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

<b>set </b><font color="Maroon">c<sub>2</sub></font> =  <a href="group_1.html#G1">HGrStr</a>(# <span class="p1">(<span class="default"><a href="autgroup.html#K1" target="_self">Aut</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="autgroup.html#K2" target="_self">AutComp</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #);<br/>
<a NAME="E1:12_1_1"/>
(  <a href="group_1.html#G1">HGrStr</a>(# <span class="p1">(<span class="default"><a href="autgroup.html#K1" target="_self">Aut</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="autgroup.html#K2" target="_self">AutComp</a> <font color="Maroon">c<sub>1</sub></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="autgroup.html#K1" target="_self">Aut</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="autgroup.html#K2" target="_self">AutComp</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #) is <a href="group_1.html#V3">Group-like</a> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:12_1_1_1"/><i><font color="Green">E11</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></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="autgroup.html#K1" target="_self">Aut</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="autgroup.html#K2" target="_self">AutComp</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #)<br/>for <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font> being   <a href="fraenkel.html#M2">Element</a> of  <a href="autgroup.html#K1" target="_self">Aut</a> <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>4</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="group_1.html#K1">*</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> <a href="partfun1.html#K1">*</a> <font color="Olive">b<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="autgroup.html#D2" target="_self">Def2</a></i>;<br/>
<b>thus </b><a NAME="E2:12_1_1_1"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></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="autgroup.html#K1" target="_self">Aut</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="autgroup.html#K2" target="_self">AutComp</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #) holds <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="group_1.html#K1">*</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="group_1.html#K1">*</a> <font color="Olive">b<sub>3</sub></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="12_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></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="autgroup.html#K1" target="_self">Aut</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="autgroup.html#K2" target="_self">AutComp</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #);<br/>

<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>5</sub></font> as    <a href="fraenkel.html#M2">Element</a> of  <a href="autgroup.html#K1" target="_self">Aut</a> <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E2:12_1_1_1_1"/><i><font color="Green">E12</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> <font color="Maroon">c<sub>6</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="autgroup.html#E1:12_1_1_1"><i><font color="Green">E11</font></i></a></i>;<br/>
<a NAME="E3:12_1_1_1_1"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="autgroup.html#E1:12_1_1_1"><i><font color="Green">E11</font></i></a></i>;<br/>
<b>thus </b><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="group_1.html#K1">*</a> <font color="Maroon">c<sub>5</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="txt" href="autgroup.html#E1:12_1_1_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="autgroup.html#E2:12_1_1_1_1"><i><font color="Green">E12</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>6</sub></font> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T55">RELAT_1:55</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <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#K1">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="autgroup.html#E1:12_1_1_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="autgroup.html#E3:12_1_1_1_1"><i><font color="Green">E13</font></i></a></i>
;<br/>


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

<b>thus </b><a NAME="E3:12_1_1_1"/>
ex <font color="Olive">b<sub>1</sub></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="autgroup.html#K1" target="_self">Aut</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="autgroup.html#K2" target="_self">AutComp</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #) st <br/>for <font color="Olive">b<sub>2</sub></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="autgroup.html#K1" target="_self">Aut</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="autgroup.html#K2" target="_self">AutComp</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #) holds <br/>( <font color="Olive">b<sub>2</sub></font> <a href="group_1.html#K1">*</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="group_1.html#K1">*</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> &amp; ex <font color="Olive">b<sub>3</sub></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="autgroup.html#K1" target="_self">Aut</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="autgroup.html#K2" target="_self">AutComp</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #) st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="group_1.html#K1">*</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="group_1.html#K1">*</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></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="12_1_1_1_2"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">c<sub>3</sub></font> =  <a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></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="autgroup.html#K1" target="_self">Aut</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="autgroup.html#K2" target="_self">AutComp</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #) <b>by </b><i><a class="ref" href="autgroup.html#T4" target="_self">Th4</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>3</sub></font>
;<br/>

<b>let </b><font color="Maroon">c<sub>4</sub></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="autgroup.html#K1" target="_self">Aut</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="autgroup.html#K2" target="_self">AutComp</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #);<br/>

<b>consider </b><font color="Maroon">c<sub>5</sub></font> being    <a href="fraenkel.html#M2">Element</a> of  <a href="autgroup.html#K1" target="_self">Aut</a> <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:12_1_1_1_2"/><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>
<font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>3</sub></font> = 
<font color="Maroon">c<sub>5</sub></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">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="autgroup.html#E1:12_1_1_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="autgroup.html#E3:12_1_1_1_2"><i><font color="Green">E12</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="ref" href="funct_2.html#T23">FUNCT_2:23</a></i>
;<br/>
<b>hence </b><a NAME="E5:12_1_1_1_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="autgroup.html#E3:12_1_1_1_2"><i><font color="Green">E12</font></i></a></i>;<br/>

<font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>4</sub></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">c<sub>1</sub></font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="autgroup.html#E1:12_1_1_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="autgroup.html#E3:12_1_1_1_2"><i><font color="Green">E12</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="ref" href="funct_2.html#T23">FUNCT_2:23</a></i>
;<br/>
<b>hence </b><a NAME="E7:12_1_1_1_2"/>
<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> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="autgroup.html#E3:12_1_1_1_2"><i><font color="Green">E12</font></i></a></i>;<br/>

<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></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="autgroup.html#K1" target="_self">Aut</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="autgroup.html#K2" target="_self">AutComp</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #) <b>by </b><i><a class="ref" href="autgroup.html#T7" target="_self">Th7</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>6</sub></font>
;<br/>

<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>5</sub></font> as   <a href="group_6.html#NM2">Homomorphism</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="autgroup.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E10:12_1_1_1_2"/><i><font color="Green">E13</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> is <a href="funct_1.html#V2">one-to-one</a> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="group_6.html#NR2" target="_self">is_epimorphism</a>  )
 
<b>by </b><i><a class="ref" href="autgroup.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E11:12_1_1_1_2"/><b>then </b><i><font color="Green">E14</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="group_6.html#D13">GROUP_6:def 13</a></i>;<br/>
<b>thus </b><font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>6</sub></font> = 
<font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K2">"</a> </span>)</span>
<b>by </b><i><a class="txt" href="autgroup.html#E1:12_1_1_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="autgroup.html#E3:12_1_1_1_2"><i><font color="Green">E12</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="autgroup.html#E10:12_1_1_1_2"><i><font color="Green">E13</font></i></a>, <a class="txt" href="autgroup.html#E11:12_1_1_1_2"><i><font color="Green">E14</font></i></a>, <a class="ref" href="funct_2.html#T35">FUNCT_2:35</a></i>
;<br/>

<b>thus </b><font color="Maroon">c<sub>6</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="txt" href="autgroup.html#E1:12_1_1_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="autgroup.html#E3:12_1_1_1_2"><i><font color="Green">E12</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="autgroup.html#E10:12_1_1_1_2"><i><font color="Green">E13</font></i></a>, <a class="txt" href="autgroup.html#E11:12_1_1_1_2"><i><font color="Green">E14</font></i></a>, <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="E2:12_1_1"/>
 <a href="group_1.html#G1">HGrStr</a>(# <span class="p1">(<span class="default"><a href="autgroup.html#K1" target="_self">Aut</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="autgroup.html#K2" target="_self">AutComp</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #) is  <a href="group_1.html#V1">strict</a> <a href="group_1.html#NM1">Group</a>
 ;<br/>


</div>
