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

<a NAME="E1:29_1_1"/>
( <font color="Maroon">G</font> <a href="group_6.html#K6" target="_self">./.</a> <font color="Maroon">N</font> is <a href="group_1.html#V4">associative</a> &amp; <font color="Maroon">G</font> <a href="group_6.html#K6" target="_self">./.</a> <font color="Maroon">N</font> is <a href="group_1.html#V3">Group-like</a> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:29_1_1_1"/>
for <font color="Olive">f</font>, <font color="Olive">g</font>, <font color="Olive">h</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="group_6.html#K6" target="_self">./.</a> <font color="Maroon">N</font></span>)</span> holds  <font color="Olive">f</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Olive">g</font> <a href="group_1.html#K1">*</a> <font color="Olive">h</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">f</font> <a href="group_1.html#K1">*</a> <font color="Olive">g</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Olive">h</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="group_1.html#D4">GROUP_1:def 4</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">f</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="group_6.html#K6" target="_self">./.</a> <font color="Maroon">N</font></span>)</span>, <font color="Maroon">g</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="group_6.html#K6" target="_self">./.</a> <font color="Maroon">N</font></span>)</span>, <font color="Maroon">h</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="group_6.html#K6" target="_self">./.</a> <font color="Maroon">N</font></span>)</span>;<br/>

<b>consider </b><font color="Maroon">a</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E2:29_1_1_1_1"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="group_2.html#K12">*</a> <font color="Maroon">N</font> &amp; <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">N</font> <a href="group_2.html#K13">*</a> <font color="Maroon">a</font> )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">c</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E4:29_1_1_1_1"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> <a href="group_2.html#K12">*</a> <font color="Maroon">N</font> &amp; <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">N</font> <a href="group_2.html#K13">*</a> <font color="Maroon">c</font> )
 <b>by </b><i/>;<br/>
<b>thus </b><font color="Maroon">f</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="group_1.html#K1">*</a> <font color="Maroon">h</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="group_6.html#K7" target="_self">@</a> <font color="Maroon">f</font></span>)</span> <a href="group_2.html#K2">*</a> <span class="p1">(<span class="default"><a href="group_6.html#K7" target="_self">@</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="group_1.html#K1">*</a> <font color="Maroon">h</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_2.html#K12">*</a> <font color="Maroon">N</font></span>)</span> <a href="group_2.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_6.html#K7" target="_self">@</a> <font color="Maroon">g</font></span>)</span> <a href="group_2.html#K2">*</a> <span class="p2">(<span class="default"><a href="group_6.html#K7" target="_self">@</a> <font color="Maroon">h</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_6.html#T1" target="_self">Th1</a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_6.html#K7" target="_self">@</a> <font color="Maroon">f</font></span>)</span> <a href="group_2.html#K2">*</a> <span class="p2">(<span class="default"><a href="group_6.html#K7" target="_self">@</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="group_2.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="group_2.html#K12">*</a> <font color="Maroon">N</font></span>)</span>
<b>by </b><i><a class="ref" href="group_6.html#T1" target="_self">Th1</a>, , <a class="ref" href="group_2.html#T14">GROUP_2:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_6.html#K7" target="_self">@</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="group_1.html#K1">*</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="group_2.html#K2">*</a> <span class="p1">(<span class="default"><a href="group_6.html#K7" target="_self">@</a> <font color="Maroon">h</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="group_1.html#K1">*</a> <font color="Maroon">g</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">h</font>
<b>by </b><i/>
;<br/>


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

<a NAME="E2:29_1_1_1"/>
(  <a href="group_2.html#K7">carr</a> <font color="Maroon">N</font> <a href="hidden.html#R2">in</a>  <a href="group_2.html#K14">Left_Cosets</a> <font color="Maroon">N</font> &amp;  <a href="group_6.html#K4" target="_self">Cosets</a> <font color="Maroon">N</font> <a href="hidden.html#R1">=</a>  <a href="group_2.html#K14">Left_Cosets</a> <font color="Maroon">N</font> )
 
<b>by </b><i><a class="ref" href="group_2.html#T165">GROUP_2:165</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">e</font> =  <a href="group_2.html#K7">carr</a> <font color="Maroon">N</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="group_6.html#K6" target="_self">./.</a> <font color="Maroon">N</font></span>)</span> ;<br/>
<b>take </b>
<font color="Maroon">e</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="group_1.html#D3">GROUP_1:def 3</a><br/>

<b>let </b><font color="Maroon">h</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="group_6.html#K6" target="_self">./.</a> <font color="Maroon">N</font></span>)</span>;<br/>

<b>consider </b><font color="Maroon">a</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E5:29_1_1_1"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="group_2.html#K12">*</a> <font color="Maroon">N</font> &amp; <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">N</font> <a href="group_2.html#K13">*</a> <font color="Maroon">a</font> )
 <b>by </b><i/>;<br/>
<a NAME="E6:29_1_1_1"/><i><font color="Green">E52</font></i>: 
(  <a href="group_6.html#K7" target="_self">@</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> &amp;  <a href="group_6.html#K7" target="_self">@</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e</font> )
 
;<br/>
<b>thus </b><font color="Maroon">h</font> <a href="group_1.html#K1">*</a> <font color="Maroon">e</font> = 
<span class="p1">(<span class="default"><a href="group_6.html#K7" target="_self">@</a> <font color="Maroon">h</font></span>)</span> <a href="group_2.html#K2">*</a> <span class="p1">(<span class="default"><a href="group_6.html#K7" target="_self">@</a> <font color="Maroon">e</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_2.html#K12">*</a> <font color="Maroon">N</font></span>)</span> <a href="group_2.html#K10">*</a> <font color="Maroon">N</font>
<b>by </b><i><a class="ref" href="group_6.html#T1" target="_self">Th1</a></i>
<br/>.= 
<font color="Maroon">a</font> <a href="group_2.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">N</font> <a href="group_4.html#K7">*</a> <font color="Maroon">N</font></span>)</span>
<b>by </b><i><a class="ref" href="group_4.html#T60">GROUP_4:60</a></i>
<br/>.= 
<font color="Maroon">h</font>
<b>by </b><i><a class="ref" href="group_4.html#T58">GROUP_4:58</a>, <a class="ref" href="group_6.html#T1" target="_self">Th1</a></i>
;<br/>

<b>thus </b><font color="Maroon">e</font> <a href="group_1.html#K1">*</a> <font color="Maroon">h</font> = 
<span class="p1">(<span class="default"><a href="group_6.html#K7" target="_self">@</a> <font color="Maroon">e</font></span>)</span> <a href="group_2.html#K2">*</a> <span class="p1">(<span class="default"><a href="group_6.html#K7" target="_self">@</a> <font color="Maroon">h</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">N</font> <a href="group_2.html#K11">*</a> <span class="p1">(<span class="default"><font color="Maroon">N</font> <a href="group_2.html#K13">*</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i><a class="ref" href="group_6.html#T1" target="_self">Th1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">N</font> <a href="group_4.html#K7">*</a> <font color="Maroon">N</font></span>)</span> <a href="group_2.html#K4">*</a> <font color="Maroon">a</font>
<b>by </b><i><a class="ref" href="group_4.html#T61">GROUP_4:61</a></i>
<br/>.= 
<font color="Maroon">h</font>
<b>by </b><i><a class="ref" href="group_4.html#T58">GROUP_4:58</a>, <a class="ref" href="group_6.html#T1" target="_self">Th1</a></i>
;<br/>

<b>reconsider </b><font color="Maroon">g</font> = <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_2.html#K12">*</a> <font color="Maroon">N</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="group_6.html#K6" target="_self">./.</a> <font color="Maroon">N</font></span>)</span> <b>by </b><i><a class="ref" href="group_2.html#D15">GROUP_2:def 15</a></i>;<br/>
<a NAME="E10:29_1_1_1"/><i><font color="Green">E53</font></i>: 
 <a href="group_6.html#K7" target="_self">@</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font>
 
;<br/>
<b>take </b>
<font color="Maroon">g</font>
;<br/>

<b>thus </b><font color="Maroon">h</font> <a href="group_1.html#K1">*</a> <font color="Maroon">g</font> = 
<span class="p1">(<span class="default"><font color="Maroon">N</font> <a href="group_2.html#K13">*</a> <font color="Maroon">a</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">a</font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_2.html#K12">*</a> <font color="Maroon">N</font></span>)</span>
<b>by </b><i><a class="ref" href="group_6.html#T1" target="_self">Th1</a>, , , </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">N</font> <a href="group_2.html#K13">*</a> <font color="Maroon">a</font></span>)</span> <a href="group_2.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span> <a href="group_2.html#K10">*</a> <font color="Maroon">N</font>
<b>by </b><i><a class="ref" href="group_3.html#T10">GROUP_3:10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">N</font> <a href="group_2.html#K13">*</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span></span>)</span> <a href="group_2.html#K10">*</a> <font color="Maroon">N</font>
<b>by </b><i><a class="ref" href="group_2.html#T129">GROUP_2:129</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">N</font> <a href="group_2.html#K13">*</a> <span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">G</font></span>)</span></span>)</span> <a href="group_2.html#K10">*</a> <font color="Maroon">N</font>
<b>by </b><i><a class="ref" href="group_1.html#D6">GROUP_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">N</font></span>)</span> <a href="group_2.html#K2">*</a> <span class="p1">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">N</font></span>)</span>
<b>by </b><i><a class="ref" href="group_2.html#T132">GROUP_2:132</a></i>
<br/>.= 
<font color="Maroon">e</font>
<b>by </b><i><a class="ref" href="group_2.html#T91">GROUP_2:91</a></i>
;<br/>

<b>thus </b><font color="Maroon">g</font> <a href="group_1.html#K1">*</a> <font color="Maroon">h</font> = 
<span class="p1">(<span class="default"><a href="group_6.html#K7" target="_self">@</a> <font color="Maroon">g</font></span>)</span> <a href="group_2.html#K2">*</a> <span class="p1">(<span class="default"><a href="group_6.html#K7" target="_self">@</a> <font color="Maroon">h</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_2.html#K12">*</a> <font color="Maroon">N</font></span>)</span> <a href="group_2.html#K4">*</a> <font color="Maroon">a</font></span>)</span> <a href="group_2.html#K10">*</a> <font color="Maroon">N</font>
<b>by </b><i><a class="ref" href="group_6.html#T1" target="_self">Th1</a>, <a class="ref" href="group_3.html#T10">GROUP_3:10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</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">N</font> <a href="group_2.html#K13">*</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="group_2.html#K10">*</a> <font color="Maroon">N</font>
<b>by </b><i><a class="ref" href="group_2.html#T128">GROUP_2:128</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</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">a</font> <a href="group_2.html#K12">*</a> <font color="Maroon">N</font></span>)</span></span>)</span> <a href="group_2.html#K10">*</a> <font color="Maroon">N</font>
<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">a</font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">a</font></span>)</span> <a href="group_2.html#K12">*</a> <font color="Maroon">N</font></span>)</span> <a href="group_2.html#K10">*</a> <font color="Maroon">N</font>
<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"><a href="group_1.html#K2">1.</a> <font color="Maroon">G</font></span>)</span> <a href="group_2.html#K12">*</a> <font color="Maroon">N</font></span>)</span> <a href="group_2.html#K10">*</a> <font color="Maroon">N</font>
<b>by </b><i><a class="ref" href="group_1.html#D6">GROUP_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">G</font></span>)</span> <a href="group_2.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">N</font> <a href="group_4.html#K7">*</a> <font color="Maroon">N</font></span>)</span>
<b>by </b><i><a class="ref" href="group_4.html#T60">GROUP_4:60</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">G</font></span>)</span> <a href="group_2.html#K3">*</a> <span class="p1">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">N</font></span>)</span>
<b>by </b><i><a class="ref" href="group_4.html#T58">GROUP_4:58</a></i>
<br/>.= 
<font color="Maroon">e</font>
<b>by </b><i><a class="ref" href="group_2.html#T43">GROUP_2:43</a></i>
;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:29_1_1"/>
( <font color="Maroon">G</font> <a href="group_6.html#K6" target="_self">./.</a> <font color="Maroon">N</font> is <a href="group_1.html#V4">associative</a> &amp; <font color="Maroon">G</font> <a href="group_6.html#K6" target="_self">./.</a> <font color="Maroon">N</font> is <a href="group_1.html#V3">Group-like</a> )
 ;<br/>


</div>
