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

<span class="kw">let </span><font color="Maroon" title="c1">G</font> be   <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">H</font> being   <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font><br/> for <font color="Olive" title="b2">N</font> being  <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">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> holds <br/> (  <a href="group_5.html#K5" title="GROUP_5:func.5">commutators</a> <font color="Olive" title="b1">H</font>,<font color="Olive" title="b2">N</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Olive" title="b2">N</font> &amp;  <a href="group_5.html#K5" title="GROUP_5:func.5">commutators</a> <font color="Olive" title="b2">N</font>,<font color="Olive" title="b1">H</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Olive" title="b2">N</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">H</font> be    <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">N</font> being  <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">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> holds <br/> (  <a href="group_5.html#K5" title="GROUP_5:func.5">commutators</a> <font color="Maroon" title="c2">H</font>,<font color="Olive" title="b1">N</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Olive" title="b1">N</font> &amp;  <a href="group_5.html#K5" title="GROUP_5:func.5">commutators</a> <font color="Olive" title="b1">N</font>,<font color="Maroon" title="c2">H</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Olive" title="b1">N</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">N</font> be   <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="group_5.html#K5" title="GROUP_5:func.5">commutators</a> <font color="Maroon" title="c2">H</font>,<font color="Maroon" title="c3">N</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c3">N</font> &amp;  <a href="group_5.html#K5" title="GROUP_5:func.5">commutators</a> <font color="Maroon" title="c3">N</font>,<font color="Maroon" title="c2">H</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c3">N</font> )</span><br/>





<span class="kw">thus </span><a NAME="E1:65"/>
 <a href="group_5.html#K5" title="GROUP_5:func.5">commutators</a> <font color="Maroon" title="c2">H</font>,<font color="Maroon" title="c3">N</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c3">N</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="group_5.html#K5" title="GROUP_5:func.5">commutators</a> <font color="Maroon" title="c3">N</font>,<font color="Maroon" title="c2">H</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c3">N</font></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c4">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> <font color="Maroon" title="c2">H</font>,<font color="Maroon" title="c3">N</font> or <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c3">N</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:65_1"/>
<font color="Maroon" title="c4">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> <font color="Maroon" title="c2">H</font>,<font color="Maroon" title="c3">N</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c3">N</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c6">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font><span class="kw"> such that </span><br/><a NAME="E3:65_1"/><span class="lab"><font color="Green" title="E31">A1</font></span>: 
<font color="Maroon" title="c4">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="c5">a</font>,<font color="Maroon" title="c6">b</font></span><a href="group_5.html#K2" title="GROUP_5:func.2">.]</a></span>
 <span class="kw">and </span><br/><a NAME="E4:65_1"/>
<font color="Maroon" title="c5">a</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">H</font>
 <span class="kw">and </span><br/><a NAME="E5:65_1"/><span class="lab"><font color="Green" title="E32">A2</font></span>: 
<font color="Maroon" title="c6">b</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">N</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_5.html#T58" target="_self" title="GROUP_5:th.58">Th58</a></span>;<br/>
<a NAME="E6:65_1"/>
<font color="Maroon" title="c6">b</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a>  <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">N</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:65_1"><span class="lab"><font color="Green" title="E32">A2</font></span></a>, <a class="ref" href="group_2.html#T60" title="GROUP_2:th.60">GROUP_2:60</a></span>;<br/>
<a NAME="E7:65_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">b</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span> <a href="group_3.html#K2" title="GROUP_3:func.2">|^</a> <font color="Maroon" title="c5">a</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">N</font> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Maroon" title="c5">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_3.html#T70" title="GROUP_3:th.70">GROUP_3:70</a></span>;<br/>
<a NAME="E8:65_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">b</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span> <a href="group_3.html#K2" title="GROUP_3:func.2">|^</a> <font color="Maroon" title="c5">a</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">N</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_3.html#D13" title="GROUP_3:def.13">GROUP_3:def 13</a></span>;<br/>
<a NAME="E9:65_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">N</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:65_1"><span class="lab"><font color="Green" title="E31">A1</font></span></a>, <a class="txt" href="#E5:65_1"><span class="lab"><font color="Green" title="E32">A2</font></span></a>, <a class="ref" href="group_2.html#T59" title="GROUP_2:th.59">GROUP_2:59</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:65_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c3">N</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">let </span><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c4">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> <font color="Maroon" title="c3">N</font>,<font color="Maroon" title="c2">H</font> or <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c3">N</font> )</span><br/>

<span class="kw">assume </span><a NAME="E2:65"/>
<font color="Maroon" title="c4">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> <font color="Maroon" title="c3">N</font>,<font color="Maroon" title="c2">H</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c3">N</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c6">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font><span class="kw"> such that </span><br/><a NAME="E4:65"/><span class="lab"><font color="Green" title="E31">A3</font></span>: 
<font color="Maroon" title="c4">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="c5">a</font>,<font color="Maroon" title="c6">b</font></span><a href="group_5.html#K2" title="GROUP_5:func.2">.]</a></span>
 <span class="kw">and </span><br/><a NAME="E5:65"/><span class="lab"><font color="Green" title="E32">A4</font></span>: 
<font color="Maroon" title="c5">a</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">N</font>
 <span class="kw">and </span><br/><a NAME="E6:65"/>
<font color="Maroon" title="c6">b</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">H</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_5.html#T58" target="_self" title="GROUP_5:th.58">Th58</a></span>;<br/>
<a NAME="E7:65"/>
<font color="Maroon" title="c5">a</font> <a href="group_3.html#K2" title="GROUP_3:func.2">|^</a> <font color="Maroon" title="c6">b</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">N</font> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Maroon" title="c6">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:65"><span class="lab"><font color="Green" title="E32">A4</font></span></a>, <a class="ref" href="group_3.html#T70" title="GROUP_3:th.70">GROUP_3:70</a></span>;<br/>
<a NAME="E8:65"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A5</font></span>: 
<font color="Maroon" title="c5">a</font> <a href="group_3.html#K2" title="GROUP_3:func.2">|^</a> <font color="Maroon" title="c6">b</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">N</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_3.html#D13" title="GROUP_3:def.13">GROUP_3:def 13</a></span>;<br/>
<a NAME="E9:65"/>
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="group_3.html#K2" title="GROUP_3:func.2">|^</a> <font color="Maroon" title="c6">b</font></span>)</span> &amp; <font color="Maroon" title="c5">a</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a>  <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">N</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:65"><span class="lab"><font color="Green" title="E31">A3</font></span></a>, <a class="txt" href="#E5:65"><span class="lab"><font color="Green" title="E32">A4</font></span></a>, <a class="ref" href="group_5.html#T19" target="_self" title="GROUP_5:th.19">Th19</a>, <a class="ref" href="group_2.html#T60" title="GROUP_2:th.60">GROUP_2:60</a></span>;<br/>
<a NAME="E10:65"/><span class="kw">then </span>
<font color="Maroon" title="c4">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">N</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:65"><span class="lab"><font color="Green" title="E33">A5</font></span></a>, <a class="ref" href="group_2.html#T59" title="GROUP_2:th.59">GROUP_2:59</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:65"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_2.html#K8" title="GROUP_2:func.8">carr</a> <font color="Maroon" title="c3">N</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
