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

<span class="kw">set </span><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="autalg_1.html#K1" title="AUTALG_1:func.1">UAAut</a> <font color="Maroon" title="c1">UA</font></span>)</span>,<span class="p1">(<span class="default"><a href="autalg_1.html#K2" title="AUTALG_1:func.2">UAAutComp</a> <font color="Maroon" title="c1">UA</font></span>)</span> #);<br/>
<a NAME="E1:11_1_1"/><span class="lab"><font color="Green" title="E8">A1</font></span>: 
 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="autalg_1.html#K1" title="AUTALG_1:func.1">UAAut</a> <font color="Maroon" title="c1">UA</font></span>)</span>,<span class="p1">(<span class="default"><a href="autalg_1.html#K2" title="AUTALG_1:func.2">UAAutComp</a> <font color="Maroon" title="c1">UA</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="autalg_1.html#K1" title="AUTALG_1:func.1">UAAut</a> <font color="Maroon" title="c1">UA</font></span>)</span>,<span class="p1">(<span class="default"><a href="autalg_1.html#K2" title="AUTALG_1:func.2">UAAutComp</a> <font color="Maroon" title="c1">UA</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="autalg_1.html#K1" title="AUTALG_1:func.1">UAAut</a> <font color="Maroon" title="c1">UA</font></span>)</span>,<span class="p1">(<span class="default"><a href="autalg_1.html#K2" title="AUTALG_1:func.2">UAAutComp</a> <font color="Maroon" title="c1">UA</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> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">reconsider </span><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">UA</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="autalg_1.html#K1" title="AUTALG_1:func.1">UAAut</a> <font color="Maroon" title="c1">UA</font></span>)</span>,<span class="p1">(<span class="default"><a href="autalg_1.html#K2" title="AUTALG_1:func.2">UAAutComp</a> <font color="Maroon" title="c1">UA</font></span>)</span> #) <span class="kw">by</span> <span class="lab"><a class="ref" href="autalg_1.html#T4" target="_self" title="AUTALG_1:th.4">Th4</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c3">e</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="autalg_1.html#K1" title="AUTALG_1:func.1">UAAut</a> <font color="Maroon" title="c1">UA</font></span>)</span>,<span class="p1">(<span class="default"><a href="autalg_1.html#K2" title="AUTALG_1:func.2">UAAutComp</a> <font color="Maroon" title="c1">UA</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="autalg_1.html#K1" title="AUTALG_1:func.1">UAAut</a> <font color="Maroon" title="c1">UA</font></span>)</span>,<span class="p1">(<span class="default"><a href="autalg_1.html#K2" title="AUTALG_1:func.2">UAAutComp</a> <font color="Maroon" title="c1">UA</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/>

<span class="kw">let </span><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="autalg_1.html#K1" title="AUTALG_1:func.1">UAAut</a> <font color="Maroon" title="c1">UA</font></span>)</span>,<span class="p1">(<span class="default"><a href="autalg_1.html#K2" title="AUTALG_1:func.2">UAAutComp</a> <font color="Maroon" title="c1">UA</font></span>)</span> #); <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="autalg_1.html#K1" title="AUTALG_1:func.1">UAAut</a> <font color="Maroon" title="c1">UA</font></span>)</span>,<span class="p1">(<span class="default"><a href="autalg_1.html#K2" title="AUTALG_1:func.2">UAAutComp</a> <font color="Maroon" title="c1">UA</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/>

<span class="kw">consider </span><font color="Maroon" title="c5">A</font> being    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="autalg_1.html#K1" title="AUTALG_1:func.1">UAAut</a> <font color="Maroon" title="c1">UA</font><span class="kw"> such that </span><br/><a NAME="E3:11_1_1_1"/><span class="lab"><font color="Green" title="E9">A2</font></span>: 
<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> = 
<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">UA</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c5">A</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:11_1_1_1"><span class="lab"><font color="Green" title="E9">A2</font></span></a>, <a class="ref" href="autalg_1.html#D2" target="_self" title="AUTALG_1:def.2">Def2</a></span>
<br/>.= 
<font color="Maroon" title="c5">A</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T23" title="FUNCT_2:th.23">FUNCT_2:23</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E5:11_1_1_1"/>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:11_1_1_1"><span class="lab"><font color="Green" title="E9">A2</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="autalg_1.html#K1" title="AUTALG_1:func.1">UAAut</a> <font color="Maroon" title="c1">UA</font></span>)</span>,<span class="p1">(<span class="default"><a href="autalg_1.html#K2" title="AUTALG_1:func.2">UAAutComp</a> <font color="Maroon" title="c1">UA</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> = 
<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">UA</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:11_1_1_1"><span class="lab"><font color="Green" title="E9">A2</font></span></a>, <a class="ref" href="autalg_1.html#D2" target="_self" title="AUTALG_1:def.2">Def2</a></span>
<br/>.= 
<font color="Maroon" title="c5">A</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T23" title="FUNCT_2:th.23">FUNCT_2:23</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E7:11_1_1_1"/>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:11_1_1_1"><span class="lab"><font color="Green" title="E9">A2</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="autalg_1.html#K1" title="AUTALG_1:func.1">UAAut</a> <font color="Maroon" title="c1">UA</font></span>)</span>,<span class="p1">(<span class="default"><a href="autalg_1.html#K2" title="AUTALG_1:func.2">UAAutComp</a> <font color="Maroon" title="c1">UA</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/>

<span class="kw">reconsider </span><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="autalg_1.html#K1" title="AUTALG_1:func.1">UAAut</a> <font color="Maroon" title="c1">UA</font></span>)</span>,<span class="p1">(<span class="default"><a href="autalg_1.html#K2" title="AUTALG_1:func.2">UAAutComp</a> <font color="Maroon" title="c1">UA</font></span>)</span> #) <span class="kw">by</span> <span class="lab"><a class="ref" href="autalg_1.html#T6" target="_self" title="AUTALG_1:th.6">Th6</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c6">g</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/>

<a NAME="E9:11_1_1_1"/><span class="lab"><font color="Green" title="E10">A3</font></span>: 
<font color="Maroon" title="c5">A</font> <a href="alg_1.html#R4" title="ALG_1:pred.4">is_isomorphism</a> <font color="Maroon" title="c1">UA</font>,<font color="Maroon" title="c1">UA</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="autalg_1.html#D1" target="_self" title="AUTALG_1:def.1">Def1</a></span>;<br/>
<a NAME="E10:11_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E11">A4</font></span>: 
<font color="Maroon" title="c5">A</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="alg_1.html#T8" title="ALG_1:th.8">ALG_1:8</a></span>;<br/>
<a NAME="E11:11_1_1_1"/>
<font color="Maroon" title="c5">A</font> <a href="alg_1.html#R3" title="ALG_1:pred.3">is_epimorphism</a> <font color="Maroon" title="c1">UA</font>,<font color="Maroon" title="c1">UA</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:11_1_1_1"><span class="lab"><font color="Green" title="E10">A3</font></span></a>, <a class="ref" href="alg_1.html#D4" title="ALG_1:def.4">ALG_1:def 4</a></span>;<br/>
<a NAME="E12:11_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E12">A5</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">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">UA</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="alg_1.html#D3" title="ALG_1:def.3">ALG_1:def 3</a></span>;<br/>
<span class="kw">thus </span><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> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">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="c5">A</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:11_1_1_1"><span class="lab"><font color="Green" title="E9">A2</font></span></a>, <a class="ref" href="autalg_1.html#D2" target="_self" title="AUTALG_1:def.2">Def2</a></span>
<br/>.= 
<font color="Maroon" title="c3">e</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:11_1_1_1"><span class="lab"><font color="Green" title="E11">A4</font></span></a>, <a class="txt" href="#E12:11_1_1_1"><span class="lab"><font color="Green" title="E12">A5</font></span></a>, <a class="ref" href="funct_2.html#T35" title="FUNCT_2:th.35">FUNCT_2:35</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/>

<span class="kw">thus </span><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> = 
<font color="Maroon" title="c5">A</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:11_1_1_1"><span class="lab"><font color="Green" title="E9">A2</font></span></a>, <a class="ref" href="autalg_1.html#D2" target="_self" title="AUTALG_1:def.2">Def2</a></span>
<br/>.= 
<font color="Maroon" title="c3">e</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:11_1_1_1"><span class="lab"><font color="Green" title="E11">A4</font></span></a>, <a class="txt" href="#E12:11_1_1_1"><span class="lab"><font color="Green" title="E12">A5</font></span></a>, <a class="ref" href="funct_2.html#T35" title="FUNCT_2:th.35">FUNCT_2:35</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E2:11_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="autalg_1.html#K1" title="AUTALG_1:func.1">UAAut</a> <font color="Maroon" title="c1">UA</font></span>)</span>,<span class="p1">(<span class="default"><a href="autalg_1.html#K2" title="AUTALG_1:func.2">UAAutComp</a> <font color="Maroon" title="c1">UA</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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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="autalg_1.html#K1" title="AUTALG_1:func.1">UAAut</a> <font color="Maroon" title="c1">UA</font></span>)</span>,<span class="p1">(<span class="default"><a href="autalg_1.html#K2" title="AUTALG_1:func.2">UAAutComp</a> <font color="Maroon" title="c1">UA</font></span>)</span> #); <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/>

<span class="kw">reconsider </span><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="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="autalg_1.html#K1" title="AUTALG_1:func.1">UAAut</a> <font color="Maroon" title="c1">UA</font> ;<br/>
<a NAME="E2:11_1_1_2"/><span class="lab"><font color="Green" title="E9">A6</font></span>: 
<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="c8">C</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c7">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="autalg_1.html#D2" target="_self" title="AUTALG_1:def.2">Def2</a></span>;<br/>
<a NAME="E3:11_1_1_2"/>
<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="c7">B</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c6">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="autalg_1.html#D2" target="_self" title="AUTALG_1:def.2">Def2</a></span>;<br/>
<span class="kw">hence </span><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> = 
<font color="Maroon" title="c8">C</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="c6">A</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="autalg_1.html#D2" target="_self" title="AUTALG_1:def.2">Def2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">C</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="c6">A</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T55" title="RELAT_1:th.55">RELAT_1:55</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:11_1_1_2"><span class="lab"><font color="Green" title="E9">A6</font></span></a>, <a class="ref" href="autalg_1.html#D2" target="_self" title="AUTALG_1:def.2">Def2</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:11_1_1"/>
 <a href="algstr_0.html#G3" title="ALGSTR_0:aggr.3">multMagma</a>(# <span class="p1">(<span class="default"><a href="autalg_1.html#K1" title="AUTALG_1:func.1">UAAut</a> <font color="Maroon" title="c1">UA</font></span>)</span>,<span class="p1">(<span class="default"><a href="autalg_1.html#K2" title="AUTALG_1:func.2">UAAutComp</a> <font color="Maroon" title="c1">UA</font></span>)</span> #) is   <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:11_1_1"><span class="lab"><font color="Green" title="E8">A1</font></span></a>, <a class="ref" href="group_1.html#D3" title="GROUP_1:def.3">GROUP_1:def 3</a>, <a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">GROUP_1:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
