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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="group_1.html#NM1">Group</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be  <a href="group_1.html#V1">strict</a> <a href="group_3.html#V1">normal</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">c<sub>1</sub></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">c<sub>4</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></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">c<sub>5</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>assume </b><a NAME="E1:73_1_1"/>
<font color="Maroon">c<sub>5</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="group_5.html#K8" target="_self">.]</a></span> <a href="group_3.html#K6">|^</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:73_1_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="group_3.html#K2">|^</a> <font color="Maroon">c<sub>4</sub></font>
 <b>and </b><br/><a NAME="E4:73_1_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>6</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="group_5.html#K8" target="_self">.]</a></span>
 <b>by </b><i><a class="ref" href="group_3.html#T70">GROUP_3:70</a></i>;<br/><a NAME="E5:73_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="rlvect_1.html#R1">in</a> <span class="p1"><a href="group_5.html#K7" target="_self">[.</a><span class="default"><span class="p2">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="group_5.html#K7" target="_self">.]</a></span>
 <b>by </b><i><a class="txt" href="group_5.html#E4:73_1_1"><i><font color="Green">E41</font></i></a></i>;<br/><a NAME="E6:73_1_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="rlvect_1.html#R1">in</a>  <a href="group_4.html#K5">gr</a> <span class="p1">(<span class="default"><a href="group_5.html#K4" target="_self">commutators</a> <span class="p2">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>7</sub></font> being    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>8</sub></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="E8:73_1_1"/><i><font color="Green">E42</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>8</sub></font>
 <b>and </b><br/><a NAME="E9:73_1_1"/><i><font color="Green">E43</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">c=</a>  <a href="group_5.html#K4" target="_self">commutators</a> <span class="p1">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E10:73_1_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="group_4.html#K3">Product</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="group_4.html#K4">|^</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="group_4.html#T37">GROUP_4:37</a></i>;<br/><i><font color="Green">E45</font></i>: <font color="Maroon">c<sub>5</sub></font> = 
 <a href="group_4.html#K3">Product</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="group_4.html#K4">|^</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="group_5.html#K1" target="_self">|^</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="group_5.html#E3:73_1_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="group_5.html#E10:73_1_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="group_5.html#T17" target="_self">Th17</a></i>
<br/>.= 
 <a href="group_4.html#K3">Product</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="group_5.html#K1" target="_self">|^</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="group_4.html#K4">|^</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="group_5.html#T18" target="_self">Th18</a></i>
;<br/><a NAME="E12:73_1_1"/><i><font color="Green">E46</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="group_5.html#K1" target="_self">|^</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="ref" href="group_5.html#D1" target="_self">Def1</a></i>;<br/><a NAME="E13:73_1_1"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="group_5.html#K1" target="_self">|^</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="group_5.html#K4" target="_self">commutators</a> <span class="p1">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>3</sub></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">c<sub>9</sub></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="E1:73_1_1_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="group_5.html#K1" target="_self">|^</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>10</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:73_1_1_2"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="group_5.html#K1" target="_self">|^</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E4:73_1_1_2"/><i><font color="Green">E48</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="group_5.html#K1" target="_self">|^</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></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">c<sub>11</sub></font> = <font color="Maroon">c<sub>10</sub></font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="group_5.html#E3:73_1_1_2"><i><font color="Green">E47</font></i></a></i>;<br/>
<a NAME="E6:73_1_1_2"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="group_5.html#E12:73_1_1"><i><font color="Green">E46</font></i></a>, <a class="txt" href="group_5.html#E3:73_1_1_2"><i><font color="Green">E47</font></i></a>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E7:73_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>7</sub></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">c<sub>12</sub></font>, <font color="Maroon">c<sub>13</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E9:73_1_1_2"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></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<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font></span><a href="group_5.html#K2" target="_self">.]</a></span>
 <b>and </b><br/><a NAME="E10:73_1_1_2"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="group_5.html#E9:73_1_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="group_5.html#T52" target="_self">Th52</a></i>;<br/>
<a NAME="E11:73_1_1_2"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="group_5.html#E12:73_1_1"><i><font color="Green">E46</font></i></a>, <a class="txt" href="group_5.html#E3:73_1_1_2"><i><font color="Green">E47</font></i></a>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E12:73_1_1_2"/><b>then </b>
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="group_3.html#K2">|^</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>11</sub></font> )
 
<b>by </b><i><a class="txt" href="group_5.html#E4:73_1_1_2"><i><font color="Green">E48</font></i></a>, <a class="ref" href="group_5.html#D1" target="_self">Def1</a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<a NAME="E13:73_1_1_2"/><b>then </b><i><font color="Green">E51</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="group_5.html#K2" target="_self">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="group_3.html#K2">|^</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="group_3.html#K2">|^</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="group_5.html#K2" target="_self">.]</a></span>
 
<b>by </b><i><a class="txt" href="group_5.html#E9:73_1_1_2"><i><font color="Green">E49</font></i></a>, <a class="ref" href="group_5.html#T26" target="_self">Th26</a></i>;<br/>
<a NAME="E14:73_1_1_2"/>
( <font color="Maroon">c<sub>12</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="txt" href="group_5.html#E10:73_1_1_2"><i><font color="Green">E50</font></i></a>, <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<a NAME="E15:73_1_1_2"/><b>then </b>
( <font color="Maroon">c<sub>12</sub></font> <a href="group_3.html#K2">|^</a> <font color="Maroon">c<sub>4</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="group_3.html#K6">|^</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="group_3.html#K2">|^</a> <font color="Maroon">c<sub>4</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="group_3.html#K6">|^</a> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="ref" href="group_3.html#T70">GROUP_3:70</a></i>;<br/>
<a NAME="E16:73_1_1_2"/><b>then </b>
( <font color="Maroon">c<sub>12</sub></font> <a href="group_3.html#K2">|^</a> <font color="Maroon">c<sub>4</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="group_3.html#K2">|^</a> <font color="Maroon">c<sub>4</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="ref" href="group_3.html#D13">GROUP_3:def 13</a></i>;<br/>
<a NAME="E17:73_1_1_2"/><b>then </b>
( <font color="Maroon">c<sub>12</sub></font> <a href="group_3.html#K2">|^</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="group_3.html#K2">|^</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E18:73_1_1_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="group_5.html#K4" target="_self">commutators</a> <span class="p1">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="group_5.html#E13:73_1_1_2"><i><font color="Green">E51</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E14:73_1_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="rlvect_1.html#R1">in</a>  <a href="group_4.html#K5">gr</a> <span class="p1">(<span class="default"><a href="group_5.html#K4" target="_self">commutators</a> <span class="p2">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="group_5.html#E8:73_1_1"><i><font color="Green">E42</font></i></a>, <a class="txt" href="group_5.html#E11:73_1_1"><i><font color="Green">E45</font></i></a>, <a class="txt" href="group_5.html#E12:73_1_1"><i><font color="Green">E46</font></i></a>, <a class="ref" href="group_4.html#T37">GROUP_4:37</a></i>;<br/><a NAME="E15:73_1_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="rlvect_1.html#R1">in</a> <span class="p1"><a href="group_5.html#K7" target="_self">[.</a><span class="default"><span class="p2">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="group_5.html#K7" target="_self">.]</a></span>
 ;<br/><b>hence </b><a NAME="E16:73_1_1"/>
<font color="Maroon">c<sub>5</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="group_5.html#K8" target="_self">.]</a></span>
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E2:73_1"/>
<span class="p1"><a href="group_5.html#K8" target="_self">[.</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="group_5.html#K8" target="_self">.]</a></span> <a href="group_3.html#K6">|^</a> <font color="Maroon">c<sub>4</sub></font> 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">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></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><b>end;</b></div>
<b>hence </b><a NAME="E2:73"/>
<span class="p1"><a href="group_5.html#K8" target="_self">[.</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="group_5.html#K8" target="_self">.]</a></span> is  <a href="group_3.html#V1">normal</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="group_3.html#T145">GROUP_3:145</a></i>;<br/>


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