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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="group_1.html#V1">strict</a> <a href="group_1.html#NM1">Group</a>;<br/><b>let </b><font color="Maroon">c<sub>2</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/>



<b>thus </b><a NAME="E1:36"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="group_6.html#K6" target="_self">./.</a> <font color="Maroon">c<sub>2</sub></font> is  <a href="group_1.html#V7">commutative</a> <a href="group_1.html#NM1">Group</a> implies <font color="Maroon">c<sub>1</sub></font> <a href="group_5.html#K9">`</a>  is    <a href="group_6.html#M1" target="_self">Subgroup</a> of <font color="Maroon">c<sub>2</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:36_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="group_6.html#K6" target="_self">./.</a> <font color="Maroon">c<sub>2</sub></font> is  <a href="group_1.html#V7">commutative</a> <a href="group_1.html#NM1">Group</a>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>3</sub></font>, <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/><b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>4</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>2</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="group_6.html#K6" target="_self">./.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <b>by </b><i><a class="ref" href="group_6.html#T27" target="_self">Th27</a></i>;<br/><a NAME="E2:36_1_1"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="group_1.html#K3">"</a>  <a href="hidden.html#R1">=</a>  <a href="group_6.html#K7" target="_self">@</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="group_1.html#K3">"</a> </span>)</span> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="group_1.html#K3">"</a>  <a href="hidden.html#R1">=</a>  <a href="group_6.html#K7" target="_self">@</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="group_1.html#K3">"</a> </span>)</span> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="group_1.html#K3">"</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="group_1.html#K3">"</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="ref" href="group_6.html#T30" target="_self">Th30</a></i>;<br/><a NAME="E3:36_1_1"/><b>then </b><i><font color="Green">E30</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="group_2.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="group_6.html#T24" target="_self">Th24</a></i>;<br/><a NAME="E4:36_1_1"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="group_6.html#K7" target="_self">@</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="group_6.html#K7" target="_self">@</a> <font color="Maroon">c<sub>6</sub></font> )
 ;<br/><a NAME="E5:36_1_1"/><b>then </b><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="group_2.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="group_6.html#T24" target="_self">Th24</a></i>;<br/><a NAME="E6:36_1_1"/><i><font color="Green">E32</font></i>: 
(  <a href="group_6.html#K7" target="_self">@</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="group_1.html#K3">"</a> </span>)</span> &amp;  <a href="group_6.html#K7" target="_self">@</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>6</sub></font> )
 ;<br/><a NAME="E7:36_1_1"/>
<span class="p1"><a href="group_5.html#K2">[.</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="group_5.html#K2">.]</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="group_5.html#T19">GROUP_5:19</a></i>;<br/><a NAME="E8:36_1_1"/><b>then </b>
( <span class="p1"><a href="group_5.html#K2">[.</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="group_5.html#K2">.]</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="group_2.html#K2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="group_2.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="group_2.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> &amp; <span class="p1"><a href="group_5.html#K2">[.</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="group_5.html#K2">.]</a></span> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="group_6.html#K6" target="_self">./.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp;  <a href="group_1.html#K2">1.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="group_6.html#K6" target="_self">./.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="group_6.html#E1:36_1"><i><font color="Green">E29</font></i></a>, <a class="txt" href="group_6.html#E3:36_1_1"><i><font color="Green">E30</font></i></a>, <a class="txt" href="group_6.html#E5:36_1_1"><i><font color="Green">E31</font></i></a>, <a class="txt" href="group_6.html#E6:36_1_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="group_6.html#T24" target="_self">Th24</a>, <a class="ref" href="group_6.html#T29" target="_self">Th29</a>, <a class="ref" href="group_5.html#T40">GROUP_5:40</a></i>;<br/><b>then </b> <a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>2</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="group_2.html#K2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="group_2.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_2.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_2.html#K11">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_3.html#T11">GROUP_3:11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="group_2.html#K2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="group_2.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_2.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_2.html#K13">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="group_2.html#K10">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_3.html#T15">GROUP_3:15</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="group_2.html#K2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="group_2.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_2.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="group_2.html#K10">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_3.html#T140">GROUP_3:140</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="group_2.html#K2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="group_2.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_2.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_6.html#T6" target="_self">Th6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="group_2.html#K2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="group_2.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="group_2.html#T127">GROUP_2:127</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_2.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_2.html#K11">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="group_2.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="group_3.html#T11">GROUP_3:11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_2.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_2.html#K13">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span> <a href="group_2.html#K10">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="group_2.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="group_3.html#T15">GROUP_3:15</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_2.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="group_2.html#K10">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="group_2.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="group_3.html#T140">GROUP_3:140</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_2.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="group_2.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="group_6.html#T6" target="_self">Th6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="group_2.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="group_2.html#T127">GROUP_2:127</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span> <a href="group_2.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_2.html#K11">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_3.html#T11">GROUP_3:11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span> <a href="group_2.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_2.html#K13">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="group_2.html#K10">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="group_3.html#T15">GROUP_3:15</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span> <a href="group_2.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="group_2.html#K10">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="group_3.html#T140">GROUP_3:140</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span> <a href="group_2.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="group_6.html#T6" target="_self">Th6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="group_2.html#T127">GROUP_2:127</a></i>
<br/>.= 
<span class="p1"><a href="group_5.html#K2">[.</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#K2">.]</a></span> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="group_5.html#T19">GROUP_5:19</a></i>
;<br/><b>hence </b><a NAME="E10:36_1_1"/>
<span class="p1"><a href="group_5.html#K2">[.</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#K2">.]</a></span> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="group_2.html#T136">GROUP_2:136</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:36_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="group_5.html#K9">`</a>  is    <a href="group_6.html#M1" target="_self">Subgroup</a> of <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="group_6.html#T8" target="_self">Th8</a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:36"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="group_5.html#K9">`</a>  is    <a href="group_6.html#M1" target="_self">Subgroup</a> of <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="group_6.html#K6" target="_self">./.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>;<br/><b>consider </b><font color="Maroon">c<sub>5</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="E2:36_2"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="group_2.html#K13">*</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="ref" href="group_6.html#T26" target="_self">Th26</a></i>;<br/><b>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="E4:36_2"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="group_2.html#K13">*</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="ref" href="group_6.html#T26" target="_self">Th26</a></i>;<br/><a NAME="E5:36_2"/>
<span class="p1"><a href="group_5.html#K2">[.</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="group_5.html#K2">.]</a></span> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="group_6.html#E2:36"><i><font color="Green">E29</font></i></a>, <a class="ref" href="group_6.html#T8" target="_self">Th8</a></i>;<br/><b>then </b><i><font color="Green">E32</font></i>:  <a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>2</sub></font> = 
<span class="p1"><a href="group_5.html#K2">[.</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="group_5.html#K2">.]</a></span> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="group_2.html#T136">GROUP_2:136</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="group_5.html#T19">GROUP_5:19</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span> <a href="group_2.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="group_2.html#T127">GROUP_2:127</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span> <a href="group_2.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="group_2.html#K10">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="group_6.html#T6" target="_self">Th6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span> <a href="group_2.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_2.html#K13">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="group_2.html#K10">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="group_3.html#T140">GROUP_3:140</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span> <a href="group_2.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_2.html#K11">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_3.html#T15">GROUP_3:15</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="group_2.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="group_3.html#T11">GROUP_3:11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_2.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="group_2.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="group_2.html#T127">GROUP_2:127</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_2.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="group_2.html#K10">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="group_2.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="group_6.html#T6" target="_self">Th6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_2.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_2.html#K13">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span> <a href="group_2.html#K10">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="group_2.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="group_3.html#T140">GROUP_3:140</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_2.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_2.html#K11">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="group_2.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="group_3.html#T15">GROUP_3:15</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="group_2.html#K2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="group_2.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="group_3.html#T11">GROUP_3:11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="group_2.html#K2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="group_2.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="group_2.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_2.html#T127">GROUP_2:127</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="group_2.html#K2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="group_2.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="group_2.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="group_2.html#K10">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_6.html#T6" target="_self">Th6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="group_2.html#K2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="group_2.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="group_2.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_2.html#K13">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="group_2.html#K10">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_3.html#T140">GROUP_3:140</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="group_2.html#K2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="group_2.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="group_2.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_2.html#K11">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_3.html#T15">GROUP_3:15</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="group_2.html#K2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="group_2.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="group_2.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_3.html#T11">GROUP_3:11</a></i>
;<br/><a NAME="E7:36_2"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K3">"</a>  <a href="hidden.html#R1">=</a>  <a href="group_6.html#K7" target="_self">@</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K3">"</a> </span>)</span> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K3">"</a>  <a href="hidden.html#R1">=</a>  <a href="group_6.html#K7" target="_self">@</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K3">"</a> </span>)</span> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K3">"</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K3">"</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="group_6.html#E2:36_2"><i><font color="Green">E30</font></i></a>, <a class="txt" href="group_6.html#E4:36_2"><i><font color="Green">E31</font></i></a>, <a class="ref" href="group_6.html#T30" target="_self">Th30</a></i>;<br/><a NAME="E8:36_2"/><b>then </b><i><font color="Green">E33</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="group_2.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="group_6.html#T24" target="_self">Th24</a></i>;<br/><a NAME="E9:36_2"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="group_6.html#K7" target="_self">@</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="group_6.html#K7" target="_self">@</a> <font color="Maroon">c<sub>4</sub></font> )
 ;<br/><a NAME="E10:36_2"/><b>then </b><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="group_2.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="group_6.html#E2:36_2"><i><font color="Green">E30</font></i></a>, <a class="txt" href="group_6.html#E4:36_2"><i><font color="Green">E31</font></i></a>, <a class="ref" href="group_6.html#T24" target="_self">Th24</a></i>;<br/><a NAME="E11:36_2"/><i><font color="Green">E35</font></i>: 
(  <a href="group_6.html#K7" target="_self">@</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K3">"</a> </span>)</span> &amp;  <a href="group_6.html#K7" target="_self">@</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font> )
 ;<br/><a NAME="E12:36_2"/>
<span class="p1"><a href="group_5.html#K2">[.</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#K2">.]</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="group_5.html#T19">GROUP_5:19</a></i>;<br/><a NAME="E13:36_2"/><b>then </b>
( <span class="p1"><a href="group_5.html#K2">[.</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#K2">.]</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="group_2.html#K2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="group_2.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="group_2.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="group_2.html#K12">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> &amp;  <a href="group_1.html#K2">1.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="group_6.html#K6" target="_self">./.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="group_2.html#K7">carr</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="group_6.html#E8:36_2"><i><font color="Green">E33</font></i></a>, <a class="txt" href="group_6.html#E10:36_2"><i><font color="Green">E34</font></i></a>, <a class="txt" href="group_6.html#E11:36_2"><i><font color="Green">E35</font></i></a>, <a class="ref" href="group_6.html#T24" target="_self">Th24</a>, <a class="ref" href="group_6.html#T29" target="_self">Th29</a></i>;<br/><b>hence </b><a NAME="E14:36_2"/>
<span class="p1"><a href="group_5.html#K2">[.</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#K2">.]</a></span> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="group_6.html#K6" target="_self">./.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="group_6.html#E6:36_2"><i><font color="Green">E32</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:36"/>
<font color="Maroon">c<sub>1</sub></font> <a href="group_6.html#K6" target="_self">./.</a> <font color="Maroon">c<sub>2</sub></font> is  <a href="group_1.html#V7">commutative</a> <a href="group_1.html#NM1">Group</a>
 <b>by </b><i><a class="ref" href="group_5.html#T40">GROUP_5:40</a></i>;<br/>


</div>
