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

<b>set </b><font color="Maroon" title="c2">H</font> =  <a href="algstr_0.html#G3" title="ALGSTR_0:aggr.3">multMagma</a>(# <span class="p1">(<span class="default"><a href="autgroup.html#K1" title="AUTGROUP:func.1">Aut</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p1">(<span class="default"><a href="autgroup.html#K2" title="AUTGROUP:func.2">AutComp</a> <font color="Maroon" title="c1">G</font></span>)</span> #);<br/>
<a NAME="E1:12_1_1"/>
(  <a href="algstr_0.html#G3" title="ALGSTR_0:aggr.3">multMagma</a>(# <span class="p1">(<span class="default"><a href="autgroup.html#K1" title="AUTGROUP:func.1">Aut</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p1">(<span class="default"><a href="autgroup.html#K2" title="AUTGROUP:func.2">AutComp</a> <font color="Maroon" title="c1">G</font></span>)</span> #) is <a href="group_1.html#V3" title="GROUP_1:attr.3">associative</a> &amp;  <a href="algstr_0.html#G3" title="ALGSTR_0:aggr.3">multMagma</a>(# <span class="p1">(<span class="default"><a href="autgroup.html#K1" title="AUTGROUP:func.1">Aut</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p1">(<span class="default"><a href="autgroup.html#K2" title="AUTGROUP:func.2">AutComp</a> <font color="Maroon" title="c1">G</font></span>)</span> #) is <a href="group_1.html#V2" title="GROUP_1:attr.2">Group-like</a> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:12_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="algstr_0.html#G3" title="ALGSTR_0:aggr.3">multMagma</a>(# <span class="p1">(<span class="default"><a href="autgroup.html#K1" title="AUTGROUP:func.1">Aut</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p1">(<span class="default"><a href="autgroup.html#K2" title="AUTGROUP:func.2">AutComp</a> <font color="Maroon" title="c1">G</font></span>)</span> #) holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b2">g</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</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="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">g</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</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> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="algstr_0.html#G3" title="ALGSTR_0:aggr.3">multMagma</a>(# <span class="p1">(<span class="default"><a href="autgroup.html#K1" title="AUTGROUP:func.1">Aut</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p1">(<span class="default"><a href="autgroup.html#K2" title="AUTGROUP:func.2">AutComp</a> <font color="Maroon" title="c1">G</font></span>)</span> #) is <a href="group_1.html#V2" title="GROUP_1:attr.2">Group-like</a></span><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" title="c3">f</font>, <font color="Maroon" title="c4">g</font>, <font color="Maroon" title="c5">h</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="algstr_0.html#G3" title="ALGSTR_0:aggr.3">multMagma</a>(# <span class="p1">(<span class="default"><a href="autgroup.html#K1" title="AUTGROUP:func.1">Aut</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p1">(<span class="default"><a href="autgroup.html#K2" title="AUTGROUP:func.2">AutComp</a> <font color="Maroon" title="c1">G</font></span>)</span> #); <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">h</font></span>)</span></span><br/>

<b>reconsider </b><font color="Maroon" title="c6">A</font> = <font color="Maroon" title="c3">f</font>, <font color="Maroon" title="c7">B</font> = <font color="Maroon" title="c4">g</font>, <font color="Maroon" title="c8">C</font> = <font color="Maroon" title="c5">h</font> as    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="autgroup.html#K1" title="AUTGROUP:func.1">Aut</a> <font color="Maroon" title="c1">G</font> ;<br/>
<a NAME="E2:12_1_1_1_1"/><i><font color="Green" title="E11">A1</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">A</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c7">B</font>
 
<b>by </b><i><a class="ref" href="autgroup.html#D2" target="_self" title="AUTGROUP:def.2">Def2</a></i>;<br/>
<a NAME="E3:12_1_1_1_1"/><i><font color="Green" title="E12">A2</font></i>: 
<font color="Maroon" title="c4">g</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">B</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c8">C</font>
 
<b>by </b><i><a class="ref" href="autgroup.html#D2" target="_self" title="AUTGROUP:def.2">Def2</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">h</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">A</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c7">B</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c8">C</font>
<b>by </b><i><a class="txt" href="autgroup.html#E2:12_1_1_1_1"><i><font color="Green" title="E11">A1</font></i></a>, <a class="ref" href="autgroup.html#D2" target="_self" title="AUTGROUP:def.2">Def2</a></i>
<br/>.= 
<font color="Maroon" title="c6">A</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">B</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c8">C</font></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T55" title="RELAT_1:th.55">RELAT_1:55</a></i>
<br/>.= 
<font color="Maroon" title="c3">f</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">h</font></span>)</span>
<b>by </b><i><a class="txt" href="autgroup.html#E3:12_1_1_1_1"><i><font color="Green" title="E12">A2</font></i></a>, <a class="ref" href="autgroup.html#D2" target="_self" title="AUTGROUP:def.2">Def2</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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

<b>thus </b><a NAME="E2:12_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="algstr_0.html#G3" title="ALGSTR_0:aggr.3">multMagma</a>(# <span class="p1">(<span class="default"><a href="autgroup.html#K1" title="AUTGROUP:func.1">Aut</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p1">(<span class="default"><a href="autgroup.html#K2" title="AUTGROUP:func.2">AutComp</a> <font color="Maroon" title="c1">G</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="algstr_0.html#G3" title="ALGSTR_0:aggr.3">multMagma</a>(# <span class="p1">(<span class="default"><a href="autgroup.html#K1" title="AUTGROUP:func.1">Aut</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p1">(<span class="default"><a href="autgroup.html#K2" title="AUTGROUP:func.2">AutComp</a> <font color="Maroon" title="c1">G</font></span>)</span> #) holds <br/> ( <font color="Olive" title="b2">h</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</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="algstr_0.html#K6" title="ALGSTR_0:func.6">*</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="algstr_0.html#G3" title="ALGSTR_0:aggr.3">multMagma</a>(# <span class="p1">(<span class="default"><a href="autgroup.html#K1" title="AUTGROUP:func.1">Aut</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p1">(<span class="default"><a href="autgroup.html#K2" title="AUTGROUP:func.2">AutComp</a> <font color="Maroon" title="c1">G</font></span>)</span> #) st <br/>( <font color="Olive" title="b2">h</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</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="algstr_0.html#K6" title="ALGSTR_0:func.6">*</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> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><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" title="c3">e</font> =  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="algstr_0.html#G3" title="ALGSTR_0:aggr.3">multMagma</a>(# <span class="p1">(<span class="default"><a href="autgroup.html#K1" title="AUTGROUP:func.1">Aut</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p1">(<span class="default"><a href="autgroup.html#K2" title="AUTGROUP:func.2">AutComp</a> <font color="Maroon" title="c1">G</font></span>)</span> #) <b>by </b><i><a class="ref" href="autgroup.html#T4" target="_self" title="AUTGROUP:th.4">Th4</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c3">e</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">h</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="algstr_0.html#G3" title="ALGSTR_0:aggr.3">multMagma</a>(# <span class="p1">(<span class="default"><a href="autgroup.html#K1" title="AUTGROUP:func.1">Aut</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p1">(<span class="default"><a href="autgroup.html#K2" title="AUTGROUP:func.2">AutComp</a> <font color="Maroon" title="c1">G</font></span>)</span> #) holds <br/> ( <font color="Olive" title="b1">h</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">h</font> &amp; <font color="Maroon" title="c3">e</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b1">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">h</font> &amp;  ex <font color="Olive" title="b2">g</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="algstr_0.html#G3" title="ALGSTR_0:aggr.3">multMagma</a>(# <span class="p1">(<span class="default"><a href="autgroup.html#K1" title="AUTGROUP:func.1">Aut</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p1">(<span class="default"><a href="autgroup.html#K2" title="AUTGROUP:func.2">AutComp</a> <font color="Maroon" title="c1">G</font></span>)</span> #) st <br/>( <font color="Olive" title="b1">h</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">e</font> &amp; <font color="Olive" title="b2">g</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b1">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">e</font> ) )</span><br/>

<b>let </b><font color="Maroon" title="c4">h</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="algstr_0.html#G3" title="ALGSTR_0:aggr.3">multMagma</a>(# <span class="p1">(<span class="default"><a href="autgroup.html#K1" title="AUTGROUP:func.1">Aut</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p1">(<span class="default"><a href="autgroup.html#K2" title="AUTGROUP:func.2">AutComp</a> <font color="Maroon" title="c1">G</font></span>)</span> #); <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">h</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">h</font> &amp; <font color="Maroon" title="c3">e</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">h</font> &amp;  ex <font color="Olive" title="b1">g</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="algstr_0.html#G3" title="ALGSTR_0:aggr.3">multMagma</a>(# <span class="p1">(<span class="default"><a href="autgroup.html#K1" title="AUTGROUP:func.1">Aut</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p1">(<span class="default"><a href="autgroup.html#K2" title="AUTGROUP:func.2">AutComp</a> <font color="Maroon" title="c1">G</font></span>)</span> #) st <br/>( <font color="Maroon" title="c4">h</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">e</font> &amp; <font color="Olive" title="b1">g</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">e</font> ) )</span><br/>

<b>consider </b><font color="Maroon" title="c5">A</font> being    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="autgroup.html#K1" title="AUTGROUP:func.1">Aut</a> <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E3:12_1_1_1_2"/><i><font color="Green" title="E11">A3</font></i>: 
<font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">h</font>
 ;<br/>
<font color="Maroon" title="c4">h</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">e</font> = 
<font color="Maroon" title="c5">A</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font></span>)</span>
<b>by </b><i><a class="txt" href="autgroup.html#E3:12_1_1_1_2"><i><font color="Green" title="E11">A3</font></i></a>, <a class="ref" href="autgroup.html#D2" target="_self" title="AUTGROUP:def.2">Def2</a></i>
<br/>.= 
<font color="Maroon" title="c5">A</font>
<b>by </b><i><a class="ref" href="funct_2.html#T23" title="FUNCT_2:th.23">FUNCT_2:23</a></i>
;<br/>
<b>hence </b><a NAME="E5:12_1_1_1_2"/>
<font color="Maroon" title="c4">h</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">h</font>
 <b>by </b><i><a class="txt" href="autgroup.html#E3:12_1_1_1_2"><i><font color="Green" title="E11">A3</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">e</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">h</font> &amp;  ex <font color="Olive" title="b1">g</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="algstr_0.html#G3" title="ALGSTR_0:aggr.3">multMagma</a>(# <span class="p1">(<span class="default"><a href="autgroup.html#K1" title="AUTGROUP:func.1">Aut</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p1">(<span class="default"><a href="autgroup.html#K2" title="AUTGROUP:func.2">AutComp</a> <font color="Maroon" title="c1">G</font></span>)</span> #) st <br/>( <font color="Maroon" title="c4">h</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">e</font> &amp; <font color="Olive" title="b1">g</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">e</font> ) )</span><br/>

<font color="Maroon" title="c3">e</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">h</font> = 
<span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c5">A</font>
<b>by </b><i><a class="txt" href="autgroup.html#E3:12_1_1_1_2"><i><font color="Green" title="E11">A3</font></i></a>, <a class="ref" href="autgroup.html#D2" target="_self" title="AUTGROUP:def.2">Def2</a></i>
<br/>.= 
<font color="Maroon" title="c5">A</font>
<b>by </b><i><a class="ref" href="funct_2.html#T23" title="FUNCT_2:th.23">FUNCT_2:23</a></i>
;<br/>
<b>hence </b><a NAME="E7:12_1_1_1_2"/>
<font color="Maroon" title="c3">e</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">h</font>
 <b>by </b><i><a class="txt" href="autgroup.html#E3:12_1_1_1_2"><i><font color="Green" title="E11">A3</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">g</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="algstr_0.html#G3" title="ALGSTR_0:aggr.3">multMagma</a>(# <span class="p1">(<span class="default"><a href="autgroup.html#K1" title="AUTGROUP:func.1">Aut</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p1">(<span class="default"><a href="autgroup.html#K2" title="AUTGROUP:func.2">AutComp</a> <font color="Maroon" title="c1">G</font></span>)</span> #) st <br/>( <font color="Maroon" title="c4">h</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">e</font> &amp; <font color="Olive" title="b1">g</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">e</font> )</span><br/>

<b>reconsider </b><font color="Maroon" title="c6">g</font> = <font color="Maroon" title="c5">A</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a>  as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="algstr_0.html#G3" title="ALGSTR_0:aggr.3">multMagma</a>(# <span class="p1">(<span class="default"><a href="autgroup.html#K1" title="AUTGROUP:func.1">Aut</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p1">(<span class="default"><a href="autgroup.html#K2" title="AUTGROUP:func.2">AutComp</a> <font color="Maroon" title="c1">G</font></span>)</span> #) <b>by </b><i><a class="ref" href="autgroup.html#T7" target="_self" title="AUTGROUP:th.7">Th7</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c6">g</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">h</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">e</font> &amp; <font color="Maroon" title="c6">g</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">e</font> )</span><br/>

<b>reconsider </b><font color="Maroon" title="c7">A</font> = <font color="Maroon" title="c5">A</font> as   <a href="group_6.html#NM2" title="GROUP_6:NM.2">Homomorphism</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c1">G</font> <b>by </b><i><a class="ref" href="autgroup.html#D1" target="_self" title="AUTGROUP:def.1">Def1</a></i>;<br/>
<a NAME="E10:12_1_1_1_2"/><i><font color="Green" title="E12">A4</font></i>: 
( <font color="Maroon" title="c7">A</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp; <font color="Maroon" title="c7">A</font> <a href="group_6.html#NR2" title="GROUP_6:NR.2">is_epimorphism</a>  )
 
<b>by </b><i><a class="ref" href="autgroup.html#D1" target="_self" title="AUTGROUP:def.1">Def1</a></i>;<br/>
<a NAME="E11:12_1_1_1_2"/><b>then </b><i><font color="Green" title="E13">A5</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c7">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font>
 
<b>by </b><i><a class="ref" href="group_6.html#D13" title="GROUP_6:def.13">GROUP_6:def 13</a></i>;<br/>
<b>thus </b><font color="Maroon" title="c4">h</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">g</font> = 
<font color="Maroon" title="c7">A</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span>
<b>by </b><i><a class="txt" href="autgroup.html#E3:12_1_1_1_2"><i><font color="Green" title="E11">A3</font></i></a>, <a class="ref" href="autgroup.html#D2" target="_self" title="AUTGROUP:def.2">Def2</a></i>
<br/>.= 
<font color="Maroon" title="c3">e</font>
<b>by </b><i><a class="txt" href="autgroup.html#E10:12_1_1_1_2"><i><font color="Green" title="E12">A4</font></i></a>, <a class="txt" href="autgroup.html#E11:12_1_1_1_2"><i><font color="Green" title="E13">A5</font></i></a>, <a class="ref" href="funct_2.html#T35" title="FUNCT_2:th.35">FUNCT_2:35</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">g</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">e</font></span><br/>

<b>thus </b><font color="Maroon" title="c6">g</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">h</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c7">A</font>
<b>by </b><i><a class="txt" href="autgroup.html#E3:12_1_1_1_2"><i><font color="Green" title="E11">A3</font></i></a>, <a class="ref" href="autgroup.html#D2" target="_self" title="AUTGROUP:def.2">Def2</a></i>
<br/>.= 
<font color="Maroon" title="c3">e</font>
<b>by </b><i><a class="txt" href="autgroup.html#E10:12_1_1_1_2"><i><font color="Green" title="E12">A4</font></i></a>, <a class="txt" href="autgroup.html#E11:12_1_1_1_2"><i><font color="Green" title="E13">A5</font></i></a>, <a class="ref" href="funct_2.html#T35" title="FUNCT_2:th.35">FUNCT_2:35</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:12_1_1"/>
 <a href="algstr_0.html#G3" title="ALGSTR_0:aggr.3">multMagma</a>(# <span class="p1">(<span class="default"><a href="autgroup.html#K1" title="AUTGROUP:func.1">Aut</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p1">(<span class="default"><a href="autgroup.html#K2" title="AUTGROUP:func.2">AutComp</a> <font color="Maroon" title="c1">G</font></span>)</span> #) is  <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a> <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
