<?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>, <font color="Maroon">c<sub>4</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/>



<a NAME="E1:76"/><i><font color="Green">E43</font></i>: 
( <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>4</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> &amp; <span class="p1"><a href="group_5.html#K8" target="_self">[.</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</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_5.html#T77" target="_self">Th77</a></i>;<br/>
<a NAME="E2:76"/>
 <a href="group_5.html#K5" target="_self">commutators</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</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>4</sub></font></span><a href="group_5.html#K8" target="_self">.]</a></span> <a href="group_4.html#K7">*</a> <span class="p1"><a href="group_5.html#K8" target="_self">[.</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="group_5.html#K8" target="_self">.]</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">c<sub>5</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="E1:76_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="group_5.html#K5" target="_self">commutators</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</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="E3:76_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>5</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>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="group_5.html#K2" target="_self">.]</a></span>
 <b>and </b><br/><a NAME="E4:76_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">c<sub>3</sub></font>
 <b>and </b><br/><a NAME="E5:76_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="group_5.html#T58" target="_self">Th58</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</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="E7:76_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>9</sub></font>
 <b>and </b><br/><a NAME="E8:76_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E9:76_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>9</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#E4:76_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="group_5.html#T7" target="_self">Th7</a></i>;<br/>
<a NAME="E10:76_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="group_5.html#K2" target="_self">[.</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="group_5.html#K2" target="_self">.]</a></span> <a href="group_3.html#K2">|^</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1"><a href="group_5.html#K2" target="_self">[.</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="group_5.html#K2" target="_self">.]</a></span>
 
<b>by </b><i><a class="txt" href="group_5.html#E3:76_1"><i><font color="Green">E44</font></i></a>, <a class="txt" href="group_5.html#E7:76_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="group_5.html#T28" target="_self">Th28</a></i>;<br/>
<a NAME="E11:76_1"/>
<span class="p1"><a href="group_5.html#K2" target="_self">[.</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="group_5.html#K2" target="_self">.]</a></span> <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>4</sub></font></span><a href="group_5.html#K8" target="_self">.]</a></span>
 
<b>by </b><i><a class="txt" href="group_5.html#E5:76_1"><i><font color="Green">E46</font></i></a>, <a class="txt" href="group_5.html#E8:76_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="group_5.html#T74" target="_self">Th74</a></i>;<br/>
<a NAME="E12:76_1"/><b>then </b>
<span class="p1"><a href="group_5.html#K2" target="_self">[.</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="group_5.html#K2" target="_self">.]</a></span> <a href="group_3.html#K2">|^</a> <font color="Maroon">c<sub>9</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>4</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>9</sub></font>
 
<b>by </b><i><a class="ref" href="group_3.html#T70">GROUP_3:70</a></i>;<br/>
<a NAME="E13:76_1"/><b>then </b>
( <span class="p1"><a href="group_5.html#K2" target="_self">[.</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="group_5.html#K2" target="_self">.]</a></span> <a href="group_3.html#K2">|^</a> <font color="Maroon">c<sub>9</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>4</sub></font></span><a href="group_5.html#K8" target="_self">.]</a></span> &amp; <span class="p1"><a href="group_5.html#K2" target="_self">[.</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="group_5.html#K2" target="_self">.]</a></span> <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>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="group_5.html#K8" target="_self">.]</a></span> )
 
<b>by </b><i><a class="txt" href="group_5.html#E1:76"><i><font color="Green">E43</font></i></a>, <a class="txt" href="group_5.html#E5:76_1"><i><font color="Green">E46</font></i></a>, <a class="txt" href="group_5.html#E9:76_1"><i><font color="Green">E49</font></i></a>, <a class="ref" href="group_5.html#T74" target="_self">Th74</a>, <a class="ref" href="group_3.html#D13">GROUP_3:def 13</a></i>;<br/>
<b>hence </b><a NAME="E14:76_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">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>4</sub></font></span><a href="group_5.html#K8" target="_self">.]</a></span> <a href="group_4.html#K7">*</a> <span class="p1"><a href="group_5.html#K8" target="_self">[.</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="group_5.html#K8" target="_self">.]</a></span>
 <b>by </b><i><a class="txt" href="group_5.html#E10:76_1"><i><font color="Green">E50</font></i></a>, <a class="ref" href="group_5.html#T4" target="_self">Th4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:76"/><b>then </b>
 <a href="group_4.html#K5">gr</a> <span class="p1">(<span class="default"><a href="group_5.html#K5" target="_self">commutators</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font></span>)</span> is    <a href="group_2.html#M1">Subgroup</a> of  <a href="group_4.html#K5">gr</a> <span class="p1">(<span class="default"><span class="p2"><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>4</sub></font></span><a href="group_5.html#K8" target="_self">.]</a></span> <a href="group_4.html#K7">*</a> <span class="p2"><a href="group_5.html#K8" target="_self">[.</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="group_5.html#K8" target="_self">.]</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="group_4.html#T41">GROUP_4:41</a></i>;<br/>
<a NAME="E4:76"/><b>then </b>
<span class="p1"><a href="group_5.html#K8" target="_self">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font></span><a href="group_5.html#K8" target="_self">.]</a></span> is    <a href="group_2.html#M1">Subgroup</a> of  <a href="group_4.html#K5">gr</a> <span class="p1">(<span class="default"><span class="p2"><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>4</sub></font></span><a href="group_5.html#K8" target="_self">.]</a></span> <a href="group_4.html#K7">*</a> <span class="p2"><a href="group_5.html#K8" target="_self">[.</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="group_5.html#K8" target="_self">.]</a></span></span>)</span>
 
;<br/>
<a NAME="E5:76"/><b>then </b><i><font color="Green">E44</font></i>: 
<span class="p1"><a href="group_5.html#K8" target="_self">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font></span><a href="group_5.html#K8" target="_self">.]</a></span> 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>4</sub></font></span><a href="group_5.html#K8" target="_self">.]</a></span> <a href="group_4.html#K8">"\/"</a> <span class="p1"><a href="group_5.html#K8" target="_self">[.</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="group_5.html#K8" target="_self">.]</a></span>
 
<b>by </b><i><a class="ref" href="group_4.html#T68">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">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="E1:76_2"/>
<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>4</sub></font></span><a href="group_5.html#K8" target="_self">.]</a></span> <a href="group_4.html#K8">"\/"</a> <span class="p1"><a href="group_5.html#K8" target="_self">[.</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="group_5.html#K8" target="_self">.]</a></span>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</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="E3:76_2"/><i><font color="Green">E45</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_1.html#K1">*</a> <font color="Maroon">c<sub>7</sub></font>
 <b>and </b><br/><a NAME="E4:76_2"/><i><font color="Green">E46</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>4</sub></font></span><a href="group_5.html#K8" target="_self">.]</a></span>
 <b>and </b><br/><a NAME="E5:76_2"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>7</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>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="group_5.html#K8" target="_self">.]</a></span>
 <b>by </b><i><a class="txt" href="group_5.html#E1:76"><i><font color="Green">E43</font></i></a>, <a class="ref" href="group_5.html#T7" target="_self">Th7</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>8</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>9</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="E7:76_2"/><i><font color="Green">E48</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>9</sub></font>
 <b>and </b><br/><a NAME="E8:76_2"/><i><font color="Green">E49</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>8</sub></font> <a href="tarski.html#R1">c=</a>  <a href="group_5.html#K5" target="_self">commutators</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 <b>and </b><br/><a NAME="E9:76_2"/><i><font color="Green">E50</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>8</sub></font> <a href="group_4.html#K4">|^</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="group_5.html#E4:76_2"><i><font color="Green">E46</font></i></a>, <a class="ref" href="group_5.html#T73" target="_self">Th73</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>10</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>11</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="E11:76_2"/><i><font color="Green">E51</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>11</sub></font>
 <b>and </b><br/><a NAME="E12:76_2"/><i><font color="Green">E52</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>10</sub></font> <a href="tarski.html#R1">c=</a>  <a href="group_5.html#K5" target="_self">commutators</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 <b>and </b><br/><a NAME="E13:76_2"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">c<sub>7</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>10</sub></font> <a href="group_4.html#K4">|^</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="group_5.html#E5:76_2"><i><font color="Green">E47</font></i></a>, <a class="ref" href="group_5.html#T73" target="_self">Th73</a></i>;<br/><i><font color="Green">E54</font></i>:  <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>8</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="group_4.html#K4">|^</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span> = 
 <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>8</sub></font> <a href="group_4.html#K4">|^</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="group_4.html#K4">|^</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="group_5.html#E7:76_2"><i><font color="Green">E48</font></i></a>, <a class="txt" href="group_5.html#E11:76_2"><i><font color="Green">E51</font></i></a>, <a class="ref" href="group_4.html#T25">GROUP_4:25</a></i>
<br/>.= 
<font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="group_5.html#E3:76_2"><i><font color="Green">E45</font></i></a>, <a class="txt" href="group_5.html#E9:76_2"><i><font color="Green">E50</font></i></a>, <a class="txt" href="group_5.html#E13:76_2"><i><font color="Green">E53</font></i></a>, <a class="ref" href="group_4.html#T8">GROUP_4:8</a></i>
;<br/><a NAME="E15:76_2"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T44">FINSEQ_1:44</a></i>;<br/><a NAME="E16:76_2"/><b>then </b><i><font color="Green">E55</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="group_5.html#K5" target="_self">commutators</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="group_5.html#K5" target="_self">commutators</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="group_5.html#E8:76_2"><i><font color="Green">E49</font></i></a>, <a class="txt" href="group_5.html#E12:76_2"><i><font color="Green">E52</font></i></a>, <a class="ref" href="xboole_1.html#T13">XBOOLE_1:13</a></i>;<br/><a NAME="E17:76_2"/>
( <font color="Maroon">c<sub>2</sub></font> is    <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">c<sub>2</sub></font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> is    <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">c<sub>2</sub></font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> is    <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="ref" href="group_2.html#T63">GROUP_2:63</a>, <a class="ref" href="group_4.html#T78">GROUP_4:78</a></i>;<br/><a NAME="E18:76_2"/><b>then </b>
(  <a href="group_5.html#K5" target="_self">commutators</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a>  <a href="group_5.html#K5" target="_self">commutators</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font> &amp;  <a href="group_5.html#K5" target="_self">commutators</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a>  <a href="group_5.html#K5" target="_self">commutators</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="ref" href="group_5.html#T62" target="_self">Th62</a></i>;<br/><a NAME="E19:76_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="group_5.html#K5" target="_self">commutators</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="group_5.html#K5" target="_self">commutators</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="group_5.html#K5" target="_self">commutators</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/><a NAME="E20:76_2"/><b>then </b><i><font color="Green">E56</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="group_5.html#K5" target="_self">commutators</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="group_5.html#E16:76_2"><i><font color="Green">E55</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/> <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="group_5.html#E7:76_2"><i><font color="Green">E48</font></i></a>, <a class="txt" href="group_5.html#E11:76_2"><i><font color="Green">E51</font></i></a>, <a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>
;<br/><b>hence </b><a NAME="E22:76_2"/>
<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"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font></span><a href="group_5.html#K8" target="_self">.]</a></span>
 <b>by </b><i><a class="txt" href="group_5.html#E14:76_2"><i><font color="Green">E54</font></i></a>, <a class="txt" href="group_5.html#E20:76_2"><i><font color="Green">E56</font></i></a>, <a class="ref" href="group_5.html#T73" target="_self">Th73</a></i>;<br/></div><b>end;</b></div>
<a NAME="E7:76"/><b>then </b>
<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>4</sub></font></span><a href="group_5.html#K8" target="_self">.]</a></span> <a href="group_4.html#K8">"\/"</a> <span class="p1"><a href="group_5.html#K8" target="_self">[.</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="group_5.html#K8" target="_self">.]</a></span> 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"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>4</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/>
<b>hence </b><a NAME="E8:76"/>
<span class="p1"><a href="group_5.html#K8" target="_self">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font></span><a href="group_5.html#K8" target="_self">.]</a></span> <a href="group_2.html#R1">=</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>4</sub></font></span><a href="group_5.html#K8" target="_self">.]</a></span> <a href="group_4.html#K8">"\/"</a> <span class="p1"><a href="group_5.html#K8" target="_self">[.</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="group_5.html#K8" target="_self">.]</a></span>
 <b>by </b><i><a class="txt" href="group_5.html#E5:76"><i><font color="Green">E44</font></i></a>, <a class="ref" href="group_2.html#T64">GROUP_2:64</a></i>;<br/>


</div>
