<?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>set </b><font color="Maroon" title="c2">G</font> =  <a href="endalg.html#K3" title="ENDALG:func.3">UAEndMonoid</a> <font color="Maroon" title="c1">UA</font>;<br/>
<b>set </b><font color="Maroon" title="c3">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/>
<a NAME="E1:36"/>
 ex <font color="Olive" title="b1">h</font> being  <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> st <font color="Olive" title="b1">h</font> <a href="endalg.html#R3" title="ENDALG:pred.3">is_isomorphism</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1"><b>proof </b></a><div class="add">

<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <b>-&gt; </b>  <a href="relset_1.html#M2" title="RELSET_1:mode.2">M5</a>(<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">0</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">$1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>) = <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">0</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="pboole.html#K9" title="PBOOLE:func.9">--&gt;</a> $1;<br/>
<b>consider </b><font color="Maroon" title="c4">h</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E2:36_1"/><i><font color="Green" title="E28">A1</font></i>: 
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">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="c4">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> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">x</font>) ) )
 <b>from </b><i><a class="ref" href="funct_1.html#S3" title="FUNCT_1:sch.3">FUNCT_1:sch 3</a>();<br/></i>
<b>reconsider </b><font color="Maroon" title="c5">h</font> = <font color="Maroon" title="c4">h</font> as   <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#E2:36_1"><i><font color="Green" title="E28">A1</font></i></a>, <a class="ref" href="endalg.html#T18" target="_self" title="ENDALG:th.18">Th18</a></i>;<br/>
<a NAME="E4:36_1"/>
<font color="Maroon" title="c5">h</font> <a href="endalg.html#R3" title="ENDALG:pred.3">is_isomorphism</a> 
 
<b>by </b><i><a class="txt" href="endalg.html#E2:36_1"><i><font color="Green" title="E28">A1</font></i></a>, <a class="ref" href="endalg.html#T19" target="_self" title="ENDALG:th.19">Th19</a></i>;<br/>
<b>hence </b><a NAME="E5:36_1"/>
 ex <font color="Olive" title="b1">h</font> being  <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> st <font color="Olive" title="b1">h</font> <a href="endalg.html#R3" title="ENDALG:pred.3">is_isomorphism</a> 
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:36"/>
 <a href="endalg.html#K3" title="ENDALG:func.3">UAEndMonoid</a> <font color="Maroon" title="c1">UA</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> <a href="endalg.html#R4" title="ENDALG:pred.4">are_isomorphic</a> 
 <b>by </b><i><a class="ref" href="endalg.html#D12" target="_self" title="ENDALG:def.12">Def12</a></i>;<br/>


</div>
