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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="unialg_1.html#NM2">Universal_Algebra</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="endalg.html#NM1">Homomorphism</a> of <span class="p1">(<span class="default"><a href="endalg.html#K3" target="_self">UAEndMonoid</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="endalg.html#K6" target="_self">MSAEndMonoid</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K10">MSAlg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>;<br/>



<b>assume </b><a NAME="E1:35"/><i><font color="Green">E28</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="endalg.html#K1" target="_self">UAEnd</a> <font color="Maroon">c<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span> <a href="pboole.html#K9">--&gt;</a> <font color="Olive">b<sub>1</sub></font>
 ;<br/>

<b>set </b><font color="Maroon">c<sub>3</sub></font> =  <a href="endalg.html#K3" target="_self">UAEndMonoid</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>4</sub></font> =  <a href="endalg.html#K6" target="_self">MSAEndMonoid</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K10">MSAlg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> =  <a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> as    <a href="fraenkel.html#M2">Element</a> of  <a href="endalg.html#K1" target="_self">UAEnd</a> <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="endalg.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E3:35"/><i><font color="Green">E29</font></i>: 
 <a href="endalg.html#K3" target="_self">UAEndMonoid</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="vectsp_1.html#G1">multLoopStr</a>(# <span class="p1">(<span class="default"><a href="endalg.html#K1" target="_self">UAEnd</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="endalg.html#K2" target="_self">UAEndComp</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font> #)
 
<b>by </b><i><a class="ref" href="endalg.html#D3" target="_self">Def3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> =  <a href="msualg_3.html#K2">id</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msualg_1.html#K10">MSAlg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> as    <a href="autalg_1.html#M2">Element</a> of  <a href="endalg.html#K4" target="_self">MSAEnd</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K10">MSAlg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <b>by </b><i><a class="ref" href="endalg.html#T10" target="_self">Th10</a></i>;<br/>
<a NAME="E5:35"/><i><font color="Green">E30</font></i>: 
 <a href="endalg.html#K6" target="_self">MSAEndMonoid</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K10">MSAlg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="vectsp_1.html#G1">multLoopStr</a>(# <span class="p1">(<span class="default"><a href="endalg.html#K4" target="_self">MSAEnd</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K10">MSAlg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="endalg.html#K5" target="_self">MSAEndComp</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K10">MSAlg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>6</sub></font> #)
 
<b>by </b><i><a class="ref" href="endalg.html#D6" target="_self">Def6</a></i>;<br/>
<b>thus </b><a NAME="E6:35"/>
<font color="Maroon">c<sub>2</sub></font> <a href="endalg.html#R3" target="_self">is_isomorphism</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:35_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="endalg.html#R2" target="_self">is_epimorphism</a> 
  <i><font color="Red">:: according to </font></i><a class="ref" href="endalg.html#D11" target="_self">ENDALG:def 11</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:35_1_1"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="endalg.html#K1" target="_self">UAEnd</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="endalg.html#E3:35"><i><font color="Green">E29</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>hence </b><a NAME="E2:35_1_1"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="endalg.html#K6" target="_self">MSAEndMonoid</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K10">MSAlg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="endalg.html#E1:35"><i><font color="Green">E28</font></i></a>, <a class="txt" href="endalg.html#E5:35"><i><font color="Green">E30</font></i></a>, <a class="txt" href="endalg.html#E25"><i><font color="Green">Lemma19</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="endalg.html#D10" target="_self">ENDALG:def 10</a><br/>


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

<b>thus </b><a NAME="E2:35_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="endalg.html#R1" target="_self">is_monomorphism</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:35_1_2"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
  <i><font color="Red">:: according to </font></i><a class="ref" href="endalg.html#D9" target="_self">ENDALG:def 9</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:35_1_2_1"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="endalg.html#K3" target="_self">UAEndMonoid</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> st <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>2</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="endalg.html#K3" target="_self">UAEndMonoid</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:35_1_2_1_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/>

<i><font color="Green">E32</font></i>: <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font> = 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span> <a href="pboole.html#K9">--&gt;</a> <font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="txt" href="endalg.html#E1:35"><i><font color="Green">E28</font></i></a>, <a class="txt" href="endalg.html#E3:35"><i><font color="Green">E29</font></i></a></i>
<br/>.= 
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K1">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span>
<b>by </b><i><a class="ref" href="funcop_1.html#D2">FUNCOP_1:def 2</a></i>
;<br/>
<font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font> = 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span> <a href="pboole.html#K9">--&gt;</a> <font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="txt" href="endalg.html#E1:35"><i><font color="Green">E28</font></i></a>, <a class="txt" href="endalg.html#E3:35"><i><font color="Green">E29</font></i></a></i>
<br/>.= 
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>8</sub></font></span><a href="tarski.html#K1">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span>
<b>by </b><i><a class="ref" href="funcop_1.html#D2">FUNCOP_1:def 2</a></i>
;<br/>
<a NAME="E4:35_1_2_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>8</sub></font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="txt" href="endalg.html#E1:35_1_2_1_1"><i><font color="Green">E31</font></i></a>, <a class="txt" href="endalg.html#E2:35_1_2_1_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="zfmisc_1.html#T134">ZFMISC_1:134</a></i>;<br/>
<b>hence </b><a NAME="E5:35_1_2_1_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T6">ZFMISC_1:6</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:35_1_2_1"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i><a class="ref" href="group_6.html#T1">GROUP_6:1</a></i>;<br/>


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


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


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


</div>
