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<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">H</font> be    <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font>;<br/><b>let </b><font color="Maroon" title="c3">N</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="72_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/><b>assume </b><a NAME="E1:72_1"/>
<font color="Maroon" title="c4">a</font> <a href="rlvect_1.html#R1" title="RLVECT_1: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">N</font>,<font color="Maroon" title="c2">H</font></span><a href="group_5.html#K8" title="GROUP_5:func.8">.]</a></span>
 ;<br/><b>then consider </b><font color="Maroon" title="c5">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="c6">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="E3:72_1"/><i><font color="Green" title="E39">A1</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">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="c6">I</font>
 <b>and </b><br/><a NAME="E4:72_1"/><i><font color="Green" title="E40">A2</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">F</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">N</font>,<font color="Maroon" title="c2">H</font>
 <b>and </b><br/><a NAME="E5:72_1"/><i><font color="Green" title="E41">A3</font></i>: 
<font color="Maroon" title="c4">a</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="c5">F</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Maroon" title="c6">I</font></span>)</span>
 <b>by </b><i><a class="ref" href="group_5.html#T73" target="_self" title="GROUP_5:th.73">Th73</a></i>;<br/><a NAME="E6:72_1"/>
 <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#K7" title="GROUP_2:func.7">carr</a> <font color="Maroon" title="c3">N</font>
 <b>by </b><i><a class="ref" href="group_5.html#T61" target="_self" title="GROUP_5:th.61">Th61</a></i>;<br/><a NAME="E7:72_1"/><b>then </b>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <font color="Maroon" title="c3">N</font>
 <b>by </b><i><a class="txt" href="group_5.html#E4:72_1"><i><font color="Green" title="E40">A2</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><a NAME="E8:72_1"/><b>then </b>
<font color="Maroon" title="c4">a</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a>  <a href="group_4.html#K5" title="GROUP_4:func.5">gr</a> <span class="p1">(<span class="default"><a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <font color="Maroon" title="c3">N</font></span>)</span>
 <b>by </b><i><a class="txt" href="group_5.html#E3:72_1"><i><font color="Green" title="E39">A1</font></i></a>, <a class="txt" href="group_5.html#E5:72_1"><i><font color="Green" title="E41">A3</font></i></a>, <a class="ref" href="group_4.html#T37" title="GROUP_4:th.37">GROUP_4:37</a></i>;<br/><b>hence </b><a NAME="E9:72_1"/>
<font color="Maroon" title="c4">a</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c3">N</font>
 <b>by </b><i><a class="ref" href="group_4.html#T40" title="GROUP_4:th.40">GROUP_4:40</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:72"/>
<span class="p1"><a href="group_5.html#K8" title="GROUP_5:func.8">[.</a><span class="default"><font color="Maroon" title="c3">N</font>,<font color="Maroon" title="c2">H</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 <font color="Maroon" title="c3">N</font>
 <b>by </b><i><a class="ref" href="group_2.html#T67" title="GROUP_2:th.67">GROUP_2:67</a></i>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_2"><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/><b>assume </b><a NAME="E1:72_2"/>
<font color="Maroon" title="c4">a</font> <a href="rlvect_1.html#R1" title="RLVECT_1: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">H</font>,<font color="Maroon" title="c3">N</font></span><a href="group_5.html#K8" title="GROUP_5:func.8">.]</a></span>
 ;<br/><b>then consider </b><font color="Maroon" title="c5">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="c6">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="E3:72_2"/><i><font color="Green" title="E39">A4</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">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="c6">I</font>
 <b>and </b><br/><a NAME="E4:72_2"/><i><font color="Green" title="E40">A5</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">F</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">H</font>,<font color="Maroon" title="c3">N</font>
 <b>and </b><br/><a NAME="E5:72_2"/><i><font color="Green" title="E41">A6</font></i>: 
<font color="Maroon" title="c4">a</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="c5">F</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Maroon" title="c6">I</font></span>)</span>
 <b>by </b><i><a class="ref" href="group_5.html#T73" target="_self" title="GROUP_5:th.73">Th73</a></i>;<br/><a NAME="E6:72_2"/>
 <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#K7" title="GROUP_2:func.7">carr</a> <font color="Maroon" title="c3">N</font>
 <b>by </b><i><a class="ref" href="group_5.html#T61" target="_self" title="GROUP_5:th.61">Th61</a></i>;<br/><a NAME="E7:72_2"/><b>then </b>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <font color="Maroon" title="c3">N</font>
 <b>by </b><i><a class="txt" href="group_5.html#E4:72_2"><i><font color="Green" title="E40">A5</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><a NAME="E8:72_2"/><b>then </b>
<font color="Maroon" title="c4">a</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a>  <a href="group_4.html#K5" title="GROUP_4:func.5">gr</a> <span class="p1">(<span class="default"><a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <font color="Maroon" title="c3">N</font></span>)</span>
 <b>by </b><i><a class="txt" href="group_5.html#E3:72_2"><i><font color="Green" title="E39">A4</font></i></a>, <a class="txt" href="group_5.html#E5:72_2"><i><font color="Green" title="E41">A6</font></i></a>, <a class="ref" href="group_4.html#T37" title="GROUP_4:th.37">GROUP_4:37</a></i>;<br/><b>hence </b><a NAME="E9:72_2"/>
<font color="Maroon" title="c4">a</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c3">N</font>
 <b>by </b><i><a class="ref" href="group_4.html#T40" title="GROUP_4:th.40">GROUP_4:40</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:72"/>
<span class="p1"><a href="group_5.html#K8" title="GROUP_5:func.8">[.</a><span class="default"><font color="Maroon" title="c2">H</font>,<font color="Maroon" title="c3">N</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 <font color="Maroon" title="c3">N</font>
 <b>by </b><i><a class="ref" href="group_2.html#T67" title="GROUP_2:th.67">GROUP_2:67</a></i>;<br/>


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