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

<b>set </b><font color="Maroon">H</font> =  <a href="group_1.html#G1">HGrStr</a>(# <span class="p1">(<span class="default"><a href="autalg_1.html#K1" target="_self">UAAut</a> <font color="Maroon">UA</font></span>)</span>,<span class="p1">(<span class="default"><a href="autalg_1.html#K2" target="_self">UAAutComp</a> <font color="Maroon">UA</font></span>)</span> #);<br/>
<a NAME="E1:10_1_1"/>
(  <a href="group_1.html#G1">HGrStr</a>(# <span class="p1">(<span class="default"><a href="autalg_1.html#K1" target="_self">UAAut</a> <font color="Maroon">UA</font></span>)</span>,<span class="p1">(<span class="default"><a href="autalg_1.html#K2" target="_self">UAAutComp</a> <font color="Maroon">UA</font></span>)</span> #) is <a href="group_1.html#V4">associative</a> &amp;  <a href="group_1.html#G1">HGrStr</a>(# <span class="p1">(<span class="default"><a href="autalg_1.html#K1" target="_self">UAAut</a> <font color="Maroon">UA</font></span>)</span>,<span class="p1">(<span class="default"><a href="autalg_1.html#K2" target="_self">UAAutComp</a> <font color="Maroon">UA</font></span>)</span> #) is <a href="group_1.html#V3">Group-like</a> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:10_1_1_1"/><i><font color="Green">E33</font></i>: 
for <font color="Olive">f</font>, <font color="Olive">g</font> being  <a href="struct_0.html#NM1">Element</a> of <a href="group_1.html#G1">HGrStr</a>(# <span class="p1">(<span class="default"><a href="autalg_1.html#K1" target="_self">UAAut</a> <font color="Maroon">UA</font></span>)</span>,<span class="p1">(<span class="default"><a href="autalg_1.html#K2" target="_self">UAAutComp</a> <font color="Maroon">UA</font></span>)</span> #)<br/> for <font color="Olive">A</font>, <font color="Olive">B</font> being   <a href="fraenkel.html#M2">Element</a> of  <a href="autalg_1.html#K1" target="_self">UAAut</a> <font color="Maroon">UA</font>  st <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <font color="Olive">A</font> &amp; <font color="Olive">g</font> <a href="hidden.html#R1">=</a> <font color="Olive">B</font> holds <br/><font color="Olive">f</font> <a href="group_1.html#K1">*</a> <font color="Olive">g</font> <a href="hidden.html#R1">=</a> <font color="Olive">B</font> <a href="partfun1.html#K1">*</a> <font color="Olive">A</font>
 
<b>by </b><i/>;<br/>
<b>thus </b><a NAME="E2:10_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 <a href="group_1.html#G1">HGrStr</a>(# <span class="p1">(<span class="default"><a href="autalg_1.html#K1" target="_self">UAAut</a> <font color="Maroon">UA</font></span>)</span>,<span class="p1">(<span class="default"><a href="autalg_1.html#K2" target="_self">UAAutComp</a> <font color="Maroon">UA</font></span>)</span> #) holds  <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> <a href="hidden.html#R1">=</a> <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>
  <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="10_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 <a href="group_1.html#G1">HGrStr</a>(# <span class="p1">(<span class="default"><a href="autalg_1.html#K1" target="_self">UAAut</a> <font color="Maroon">UA</font></span>)</span>,<span class="p1">(<span class="default"><a href="autalg_1.html#K2" target="_self">UAAutComp</a> <font color="Maroon">UA</font></span>)</span> #), <font color="Maroon">g</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="group_1.html#G1">HGrStr</a>(# <span class="p1">(<span class="default"><a href="autalg_1.html#K1" target="_self">UAAut</a> <font color="Maroon">UA</font></span>)</span>,<span class="p1">(<span class="default"><a href="autalg_1.html#K2" target="_self">UAAutComp</a> <font color="Maroon">UA</font></span>)</span> #), <font color="Maroon">h</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="group_1.html#G1">HGrStr</a>(# <span class="p1">(<span class="default"><a href="autalg_1.html#K1" target="_self">UAAut</a> <font color="Maroon">UA</font></span>)</span>,<span class="p1">(<span class="default"><a href="autalg_1.html#K2" target="_self">UAAutComp</a> <font color="Maroon">UA</font></span>)</span> #);<br/>

<b>reconsider </b><font color="Maroon">A</font> = <font color="Maroon">f</font>, <font color="Maroon">B</font> = <font color="Maroon">g</font>, <font color="Maroon">C</font> = <font color="Maroon">h</font> as    <a href="fraenkel.html#M2">Element</a> of  <a href="autalg_1.html#K1" target="_self">UAAut</a> <font color="Maroon">UA</font> ;<br/>
<a NAME="E2:10_1_1_1_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">f</font> <a href="group_1.html#K1">*</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">A</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:10_1_1_1_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">g</font> <a href="group_1.html#K1">*</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">B</font>
 
<b>by </b><i/>;<br/>
<b>thus </b><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> = 
<font color="Maroon">C</font> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">A</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">B</font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">A</font>
<b>by </b><i><a class="ref" href="relat_1.html#T55">RELAT_1:55</a></i>
<br/>.= 
<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>
<b>by </b><i>, </i>
;<br/>


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

<b>thus </b><a NAME="E3:10_1_1_1"/>
 ex <font color="Olive">e</font> being  <a href="struct_0.html#NM1">Element</a> of <a href="group_1.html#G1">HGrStr</a>(# <span class="p1">(<span class="default"><a href="autalg_1.html#K1" target="_self">UAAut</a> <font color="Maroon">UA</font></span>)</span>,<span class="p1">(<span class="default"><a href="autalg_1.html#K2" target="_self">UAAutComp</a> <font color="Maroon">UA</font></span>)</span> #) st <br/>for <font color="Olive">h</font> being  <a href="struct_0.html#NM1">Element</a> of <a href="group_1.html#G1">HGrStr</a>(# <span class="p1">(<span class="default"><a href="autalg_1.html#K1" target="_self">UAAut</a> <font color="Maroon">UA</font></span>)</span>,<span class="p1">(<span class="default"><a href="autalg_1.html#K2" target="_self">UAAutComp</a> <font color="Maroon">UA</font></span>)</span> #) holds <br/> ( <font color="Olive">h</font> <a href="group_1.html#K1">*</a> <font color="Olive">e</font> <a href="hidden.html#R1">=</a> <font color="Olive">h</font> &amp; <font color="Olive">e</font> <a href="group_1.html#K1">*</a> <font color="Olive">h</font> <a href="hidden.html#R1">=</a> <font color="Olive">h</font> &amp;  ex <font color="Olive">g</font> being  <a href="struct_0.html#NM1">Element</a> of <a href="group_1.html#G1">HGrStr</a>(# <span class="p1">(<span class="default"><a href="autalg_1.html#K1" target="_self">UAAut</a> <font color="Maroon">UA</font></span>)</span>,<span class="p1">(<span class="default"><a href="autalg_1.html#K2" target="_self">UAAutComp</a> <font color="Maroon">UA</font></span>)</span> #) st <br/>( <font color="Olive">h</font> <a href="group_1.html#K1">*</a> <font color="Olive">g</font> <a href="hidden.html#R1">=</a> <font color="Olive">e</font> &amp; <font color="Olive">g</font> <a href="group_1.html#K1">*</a> <font color="Olive">h</font> <a href="hidden.html#R1">=</a> <font color="Olive">e</font> ) )
  <i><font color="Red">:: according to </font></i><a class="ref" href="group_1.html#D3">GROUP_1:def 3</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1_1_2"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">e</font> =  <a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">UA</font> as   <a href="struct_0.html#NM1">Element</a> of <a href="group_1.html#G1">HGrStr</a>(# <span class="p1">(<span class="default"><a href="autalg_1.html#K1" target="_self">UAAut</a> <font color="Maroon">UA</font></span>)</span>,<span class="p1">(<span class="default"><a href="autalg_1.html#K2" target="_self">UAAutComp</a> <font color="Maroon">UA</font></span>)</span> #) <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">e</font>
;<br/>

<b>let </b><font color="Maroon">h</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="group_1.html#G1">HGrStr</a>(# <span class="p1">(<span class="default"><a href="autalg_1.html#K1" target="_self">UAAut</a> <font color="Maroon">UA</font></span>)</span>,<span class="p1">(<span class="default"><a href="autalg_1.html#K2" target="_self">UAAutComp</a> <font color="Maroon">UA</font></span>)</span> #);<br/>

<b>consider </b><font color="Maroon">A</font> being    <a href="fraenkel.html#M2">Element</a> of  <a href="autalg_1.html#K1" target="_self">UAAut</a> <font color="Maroon">UA</font><b> such that </b><br/><a NAME="E3:10_1_1_1_2"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font>
 ;<br/>
<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="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">UA</font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">A</font>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">A</font>
<b>by </b><i><a class="ref" href="funct_2.html#T23">FUNCT_2:23</a></i>
;<br/>
<b>hence </b><a NAME="E5:10_1_1_1_2"/>
<font color="Maroon">h</font> <a href="group_1.html#K1">*</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font>
 <b>by </b><i/>;<br/>

<font color="Maroon">e</font> <a href="group_1.html#K1">*</a> <font color="Maroon">h</font> = 
<font color="Maroon">A</font> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">UA</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">A</font>
<b>by </b><i><a class="ref" href="funct_2.html#T23">FUNCT_2:23</a></i>
;<br/>
<b>hence </b><a NAME="E7:10_1_1_1_2"/>
<font color="Maroon">e</font> <a href="group_1.html#K1">*</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font>
 <b>by </b><i/>;<br/>

<b>reconsider </b><font color="Maroon">g</font> = <font color="Maroon">A</font> <a href="funct_1.html#K2">"</a>  as   <a href="struct_0.html#NM1">Element</a> of <a href="group_1.html#G1">HGrStr</a>(# <span class="p1">(<span class="default"><a href="autalg_1.html#K1" target="_self">UAAut</a> <font color="Maroon">UA</font></span>)</span>,<span class="p1">(<span class="default"><a href="autalg_1.html#K2" target="_self">UAAutComp</a> <font color="Maroon">UA</font></span>)</span> #) <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">g</font>
;<br/>

<a NAME="E9:10_1_1_1_2"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">A</font> <a href="alg_1.html#R4">is_isomorphism</a> <font color="Maroon">UA</font>,<font color="Maroon">UA</font>
 
<b>by </b><i/>;<br/>
<a NAME="E10:10_1_1_1_2"/><b>then </b><i><font color="Green">E42</font></i>: 
<font color="Maroon">A</font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i><a class="ref" href="alg_1.html#T8">ALG_1:8</a></i>;<br/>
<a NAME="E11:10_1_1_1_2"/>
<font color="Maroon">A</font> <a href="alg_1.html#R3">is_epimorphism</a> <font color="Maroon">UA</font>,<font color="Maroon">UA</font>
 
<b>by </b><i>, <a class="ref" href="alg_1.html#D4">ALG_1:def 4</a></i>;<br/>
<a NAME="E12:10_1_1_1_2"/><b>then </b><i><font color="Green">E43</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">UA</font>
 
<b>by </b><i><a class="ref" href="alg_1.html#D3">ALG_1:def 3</a></i>;<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">A</font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">A</font>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">e</font>
<b>by </b><i>, , <a class="ref" href="funct_2.html#T35">FUNCT_2:35</a></i>
;<br/>

<b>thus </b><font color="Maroon">g</font> <a href="group_1.html#K1">*</a> <font color="Maroon">h</font> = 
<font color="Maroon">A</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="funct_1.html#K2">"</a> </span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">e</font>
<b>by </b><i>, , <a class="ref" href="funct_2.html#T35">FUNCT_2:35</a></i>
;<br/>


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


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:10_1_1"/>
 <a href="group_1.html#G1">HGrStr</a>(# <span class="p1">(<span class="default"><a href="autalg_1.html#K1" target="_self">UAAut</a> <font color="Maroon">UA</font></span>)</span>,<span class="p1">(<span class="default"><a href="autalg_1.html#K2" target="_self">UAAutComp</a> <font color="Maroon">UA</font></span>)</span> #) is   <a href="group_1.html#NM1">Group</a>
 ;<br/>


</div>
