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

<b>let </b><font color="Maroon" title="c1">UA</font> be   <a href="unialg_1.html#NM2" title="UNIALG_1:NM.2">Universal_Algebra</a>;<br/><b>let </b><font color="Maroon" title="c2">h</font> be   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>;<br/>



<b>assume </b><a NAME="E1:34"/><i><font color="Green" title="E26">A1</font></i>: 
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="endalg.html#K1" title="ENDALG:func.1">UAEnd</a> <font color="Maroon" title="c1">UA</font> &amp; ( for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="endalg.html#K1" title="ENDALG:func.1">UAEnd</a> <font color="Maroon" title="c1">UA</font> holds <br/><font color="Maroon" title="c2">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="pboole.html#K9" title="PBOOLE:func.9">--&gt;</a> <font color="Olive" title="b1">x</font> ) )
 ;<br/>

<b>set </b><font color="Maroon" title="c3">G</font> =  <a href="endalg.html#K3" title="ENDALG:func.3">UAEndMonoid</a> <font color="Maroon" title="c1">UA</font>;<br/>
<b>reconsider </b><font color="Maroon" title="c4">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="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="endalg.html#K1" title="ENDALG:func.1">UAEnd</a> <font color="Maroon" title="c1">UA</font> <b>by </b><i><a class="ref" href="endalg.html#T3" target="_self" title="ENDALG:th.3">Th3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">i</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 <span class="p1">(<span class="default"><a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</a> <font color="Maroon" title="c1">UA</font></span>)</span> as    <a href="autalg_1.html#M2" title="AUTALG_1:mode.2">Element</a> of  <a href="endalg.html#K4" title="ENDALG:func.4">MSAEnd</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</a> <font color="Maroon" title="c1">UA</font></span>)</span> <b>by </b><i><a class="ref" href="endalg.html#T10" target="_self" title="ENDALG:th.10">Th10</a></i>;<br/>
<b>set </b><font color="Maroon" title="c6">H</font> =  <a href="endalg.html#K6" title="ENDALG:func.6">MSAEndMonoid</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</a> <font color="Maroon" title="c1">UA</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c7">M</font> =  <a href="vectsp_1.html#G1" title="VECTSP_1:aggr.1">multLoopStr</a>(# <span class="p1">(<span class="default"><a href="endalg.html#K4" title="ENDALG:func.4">MSAEnd</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</a> <font color="Maroon" title="c1">UA</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="endalg.html#K5" title="ENDALG:func.5">MSAEndComp</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</a> <font color="Maroon" title="c1">UA</font></span>)</span></span>)</span>,<font color="Maroon" title="c5">i</font> #);<br/>
<a NAME="E4:34"/>
 <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <a href="vectsp_1.html#G1" title="VECTSP_1:aggr.1">multLoopStr</a>(# <span class="p1">(<span class="default"><a href="endalg.html#K4" title="ENDALG:func.4">MSAEnd</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</a> <font color="Maroon" title="c1">UA</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="endalg.html#K5" title="ENDALG:func.5">MSAEndComp</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</a> <font color="Maroon" title="c1">UA</font></span>)</span></span>)</span>,<font color="Maroon" title="c5">i</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">i</font>
 
;<br/>
<a NAME="E5:34"/><b>then </b><i><font color="Green" title="E27">A2</font></i>: 
 <a href="endalg.html#K6" title="ENDALG:func.6">MSAEndMonoid</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</a> <font color="Maroon" title="c1">UA</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_1.html#G1" title="VECTSP_1:aggr.1">multLoopStr</a>(# <span class="p1">(<span class="default"><a href="endalg.html#K4" title="ENDALG:func.4">MSAEnd</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</a> <font color="Maroon" title="c1">UA</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="endalg.html#K5" title="ENDALG:func.5">MSAEndComp</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</a> <font color="Maroon" title="c1">UA</font></span>)</span></span>)</span>,<font color="Maroon" title="c5">i</font> #)
 
<b>by </b><i><a class="ref" href="endalg.html#D6" target="_self" title="ENDALG:def.6">Def6</a></i>;<br/>
<a NAME="E6:34"/><i><font color="Green" title="E28">A3</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="endalg.html#K3" title="ENDALG:func.3">UAEndMonoid</a> <font color="Maroon" title="c1">UA</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">h</font>
 
<b>by </b><i><a class="txt" href="endalg.html#E1:34"><i><font color="Green" title="E26">A1</font></i></a>, <a class="ref" href="endalg.html#D3" target="_self" title="ENDALG:def.3">Def3</a></i>;<br/>
<a NAME="E7:34"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">h</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="endalg.html#K6" title="ENDALG:func.6">MSAEndMonoid</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</a> <font color="Maroon" title="c1">UA</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="endalg.html#E1:34"><i><font color="Green" title="E26">A1</font></i></a>, <a class="txt" href="endalg.html#E5:34"><i><font color="Green" title="E27">A2</font></i></a>, <a class="txt" href="endalg.html#E25"><i><font color="Green" title="E19">Lm3</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c8">h'</font> = <font color="Maroon" title="c2">h</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="endalg.html#K3" title="ENDALG:func.3">UAEndMonoid</a> <font color="Maroon" title="c1">UA</font></span>)</span>,<span class="p1">(<span class="default"><a href="endalg.html#K6" title="ENDALG:func.6">MSAEndMonoid</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</a> <font color="Maroon" title="c1">UA</font></span>)</span></span>)</span> <b>by </b><i><a class="txt" href="endalg.html#E6:34"><i><font color="Green" title="E28">A3</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11" title="RELSET_1:th.11">RELSET_1:11</a></i>;<br/>
<a NAME="E9:34"/><i><font color="Green" title="E29">A4</font></i>: 
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="endalg.html#K3" title="ENDALG:func.3">UAEndMonoid</a> <font color="Maroon" title="c1">UA</font></span>)</span> holds  <font color="Maroon" title="c8">h'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b2">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">h'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">h'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">b</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">a</font>, <font color="Maroon" title="c10">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="endalg.html#K3" title="ENDALG:func.3">UAEndMonoid</a> <font color="Maroon" title="c1">UA</font></span>)</span>;<br/>

<b>reconsider </b><font color="Maroon" title="c11">a'</font> = <font color="Maroon" title="c9">a</font>, <font color="Maroon" title="c12">b'</font> = <font color="Maroon" title="c10">b</font> as    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="endalg.html#K1" title="ENDALG:func.1">UAEnd</a> <font color="Maroon" title="c1">UA</font> <b>by </b><i><a class="ref" href="endalg.html#D3" target="_self" title="ENDALG:def.3">Def3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c13">A</font> = <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="pboole.html#K9" title="PBOOLE:func.9">--&gt;</a> <font color="Maroon" title="c11">a'</font>, <font color="Maroon" title="c14">B</font> = <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="pboole.html#K9" title="PBOOLE:func.9">--&gt;</a> <font color="Maroon" title="c12">b'</font> as   <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <span class="p1">(<span class="default"><a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</a> <font color="Maroon" title="c1">UA</font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</a> <font color="Maroon" title="c1">UA</font></span>)</span> <b>by </b><i><a class="ref" href="endalg.html#T16" target="_self" title="ENDALG:th.16">Th16</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c15">A'</font> = <font color="Maroon" title="c13">A</font>, <font color="Maroon" title="c16">B'</font> = <font color="Maroon" title="c14">B</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="endalg.html#K6" title="ENDALG:func.6">MSAEndMonoid</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</a> <font color="Maroon" title="c1">UA</font></span>)</span></span>)</span> <b>by </b><i><a class="txt" href="endalg.html#E5:34"><i><font color="Green" title="E27">A2</font></i></a>, <a class="ref" href="endalg.html#T12" target="_self" title="ENDALG:th.12">Th12</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c17">ha</font> = <font color="Maroon" title="c8">h'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">a</font>, <font color="Maroon" title="c18">hb</font> = <font color="Maroon" title="c8">h'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">b</font> as    <a href="autalg_1.html#M2" title="AUTALG_1:mode.2">Element</a> of  <a href="endalg.html#K4" title="ENDALG:func.4">MSAEnd</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</a> <font color="Maroon" title="c1">UA</font></span>)</span> <b>by </b><i><a class="ref" href="endalg.html#D6" target="_self" title="ENDALG:def.6">Def6</a></i>;<br/>
<a NAME="E5:34_1"/><i><font color="Green" title="E30">A5</font></i>: 
( <font color="Maroon" title="c17">ha</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">A'</font> &amp; <font color="Maroon" title="c18">hb</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">B'</font> )
 
<b>by </b><i><a class="txt" href="endalg.html#E1:34"><i><font color="Green" title="E26">A1</font></i></a></i>;<br/>
<a NAME="E6:34_1"/>
<font color="Maroon" title="c12">b'</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c11">a'</font> is    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="endalg.html#K1" title="ENDALG:func.1">UAEnd</a> <font color="Maroon" title="c1">UA</font>
 
<b>by </b><i><a class="ref" href="endalg.html#T4" target="_self" title="ENDALG:th.4">Th4</a></i>;<br/>
<a NAME="E7:34_1"/><b>then </b><i><font color="Green" title="E31">A6</font></i>: 
<font color="Maroon" title="c8">h'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">b'</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c11">a'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="pboole.html#K9" title="PBOOLE:func.9">--&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">b'</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c11">a'</font></span>)</span>
 
<b>by </b><i><a class="txt" href="endalg.html#E1:34"><i><font color="Green" title="E26">A1</font></i></a></i>;<br/>
<b>thus </b><font color="Maroon" title="c8">h'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c10">b</font></span>)</span> = 
<font color="Maroon" title="c8">h'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">b'</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c11">a'</font></span>)</span>
<b>by </b><i><a class="ref" href="endalg.html#T5" target="_self" title="ENDALG:th.5">Th5</a></i>
<br/>.= 
 <a href="msuhom_1.html#K3" title="MSUHOM_1:func.3">MSAlg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">b'</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c11">a'</font></span>)</span>
<b>by </b><i><a class="txt" href="endalg.html#E7:34_1"><i><font color="Green" title="E31">A6</font></i></a>, <a class="ref" href="msuhom_1.html#D3" title="MSUHOM_1:def.3">MSUHOM_1:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="msuhom_1.html#K3" title="MSUHOM_1:func.3">MSAlg</a> <font color="Maroon" title="c12">b'</font></span>)</span> <a href="pboole.html#K12" title="PBOOLE:func.12">**</a> <span class="p1">(<span class="default"><a href="msuhom_1.html#K3" title="MSUHOM_1:func.3">MSAlg</a> <font color="Maroon" title="c11">a'</font></span>)</span>
<b>by </b><i><a class="ref" href="msuhom_1.html#T26" title="MSUHOM_1:th.26">MSUHOM_1:26</a></i>
<br/>.= 
<font color="Maroon" title="c14">B</font> <a href="pboole.html#K12" title="PBOOLE:func.12">**</a> <span class="p1">(<span class="default"><a href="msuhom_1.html#K3" title="MSUHOM_1:func.3">MSAlg</a> <font color="Maroon" title="c11">a'</font></span>)</span>
<b>by </b><i><a class="ref" href="msuhom_1.html#D3" title="MSUHOM_1:def.3">MSUHOM_1:def 3</a></i>
<br/>.= 
<font color="Maroon" title="c14">B</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c13">A</font>
<b>by </b><i><a class="ref" href="msuhom_1.html#D3" title="MSUHOM_1:def.3">MSUHOM_1:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">h'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">a</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">h'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">b</font></span>)</span>
<b>by </b><i><a class="txt" href="endalg.html#E5:34_1"><i><font color="Green" title="E30">A5</font></i></a>, <a class="ref" href="endalg.html#T13" target="_self" title="ENDALG:th.13">Th13</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E10:34"/>
<font color="Maroon" title="c8">h'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <span class="p2">(<span class="default"><a href="endalg.html#K3" title="ENDALG:func.3">UAEndMonoid</a> <font color="Maroon" title="c1">UA</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <span class="p1">(<span class="default"><a href="endalg.html#K6" title="ENDALG:func.6">MSAEndMonoid</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</a> <font color="Maroon" title="c1">UA</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_2"><b>proof </b></a><div class="add">

<a NAME="E1:34_2"/><i><font color="Green" title="E30">A7</font></i>: 
<font color="Maroon" title="c8">h'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="pboole.html#K9" title="PBOOLE:func.9">--&gt;</a> <font color="Maroon" title="c4">e</font>
 
<b>by </b><i><a class="txt" href="endalg.html#E1:34"><i><font color="Green" title="E26">A1</font></i></a></i>;<br/>
<font color="Maroon" title="c8">h'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <span class="p2">(<span class="default"><a href="endalg.html#K3" title="ENDALG:func.3">UAEndMonoid</a> <font color="Maroon" title="c1">UA</font></span>)</span></span>)</span> = 
<font color="Maroon" title="c8">h'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">e</font>
<b>by </b><i><a class="ref" href="endalg.html#T6" target="_self" title="ENDALG:th.6">Th6</a></i>
<br/>.= 
 <a href="msuhom_1.html#K3" title="MSUHOM_1:func.3">MSAlg</a> <font color="Maroon" title="c4">e</font>
<b>by </b><i><a class="txt" href="endalg.html#E1:34_2"><i><font color="Green" title="E30">A7</font></i></a>, <a class="ref" href="msuhom_1.html#D3" title="MSUHOM_1:def.3">MSUHOM_1:def 3</a></i>
<br/>.= 
<font color="Maroon" title="c5">i</font>
<b>by </b><i><a class="ref" href="msuhom_1.html#T25" title="MSUHOM_1:th.25">MSUHOM_1:25</a></i>
<br/>.= 
 <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <span class="p1">(<span class="default"><a href="endalg.html#K6" title="ENDALG:func.6">MSAEndMonoid</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</a> <font color="Maroon" title="c1">UA</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="endalg.html#T14" target="_self" title="ENDALG:th.14">Th14</a></i>
;<br/>
<b>hence </b><a NAME="E3:34_2"/>
<font color="Maroon" title="c8">h'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <span class="p2">(<span class="default"><a href="endalg.html#K3" title="ENDALG:func.3">UAEndMonoid</a> <font color="Maroon" title="c1">UA</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <span class="p1">(<span class="default"><a href="endalg.html#K6" title="ENDALG:func.6">MSAEndMonoid</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</a> <font color="Maroon" title="c1">UA</font></span>)</span></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E11:34"/>
<font color="Maroon" title="c2">h</font> is   <a href="endalg.html#NM1" title="ENDALG:NM.1">Homomorphism</a> of <span class="p1">(<span class="default"><a href="endalg.html#K3" title="ENDALG:func.3">UAEndMonoid</a> <font color="Maroon" title="c1">UA</font></span>)</span>,<span class="p1">(<span class="default"><a href="endalg.html#K6" title="ENDALG:func.6">MSAEndMonoid</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</a> <font color="Maroon" title="c1">UA</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="endalg.html#E9:34"><i><font color="Green" title="E29">A4</font></i></a>, <a class="ref" href="group_1.html#D17" title="GROUP_1:def.17">GROUP_1:def 17</a>, <a class="ref" href="group_6.html#D7" title="GROUP_6:def.7">GROUP_6:def 7</a></i>;<br/>


</div>
