="c9">d</font>,<font color="Maroon" title="c7">b</font></span><a href="group_5.html#K2" title="GROUP_5:func.2">.]</a></span> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <span class="p1"><a href="group_5.html#K8" title="GROUP_5:func.8">[.</a><span class="default"><font color="Maroon" title="c3">N2</font>,<font color="Maroon" title="c4">N3</font></span><a href="group_5.html#K8" title="GROUP_5:func.8">.]</a></span> )
 
<b>by </b><i><a class="txt" href="group_5.html#E1:76"><i><font color="Green" title="E41">A1</font></i></a>, <a class="txt" href="group_5.html#E5:76_1"><i><font color="Green" title="E44">A4</font></i></a>, <a class="txt" href="group_5.html#E9:76_1"><i><font color="Green" title="E47">A7</font></i></a>, <a class="ref" href="group_5.html#T74" target="_self" title="GROUP_5:th.74">Th74</a>, <a class="ref" href="group_3.html#D13" title="GROUP_3:def.13">GROUP_3:def 13</a></i>;<br/>
<b>hence </b><a NAME="E14:76_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="group_5.html#K8" title="GROUP_5:func.8">[.</a><span class="default"><font color="Maroon" title="c2">N1</font>,<font color="Maroon" title="c4">N3</font></span><a href="group_5.html#K8" title="GROUP_5:func.8">.]</a></span> <a href="group_4.html#K7" title="GROUP_4:func.7">*</a> <span class="p1"><a href="group_5.html#K8" title="GROUP_5:func.8">[.</a><span class="default"><font color="Maroon" title="c3">N2</font>,<font color="Maroon" title="c4">N3</font></span><a href="group_5.html#K8" title="GROUP_5:func.8">.]</a></span>
 <b>by </b><i><a class="txt" href="group_5.html#E10:76_1"><i><font color="Green" title="E48">A8</font></i></a>, <a class="ref" href="group_5.html#T4" target="_self" title="GROUP_5:th.4">Th4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:76"/><b>then </b>
 <a href="group_4.html#K5" title="GROUP_4:func.5">gr</a> <span class="p1">(<span class="default"><a href="group_5.html#K5" title="GROUP_5:func.5">commutators</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">N1</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c3">N2</font></span>)</span>,<font color="Maroon" title="c4">N3</font></span>)</span> is    <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of  <a href="group_4.html#K5" title="GROUP_4:func.5">gr</a> <span class="p1">(<span class="default"><span class="p2"><a href="group_5.html#K8" title="GROUP_5:func.8">[.</a><span class="default"><font color="Maroon" title="c2">N1</font>,<font color="Maroon" title="c4">N3</font></span><a href="group_5.html#K8" title="GROUP_5:func.8">.]</a></span> <a href="group_4.html#K7" title="GROUP_4:func.7">*</a> <span class="p2"><a href="group_5.html#K8" title="GROUP_5:func.8">[.</a><span class="default"><font color="Maroon" title="c3">N2</font>,<font color="Maroon" title="c4">N3</font></span><a href="group_5.html#K8" title="GROUP_5:func.8">.]</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="group_4.html#T41" title="GROUP_4:th.41">GROUP_4:41</a></i>;<br/>
<a NAME="E4:76"/><b>then </b><i><font color="Green" title="E42">A9</font></i>: 
<span class="p1"><a href="group_5.html#K8" title="GROUP_5:func.8">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">N1</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c3">N2</font></span>)</span>,<font color="Maroon" title="c4">N3</font></span><a href="group_5.html#K8" title="GROUP_5:func.8">.]</a></span> is    <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <span class="p1"><a href="group_5.html#K8" title="GROUP_5:func.8">[.</a><span class="default"><font color="Maroon" title="c2">N1</font>,<font color="Maroon" title="c4">N3</font></span><a href="group_5.html#K8" title="GROUP_5:func.8">.]</a></span> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <span class="p1"><a href="group_5.html#K8" title="GROUP_5:func.8">[.</a><span class="default"><font color="Maroon" title="c3">N2</font>,<font color="Maroon" title="c4">N3</font></span><a href="group_5.html#K8" title="GROUP_5:func.8">.]</a></span>
 
<b>by </b><i><a class="ref" href="group_4.html#T68" title="GROUP_4:th.68">GROUP_4:68</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font>;<br/><b>assume </b><a NAME="E1:76_2"/>
<font color="Maroon" title="c5">a</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <span class="p1"><a href="group_5.html#K8" title="GROUP_5:func.8">[.</a><span class="default"><font color="Maroon" title="c2">N1</font>,<font color="Maroon" title="c4">N3</font></span><a href="group_5.html#K8" title="GROUP_5:func.8">.]</a></span> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <span class="p1"><a href="group_5.html#K8" title="GROUP_5:func.8">[.</a><span class="default"><font color="Maroon" title="c3">N2</font>,<font color="Maroon" title="c4">N3</font></span><a href="group_5.html#K8" title="GROUP_5:func.8">.]</a></span>
 ;<br/><b>then consider </b><font color="Maroon" title="c6">b</font>, <font color="Maroon" title="c7">c</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E3:76_2"/><i><font color="Green" title="E43">A10</font></i>: 
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c7">c</font>
 <b>and </b><br/><a NAME="E4:76_2"/><i><font color="Green" title="E44">A11</font></i>: 
<font color="Maroon" title="c6">b</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <span class="p1"><a href="group_5.html#K8" title="GROUP_5:func.8">[.</a><span class="default"><font color="Maroon" title="c2">N1</font>,<font color="Maroon" title="c4">N3</font></span><a href="group_5.html#K8" title="GROUP_5:func.8">.]</a></span>
 <b>and </b><br/><a NAME="E5:76_2"/><i><font color="Green" title="E45">A12</font></i>: 
<font color="Maroon" title="c7">c</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <span class="p1"><a href="group_5.html#K8" title="GROUP_5:func.8">[.</a><span class="default"><font color="Maroon" title="c3">N2</font>,<font color="Maroon" title="c4">N3</font></span><a href="group_5.html#K8" title="GROUP_5:func.8">.]</a></span>
 <b>by </b><i><a class="txt" href="group_5.html#E1:76"><i><font color="Green" title="E41">A1</font></i></a>, <a class="ref" href="group_5.html#T7" target="_self" title="GROUP_5:th.7">Th7</a></i>;<br/><b>consider </b><font color="Maroon" title="c8">F1</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font>, <font color="Maroon" title="c9">I1</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> <b> such that </b><br/><a NAME="E7:76_2"/><i><font color="Green" title="E46">A13</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">I1</font>
 <b>and </b><br/><a NAME="E8:76_2"/><i><font color="Green" title="E47">A14</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">F1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="group_5.html#K5" title="GROUP_5:func.5">commutators</a> <font color="Maroon" title="c2">N1</font>,<font color="Maroon" title="c4">N3</font>
 <b>and </b><br/><a NAME="E9:76_2"/><i><font color="Green" title="E48">A15</font></i>: 
<font color="Maroon" title="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">F1</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Maroon" title="c9">I1</font></span>)</span>
 <b>by </b><i><a class="txt" href="group_5.html#E4:76_2"><i><font color="Green" title="E44">A11</font></i></a>, <a class="ref" href="group_5.html#T69" target="_self" title="GROUP_5:th.69">Th69</a></i>;<br/><b>consider </b><font color="Maroon" title="c10">F2</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font>, <font color="Maroon" title="c11">I2</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> <b> such that </b><br/><a NAME="E11:76_2"/><i><font color="Green" title="E49">A16</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">F2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">I2</font>
 <b>and </b><br/><a NAME="E12:76_2"/><i><font color="Green" title="E50">A17</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c10">F2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="group_5.html#K5" title="GROUP_5:func.5">commutators</a> <font color="Maroon" title="c3">N2</font>,<font color="Maroon" title="c4">N3</font>
 <b>and </b><br/><a NAME="E13:76_2"/><i><font color="Green" title="E51">A18</font></i>: 
<font color="Maroon" title="c7">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">F2</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Maroon" title="c11">I2</font></span>)</span>
 <b>by </b><i><a class="txt" href="group_5.html#E5:76_2"><i><font color="Green" title="E45">A12</font></i></a>, <a class="ref" href="group_5.html#T69" target="_self" title="GROUP_5:th.69">Th69</a></i>;<br/><i><font color="Green" title="E52">A19</font></i>:  <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">F1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c10">F2</font></span>)</span> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">I1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c11">I2</font></span>)</span></span>)</span> = 
 <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">F1</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Maroon" title="c9">I1</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">F2</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Maroon" title="c11">I2</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="group_5.html#E7:76_2"><i><font color="Green" title="E46">A13</font></i></a>, <a class="txt" href="group_5.html#E11:76_2"><i><font color="Green" title="E49">A16</font></i></a>, <a class="ref" href="group_4.html#T25" title="GROUP_4:th.25">GROUP_4:25</a></i>
<br/>.= 
<font color="Maroon" title="c5">a</font>
<b>by </b><i><a class="txt" href="group_5.html#E3:76_2"><i><font color="Green" title="E43">A10</font></i></a>, <a class="txt" href="group_5.html#E9:76_2"><i><font color="Green" title="E48">A15</font></i></a>, <a class="txt" href="group_5.html#E13:76_2"><i><font color="Green" title="E51">A18</font></i></a>, <a class="ref" href="group_4.html#T8" title="GROUP_4:th.8">GROUP_4:8</a></i>
;<br/><a NAME="E15:76_2"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">F1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c10">F2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">F1</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c10">F2</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T44" title="FINSEQ_1:th.44">FINSEQ_1:44</a></i>;<br/><a NAME="E16:76_2"/><b>then </b><i><font color="Green" title="E53">A20</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">F1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c10">F2</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="group_5.html#K5" title="GROUP_5:func.5">commutators</a> <font color="Maroon" title="c2">N1</font>,<font color="Maroon" title="c4">N3</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="group_5.html#K5" title="GROUP_5:func.5">commutators</a> <font color="Maroon" title="c3">N2</font>,<font color="Maroon" title="c4">N3</font></span>)</span>
 <b>by </b><i><a class="txt" href="group_5.html#E8:76_2"><i><font color="Green" title="E47">A14</font></i></a>, <a class="txt" href="group_5.html#E12:76_2"><i><font color="Green" title="E50">A17</font></i></a>, <a class="ref" href="xboole_1.html#T13" title="XBOOLE_1:th.13">XBOOLE_1:13</a></i>;<br/><a NAME="E17:76_2"/>
( <font color="Maroon" title="c2">N1</font> is    <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c2">N1</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c3">N2</font> &amp; <font color="Maroon" title="c3">N2</font> is    <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c2">N1</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c3">N2</font> &amp; <font color="Maroon" title="c4">N3</font> is    <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c4">N3</font> )
 <b>by </b><i><a class="ref" href="group_2.html#T63" title="GROUP_2:th.63">GROUP_2:63</a>, <a class="ref" href="group_4.html#T78" title="GROUP_4:th.78">GROUP_4:78</a></i>;<br/><a NAME="E18:76_2"/><b>then </b>
(  <a href="group_5.html#K5" title="GROUP_5:func.5">commutators</a> <font color="Maroon" title="c2">N1</font>,<font color="Maroon" title="c4">N3</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="group_5.html#K5" title="GROUP_5:func.5">commutators</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">N1</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c3">N2</font></span>)</span>,<font color="Maroon" title="c4">N3</font> &amp;  <a href="group_5.html#K5" title="GROUP_5:func.5">commutators</a> <font color="Maroon" title="c3">N2</font>,<font color="Maroon" title="c4">N3</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="group_5.html#K5" title="GROUP_5:func.5">commutators</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">N1</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c3">N2</font></span>)</span>,<font color="Maroon" title="c4">N3</font> )
 <b>by </b><i><a class="ref" href="group_5.html#T62" target="_self" title="GROUP_5:th.62">Th62</a></i>;<br/><a NAME="E19:76_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="group_5.html#K5" title="GROUP_5:func.5">commutators</a> <font color="Maroon" title="c2">N1</font>,<font color="Maroon" title="c4">N3</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="group_5.html#K5" title="GROUP_5:func.5">commutators</a> <font color="Maroon" title="c3">N2</font>,<font color="Maroon" title="c4">N3</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="group_5.html#K5" title="GROUP_5:func.5">commutators</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">N1</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c3">N2</font></span>)</span>,<font color="Maroon" title="c4">N3</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></i>;<br/><a NAME="E20:76_2"/><b>then </b><i><font color="Green" title="E54">A21</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">F1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c10">F2</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="group_5.html#K5" title="GROUP_5:func.5">commutators</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">N1</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c3">N2</font></span>)</span>,<font color="Maroon" title="c4">N3</font>
 <b>by </b><i><a class="txt" href="group_5.html#E16:76_2"><i><font color="Green" title="E53">A20</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/> <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">F1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c10">F2</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">I1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">I2</font></span>)</span>
<b>by </b><i><a class="txt" href="group_5.html#E7:76_2"><i><font color="Green" title="E46">A13</font></i></a>, <a class="txt" href="group_5.html#E11:76_2"><i><font color="Green" title="E49">A16</font></i></a>, <a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></i>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">I1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c11">I2</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></i>
;<br/><b>hence </b><a NAME="E22:76_2"/>
<font color="Maroon" title="c5">a</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <span class="p1"><a href="group_5.html#K8" title="GROUP_5:func.8">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">N1</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c3">N2</font></span>)</span>,<font color="Maroon" title="c4">N3</font></span><a href="group_5.html#K8" title="GROUP_5:func.8">.]</a></span>
 <b>by </b><i><a class="txt" href="group_5.html#E14:76_2"><i><font color="Green" title="E52">A19</font></i></a>, <a class="txt" href="group_5.html#E20:76_2"><i><font color="Green" title="E54">A21</font></i></a>, <a class="ref" href="group_5.html#T69" target="_self" title="GROUP_5:th.69">Th69</a></i>;<br/></div><b>end;</b></div>
<a NAME="E6:76"/><b>then </b>
<span class="p1"><a href="group_5.html#K8" title="GROUP_5:func.8">[.</a><span class="default"><font color="Maroon" title="c2">N1</font>,<font color="Maroon" title="c4">N3</font></span><a href="group_5.html#K8" title="GROUP_5:func.8">.]</a></span> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <span class="p1"><a href="group_5.html#K8" title="GROUP_5:func.8">[.</a><span class="default"><font color="Maroon" title="c3">N2</font>,<font color="Maroon" title="c4">N3</font></span><a href="group_5.html#K8" title="GROUP_5:func.8">.]</a></span> is    <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <span class="p1"><a href="group_5.html#K8" title="GROUP_5:func.8">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">N1</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c3">N2</font></span>)</span>,<font color="Maroon" title="c4">N3</font></span><a href="group_5.html#K8" title="GROUP_5:func.8">.]</a></span>
 
<b>by </b><i><a class="ref" href="group_2.html#T67" title="GROUP_2:th.67">GROUP_2:67</a></i>;<br/>
<b>hence </b><a NAME="E7:76"/>
<span class="p1"><a href="group_5.html#K8" title="GROUP_5:func.8">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">N1</font> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <font color="Maroon" title="c3">N2</font></span>)</span>,<font color="Maroon" title="c4">N3</font></span><a href="group_5.html#K8" title="GROUP_5:func.8">.]</a></span> <a href="group_2.html#R1" title="GROUP_2:pred.1">=</a> <span class="p1"><a href="group_5.html#K8" title="GROUP_5:func.8">[.</a><span class="default"><font color="Maroon" title="c2">N1</font>,<font color="Maroon" title="c4">N3</font></span><a href="group_5.html#K8" title="GROUP_5:func.8">.]</a></span> <a href="group_4.html#K8" title="GROUP_4:func.8">"\/"</a> <span class="p1"><a href="group_5.html#K8" title="GROUP_5:func.8">[.</a><span class="default"><font color="Maroon" title="c3">N2</font>,<font color="Maroon" title="c4">N3</font></span><a href="group_5.html#K8" title="GROUP_5:func.8">.]</a></span>
 <b>by </b><i><a class="txt" href="group_5.html#E4:76"><i><font color="Green" title="E42">A9</font></i></a>, <a class="ref" href="group_2.html#T64" title="GROUP_2:th.64">GROUP_2:64</a></i>;<br/>


</div>
