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

<b>set </b><font color="Maroon" title="c3">H</font> =  <a href="group_1.html#G1" title="GROUP_1:aggr.1">HGrStr</a>(# <span class="p1">(<span class="default"><a href="autalg_1.html#K4" title="AUTALG_1:func.4">MSAAut</a> <font color="Maroon" title="c2">U1</font></span>)</span>,<span class="p1">(<span class="default"><a href="autalg_1.html#K5" title="AUTALG_1:func.5">MSAAutComp</a> <font color="Maroon" title="c2">U1</font></span>)</span> #);<br/>
<b>set </b><font color="Maroon" title="c4">SO</font> = the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">U1</font>;<br/>
<a NAME="E1:42_1_1"/>
(  <a href="group_1.html#G1" title="GROUP_1:aggr.1">HGrStr</a>(# <span class="p1">(<span class="default"><a href="autalg_1.html#K4" title="AUTALG_1:func.4">MSAAut</a> <font color="Maroon" title="c2">U1</font></span>)</span>,<span class="p1">(<span class="default"><a href="autalg_1.html#K5" title="AUTALG_1:func.5">MSAAutComp</a> <font color="Maroon" title="c2">U1</font></span>)</span> #) is <a href="group_1.html#V4" title="GROUP_1:attr.4">associative</a> &amp;  <a href="group_1.html#G1" title="GROUP_1:aggr.1">HGrStr</a>(# <span class="p1">(<span class="default"><a href="autalg_1.html#K4" title="AUTALG_1:func.4">MSAAut</a> <font color="Maroon" title="c2">U1</font></span>)</span>,<span class="p1">(<span class="default"><a href="autalg_1.html#K5" title="AUTALG_1:func.5">MSAAutComp</a> <font color="Maroon" title="c2">U1</font></span>)</span> #) is <a href="group_1.html#V3" title="GROUP_1:attr.3">Group-like</a> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:42_1_1_1"/>
for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font>, <font color="Olive" title="b3">h</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="group_1.html#G1" title="GROUP_1:aggr.1">HGrStr</a>(# <span class="p1">(<span class="default"><a href="autalg_1.html#K4" title="AUTALG_1:func.4">MSAAut</a> <font color="Maroon" title="c2">U1</font></span>)</span>,<span class="p1">(<span class="default"><a href="autalg_1.html#K5" title="AUTALG_1:func.5">MSAAutComp</a> <font color="Maroon" title="c2">U1</font></span>)</span> #) holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b2">g</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b3">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">g</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b3">h</font></span>)</span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">GROUP_1:def 4</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">f</font>, <font color="Maroon" title="c6">g</font>, <font color="Maroon" title="c7">h</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="group_1.html#G1" title="GROUP_1:aggr.1">HGrStr</a>(# <span class="p1">(<span class="default"><a href="autalg_1.html#K4" title="AUTALG_1:func.4">MSAAut</a> <font color="Maroon" title="c2">U1</font></span>)</span>,<span class="p1">(<span class="default"><a href="autalg_1.html#K5" title="AUTALG_1:func.5">MSAAutComp</a> <font color="Maroon" title="c2">U1</font></span>)</span> #);<br/>

<b>reconsider </b><font color="Maroon" title="c8">A</font> = <font color="Maroon" title="c5">f</font>, <font color="Maroon" title="c9">B</font> = <font color="Maroon" title="c6">g</font>, <font color="Maroon" title="c10">C</font> = <font color="Maroon" title="c7">h</font> as    <a href="autalg_1.html#M2" title="AUTALG_1:mode.2">Element</a> of  <a href="autalg_1.html#K4" title="AUTALG_1:func.4">MSAAut</a> <font color="Maroon" title="c2">U1</font> ;<br/>
<a NAME="E2:42_1_1_1_1"/><i><font color="Green" title="E27">A1</font></i>: 
<font color="Maroon" title="c5">f</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">B</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c8">A</font>
 
<b>by </b><i><a class="ref" href="autalg_1.html#D8" target="_self" title="AUTALG_1:def.8">Def8</a></i>;<br/>
<a NAME="E3:42_1_1_1_1"/><i><font color="Green" title="E28">A2</font></i>: 
<font color="Maroon" title="c6">g</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c7">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">C</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c9">B</font>
 
<b>by </b><i><a class="ref" href="autalg_1.html#D8" target="_self" title="AUTALG_1:def.8">Def8</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c6">g</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c7">h</font> = 
<font color="Maroon" title="c10">C</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">B</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c8">A</font></span>)</span>
<b>by </b><i><a class="txt" href="autalg_1.html#E2:42_1_1_1_1"><i><font color="Green" title="E27">A1</font></i></a>, <a class="ref" href="autalg_1.html#D8" target="_self" title="AUTALG_1:def.8">Def8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">C</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c9">B</font></span>)</span> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c8">A</font>
<b>by </b><i><a class="ref" href="autalg_1.html#T13" target="_self" title="AUTALG_1:th.13">Th13</a></i>
<br/>.= 
<font color="Maroon" title="c5">f</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c7">h</font></span>)</span>
<b>by </b><i><a class="txt" href="autalg_1.html#E3:42_1_1_1_1"><i><font color="Green" title="E28">A2</font></i></a>, <a class="ref" href="autalg_1.html#D8" target="_self" title="AUTALG_1:def.8">Def8</a></i>
;<br/>


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

<b>thus </b><a NAME="E2:42_1_1_1"/>
 ex <font color="Olive" title="b1">e</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="group_1.html#G1" title="GROUP_1:aggr.1">HGrStr</a>(# <span class="p1">(<span class="default"><a href="autalg_1.html#K4" title="AUTALG_1:func.4">MSAAut</a> <font color="Maroon" title="c2">U1</font></span>)</span>,<span class="p1">(<span class="default"><a href="autalg_1.html#K5" title="AUTALG_1:func.5">MSAAutComp</a> <font color="Maroon" title="c2">U1</font></span>)</span> #) st <br/>for <font color="Olive" title="b2">h</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="group_1.html#G1" title="GROUP_1:aggr.1">HGrStr</a>(# <span class="p1">(<span class="default"><a href="autalg_1.html#K4" title="AUTALG_1:func.4">MSAAut</a> <font color="Maroon" title="c2">U1</font></span>)</span>,<span class="p1">(<span class="default"><a href="autalg_1.html#K5" title="AUTALG_1:func.5">MSAAutComp</a> <font color="Maroon" title="c2">U1</font></span>)</span> #) holds <br/> ( <font color="Olive" title="b2">h</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b1">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">h</font> &amp; <font color="Olive" title="b1">e</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b2">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">h</font> &amp;  ex <font color="Olive" title="b3">g</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="group_1.html#G1" title="GROUP_1:aggr.1">HGrStr</a>(# <span class="p1">(<span class="default"><a href="autalg_1.html#K4" title="AUTALG_1:func.4">MSAAut</a> <font color="Maroon" title="c2">U1</font></span>)</span>,<span class="p1">(<span class="default"><a href="autalg_1.html#K5" title="AUTALG_1:func.5">MSAAutComp</a> <font color="Maroon" title="c2">U1</font></span>)</span> #) st <br/>( <font color="Olive" title="b2">h</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b3">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">e</font> &amp; <font color="Olive" title="b3">g</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b2">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">e</font> ) )
  <i><font color="Red">:: according to </font></i><a class="ref" href="group_1.html#D3" title="GROUP_1:def.3">GROUP_1:def 3</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_1_1_2"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon" title="c5">e</font> =  <a href="msualg_3.html#K2" title="MSUALG_3:func.2">id</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">U1</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="group_1.html#G1" title="GROUP_1:aggr.1">HGrStr</a>(# <span class="p1">(<span class="default"><a href="autalg_1.html#K4" title="AUTALG_1:func.4">MSAAut</a> <font color="Maroon" title="c2">U1</font></span>)</span>,<span class="p1">(<span class="default"><a href="autalg_1.html#K5" title="AUTALG_1:func.5">MSAAutComp</a> <font color="Maroon" title="c2">U1</font></span>)</span> #) <b>by </b><i><a class="ref" href="autalg_1.html#T31" target="_self" title="AUTALG_1:th.31">Th31</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c5">e</font>
;<br/>

<b>let </b><font color="Maroon" title="c6">h</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="group_1.html#G1" title="GROUP_1:aggr.1">HGrStr</a>(# <span class="p1">(<span class="default"><a href="autalg_1.html#K4" title="AUTALG_1:func.4">MSAAut</a> <font color="Maroon" title="c2">U1</font></span>)</span>,<span class="p1">(<span class="default"><a href="autalg_1.html#K5" title="AUTALG_1:func.5">MSAAutComp</a> <font color="Maroon" title="c2">U1</font></span>)</span> #);<br/>

<b>consider </b><font color="Maroon" title="c7">A</font> being    <a href="autalg_1.html#M2" title="AUTALG_1:mode.2">Element</a> of  <a href="autalg_1.html#K4" title="AUTALG_1:func.4">MSAAut</a> <font color="Maroon" title="c2">U1</font><b> such that </b><br/><a NAME="E3:42_1_1_1_2"/><i><font color="Green" title="E27">A3</font></i>: 
<font color="Maroon" title="c7">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">h</font>
 ;<br/>
<a NAME="E4:42_1_1_1_2"/><i><font color="Green" title="E28">A4</font></i>: 
( <font color="Maroon" title="c7">A</font> is <a href="msualg_3.html#V2" title="MSUALG_3:attr.2">"onto"</a> &amp; <font color="Maroon" title="c7">A</font> is <a href="msualg_3.html#V1" title="MSUALG_3:attr.1">"1-1"</a> )
 
<b>by </b><i><a class="txt" href="autalg_1.html#E40"><i><font color="Green" title="E21">Lm3</font></i></a></i>;<br/>
<font color="Maroon" title="c6">h</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c5">e</font> = 
<span class="p1">(<span class="default"><a href="msualg_3.html#K2" title="MSUALG_3:func.2">id</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">U1</font></span>)</span> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c7">A</font>
<b>by </b><i><a class="txt" href="autalg_1.html#E3:42_1_1_1_2"><i><font color="Green" title="E27">A3</font></i></a>, <a class="ref" href="autalg_1.html#D8" target="_self" title="AUTALG_1:def.8">Def8</a></i>
<br/>.= 
<font color="Maroon" title="c7">A</font>
<b>by </b><i><a class="ref" href="msualg_3.html#T4" title="MSUALG_3:th.4">MSUALG_3:4</a></i>
;<br/>
<b>hence </b><a NAME="E6:42_1_1_1_2"/>
<font color="Maroon" title="c6">h</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c5">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">h</font>
 <b>by </b><i><a class="txt" href="autalg_1.html#E3:42_1_1_1_2"><i><font color="Green" title="E27">A3</font></i></a></i>;<br/>

<font color="Maroon" title="c5">e</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c6">h</font> = 
<font color="Maroon" title="c7">A</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <span class="p1">(<span class="default"><a href="msualg_3.html#K2" title="MSUALG_3:func.2">id</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c2">U1</font></span>)</span>
<b>by </b><i><a class="txt" href="autalg_1.html#E3:42_1_1_1_2"><i><font color="Green" title="E27">A3</font></i></a>, <a class="ref" href="autalg_1.html#D8" target="_self" title="AUTALG_1:def.8">Def8</a></i>
<br/>.= 
<font color="Maroon" title="c7">A</font>
<b>by </b><i><a class="ref" href="msualg_3.html#T3" title="MSUALG_3:th.3">MSUALG_3:3</a></i>
;<br/>
<b>hence </b><a NAME="E8:42_1_1_1_2"/>
<font color="Maroon" title="c5">e</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c6">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">h</font>
 <b>by </b><i><a class="txt" href="autalg_1.html#E3:42_1_1_1_2"><i><font color="Green" title="E27">A3</font></i></a></i>;<br/>

<b>reconsider </b><font color="Maroon" title="c8">g</font> = <font color="Maroon" title="c7">A</font> <a href="msualg_3.html#K4" title="MSUALG_3:func.4">""</a>  as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="group_1.html#G1" title="GROUP_1:aggr.1">HGrStr</a>(# <span class="p1">(<span class="default"><a href="autalg_1.html#K4" title="AUTALG_1:func.4">MSAAut</a> <font color="Maroon" title="c2">U1</font></span>)</span>,<span class="p1">(<span class="default"><a href="autalg_1.html#K5" title="AUTALG_1:func.5">MSAAutComp</a> <font color="Maroon" title="c2">U1</font></span>)</span> #) <b>by </b><i><a class="ref" href="autalg_1.html#T32" target="_self" title="AUTALG_1:th.32">Th32</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c8">g</font>
;<br/>

<b>thus </b><font color="Maroon" title="c6">h</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c8">g</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="msualg_3.html#K4" title="MSUALG_3:func.4">""</a> </span>)</span> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c7">A</font>
<b>by </b><i><a class="txt" href="autalg_1.html#E3:42_1_1_1_2"><i><font color="Green" title="E27">A3</font></i></a>, <a class="ref" href="autalg_1.html#D8" target="_self" title="AUTALG_1:def.8">Def8</a></i>
<br/>.= 
<font color="Maroon" title="c5">e</font>
<b>by </b><i><a class="txt" href="autalg_1.html#E4:42_1_1_1_2"><i><font color="Green" title="E28">A4</font></i></a>, <a class="ref" href="msualg_3.html#T5" title="MSUALG_3:th.5">MSUALG_3:5</a></i>
;<br/>

<b>thus </b><font color="Maroon" title="c8">g</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c6">h</font> = 
<font color="Maroon" title="c7">A</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="msualg_3.html#K4" title="MSUALG_3:func.4">""</a> </span>)</span>
<b>by </b><i><a class="txt" href="autalg_1.html#E3:42_1_1_1_2"><i><font color="Green" title="E27">A3</font></i></a>, <a class="ref" href="autalg_1.html#D8" target="_self" title="AUTALG_1:def.8">Def8</a></i>
<br/>.= 
<font color="Maroon" title="c5">e</font>
<b>by </b><i><a class="txt" href="autalg_1.html#E4:42_1_1_1_2"><i><font color="Green" title="E28">A4</font></i></a>, <a class="ref" href="msualg_3.html#T5" title="MSUALG_3:th.5">MSUALG_3:5</a></i>
;<br/>


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


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:42_1_1"/>
 <a href="group_1.html#G1" title="GROUP_1:aggr.1">HGrStr</a>(# <span class="p1">(<span class="default"><a href="autalg_1.html#K4" title="AUTALG_1:func.4">MSAAut</a> <font color="Maroon" title="c2">U1</font></span>)</span>,<span class="p1">(<span class="default"><a href="autalg_1.html#K5" title="AUTALG_1:func.5">MSAAutComp</a> <font color="Maroon" title="c2">U1</font></span>)</span> #) is   <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a>
 ;<br/>


</div>
