<?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> 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/>



<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">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/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">b</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="E1:73_1_1"/>
<font color="Maroon" title="c5">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="c3">N2</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="c4">a</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c6">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="E3:73_1_1"/><i><font color="Green" title="E38">A1</font></i>: 
<font color="Maroon" title="c5">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">c</font> <a href="group_3.html#K2" title="GROUP_3:func.2">|^</a> <font color="Maroon" title="c4">a</font>
 <b>and </b><br/><a NAME="E4:73_1_1"/><i><font color="Green" title="E39">A2</font></i>: 
<font color="Maroon" title="c6">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="c2">N1</font>,<font color="Maroon" title="c3">N2</font></span><a href="group_5.html#K8" title="GROUP_5:func.8">.]</a></span>
 <b>by </b><i><a class="ref" href="group_3.html#T70" title="GROUP_3:th.70">GROUP_3:70</a></i>;<br/><b>consider </b><font color="Maroon" title="c7">F</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="c8">I</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="E6:73_1_1"/><i><font color="Green" title="E40">A3</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">F</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="c8">I</font>
 <b>and </b><br/><a NAME="E7:73_1_1"/><i><font color="Green" title="E41">A4</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="group_5.html#K4" title="GROUP_5:func.4">commutators</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">N1</font></span>)</span>,<span class="p1">(<span class="default"><a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <font color="Maroon" title="c3">N2</font></span>)</span>
 <b>and </b><br/><a NAME="E8:73_1_1"/><i><font color="Green" title="E42">A5</font></i>: 
<font color="Maroon" title="c6">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="c7">F</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Maroon" title="c8">I</font></span>)</span>
 <b>by </b><i><a class="txt" href="group_5.html#E4:73_1_1"><i><font color="Green" title="E39">A2</font></i></a>, <a class="ref" href="group_4.html#T37" title="GROUP_4:th.37">GROUP_4:37</a></i>;<br/><i><font color="Green" title="E43">A6</font></i>: <font color="Maroon" title="c5">b</font> = 
 <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="c7">F</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Maroon" title="c8">I</font></span>)</span> <a href="group_5.html#K1" title="GROUP_5:func.1">|^</a> <font color="Maroon" title="c4">a</font></span>)</span>
<b>by </b><i><a class="txt" href="group_5.html#E3:73_1_1"><i><font color="Green" title="E38">A1</font></i></a>, <a class="txt" href="group_5.html#E8:73_1_1"><i><font color="Green" title="E42">A5</font></i></a>, <a class="ref" href="group_5.html#T17" target="_self" title="GROUP_5:th.17">Th17</a></i>
<br/>.= 
 <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="c7">F</font> <a href="group_5.html#K1" title="GROUP_5:func.1">|^</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Maroon" title="c8">I</font></span>)</span>
<b>by </b><i><a class="ref" href="group_5.html#T18" target="_self" title="GROUP_5:th.18">Th18</a></i>
;<br/><a NAME="E10:73_1_1"/><i><font color="Green" title="E44">A7</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="group_5.html#K1" title="GROUP_5:func.1">|^</a> <font color="Maroon" title="c4">a</font></span>)</span> <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="c7">F</font>
 <b>by </b><i><a class="ref" href="group_5.html#D1" target="_self" title="GROUP_5:def.1">Def1</a></i>;<br/><a NAME="E11:73_1_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="group_5.html#K1" title="GROUP_5:func.1">|^</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="group_5.html#K4" title="GROUP_5:func.4">commutators</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">N1</font></span>)</span>,<span class="p1">(<span class="default"><a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <font color="Maroon" title="c3">N2</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">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="E1:73_1_1_2"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="group_5.html#K1" title="GROUP_5:func.1">|^</a> <font color="Maroon" title="c4">a</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c10">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:73_1_1_2"/><i><font color="Green" title="E45">A8</font></i>: 
<font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="group_5.html#K1" title="GROUP_5:func.1">|^</a> <font color="Maroon" title="c4">a</font></span>)</span>
 <b>and </b><br/><a NAME="E4:73_1_1_2"/><i><font color="Green" title="E46">A9</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="group_5.html#K1" title="GROUP_5:func.1">|^</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">y</font> = <font color="Maroon" title="c10">y</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="group_5.html#E3:73_1_1_2"><i><font color="Green" title="E45">A8</font></i></a></i>;<br/>
<a NAME="E6:73_1_1_2"/>
<font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">F</font>
 
<b>by </b><i><a class="txt" href="group_5.html#E10:73_1_1"><i><font color="Green" title="E44">A7</font></i></a>, <a class="txt" href="group_5.html#E3:73_1_1_2"><i><font color="Green" title="E45">A8</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E7:73_1_1_2"/><b>then </b>
<font color="Maroon" title="c7">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">F</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c12">d</font>, <font color="Maroon" title="c13">e</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="E9:73_1_1_2"/><i><font color="Green" title="E47">A10</font></i>: 
<font color="Maroon" title="c7">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">y</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="c12">d</font>,<font color="Maroon" title="c13">e</font></span><a href="group_5.html#K2" title="GROUP_5:func.2">.]</a></span>
 <b>and </b><br/><a NAME="E10:73_1_1_2"/><i><font color="Green" title="E48">A11</font></i>: 
( <font color="Maroon" title="c12">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <font color="Maroon" title="c2">N1</font> &amp; <font color="Maroon" title="c13">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <font color="Maroon" title="c3">N2</font> )
 <b>by </b><i><a class="txt" href="group_5.html#E7:73_1_1"><i><font color="Green" title="E41">A4</font></i></a>, <a class="ref" href="group_5.html#T52" target="_self" title="GROUP_5:th.52">Th52</a></i>;<br/>
<a NAME="E11:73_1_1_2"/>
<font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">F</font>
 
<b>by </b><i><a class="txt" href="group_5.html#E10:73_1_1"><i><font color="Green" title="E44">A7</font></i></a>, <a class="txt" href="group_5.html#E3:73_1_1_2"><i><font color="Green" title="E45">A8</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E12:73_1_1_2"/><b>then </b>
( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">y</font></span>)</span> <a href="group_3.html#K2" title="GROUP_3:func.2">|^</a> <font color="Maroon" title="c4">a</font> &amp; <font color="Maroon" title="c7">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">y</font> )
 
<b>by </b><i><a class="txt" href="group_5.html#E4:73_1_1_2"><i><font color="Green" title="E46">A9</font></i></a>, <a class="ref" href="group_5.html#D1" target="_self" title="GROUP_5:def.1">Def1</a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/>
<a NAME="E13:73_1_1_2"/><b>then </b><i><font color="Green" title="E49">A12</font></i>: 
<font color="Maroon" title="c9">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"><span class="p2">(<span class="default"><font color="Maroon" title="c12">d</font> <a href="group_3.html#K2" title="GROUP_3:func.2">|^</a> <font color="Maroon" title="c4">a</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c13">e</font> <a href="group_3.html#K2" title="GROUP_3:func.2">|^</a> <font color="Maroon" title="c4">a</font></span>)</span></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#E9:73_1_1_2"><i><font color="Green" title="E47">A10</font></i></a>, <a class="ref" href="group_5.html#T26" target="_self" title="GROUP_5:th.26">Th26</a></i>;<br/>
<a NAME="E14:73_1_1_2"/>
( <font color="Maroon" title="c12">d</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">N1</font> &amp; <font color="Maroon" title="c13">e</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#E10:73_1_1_2"><i><font color="Green" title="E48">A11</font></i></a>, <a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/>
<a NAME="E15:73_1_1_2"/><b>then </b>
( <font color="Maroon" title="c12">d</font> <a href="group_3.html#K2" title="GROUP_3:func.2">|^</a> <font color="Maroon" title="c4">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_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Maroon" title="c4">a</font> &amp; <font color="Maroon" title="c13">e</font> <a href="group_3.html#K2" title="GROUP_3:func.2">|^</a> <font color="Maroon" title="c4">a</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">N2</font> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Maroon" title="c4">a</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="E16:73_1_1_2"/><b>then </b>
( <font color="Maroon" title="c12">d</font> <a href="group_3.html#K2" title="GROUP_3:func.2">|^</a> <font color="Maroon" title="c4">a</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">N1</font> &amp; <font color="Maroon" title="c13">e</font> <a href="group_3.html#K2" title="GROUP_3:func.2">|^</a> <font color="Maroon" title="c4">a</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="ref" href="group_3.html#D13" title="GROUP_3:def.13">GROUP_3:def 13</a></i>;<br/>
<a NAME="E17:73_1_1_2"/><b>then </b>
( <font color="Maroon" title="c12">d</font> <a href="group_3.html#K2" title="GROUP_3:func.2">|^</a> <font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <font color="Maroon" title="c2">N1</font> &amp; <font color="Maroon" title="c13">e</font> <a href="group_3.html#K2" title="GROUP_3:func.2">|^</a> <font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <font color="Maroon" title="c3">N2</font> )
 
<b>by </b><i><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/>
<b>hence </b><a NAME="E18:73_1_1_2"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_5.html#K4" title="GROUP_5:func.4">commutators</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">N1</font></span>)</span>,<span class="p1">(<span class="default"><a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <font color="Maroon" title="c3">N2</font></span>)</span>
 <b>by </b><i><a class="txt" href="group_5.html#E13:73_1_1_2"><i><font color="Green" title="E49">A12</font></i></a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E12:73_1_1"/>
<font color="Maroon" title="c5">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="c3">N2</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#E6:73_1_1"><i><font color="Green" title="E40">A3</font></i></a>, <a class="txt" href="group_5.html#E9:73_1_1"><i><font color="Green" title="E43">A6</font></i></a>, <a class="txt" href="group_5.html#E10:73_1_1"><i><font color="Green" title="E44">A7</font></i></a>, <a class="ref" href="group_4.html#T37" title="GROUP_4:th.37">GROUP_4:37</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E2:73_1"/>
<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="c3">N2</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="c4">a</font> 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="c3">N2</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/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:73"/>
<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="c3">N2</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_3.html#T145" title="GROUP_3:th.145">GROUP_3:145</a></i>;<br/>


</div>
