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

<b>set </b><font color="Maroon">SF</font> = the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>();<br/>
<b>set </b><font color="Maroon">I</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>();<br/>
<b>set </b><font color="Maroon">CC</font> = <span class="p1">{<span class="default"> <font color="Olive">C</font> where <font color="Olive">C</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K6">CongrLatt</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> :  ex <font color="Olive">R</font> being  <a href="msualg_4.html#NM5">MSCongruence</a> of <font color="Maroon">F<sub>2</sub></font>() st <br/>( <font color="Olive">R</font> <a href="hidden.html#R1">=</a> <font color="Olive">C</font> &amp; <font color="Maroon">P<sub>1</sub></font>[ <a href="msualg_4.html#K14">QuotMSAlg</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Olive">R</font>] ) </span>}</span> ;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">R</font> being  <a href="msualg_4.html#NM5">MSCongruence</a> of <font color="Maroon">F<sub>2</sub></font>() st <br/>( <font color="Olive">R</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Maroon">P<sub>1</sub></font>[ <a href="msualg_4.html#K14">QuotMSAlg</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Olive">R</font>] );<br/>
<a NAME="E1:3_1"/><i><font color="Green">E50</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">C</font> where <font color="Olive">C</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K6">CongrLatt</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">C</font>] </span>}</span>  is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K6">CongrLatt</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span>
 
<b>from </b><i><a class="ref" href="domain_1.html#S7">DOMAIN_1:sch 7</a>();<br/></i>
<div><i><font color="Green">E51</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">q</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:3_1_1"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">C</font> where <font color="Olive">C</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K6">CongrLatt</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> :  ex <font color="Olive">R</font> being  <a href="msualg_4.html#NM5">MSCongruence</a> of <font color="Maroon">F<sub>2</sub></font>() st <br/>( <font color="Olive">R</font> <a href="hidden.html#R1">=</a> <font color="Olive">C</font> &amp; <font color="Maroon">P<sub>1</sub></font>[ <a href="msualg_4.html#K14">QuotMSAlg</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Olive">R</font>] ) </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">C</font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K6">CongrLatt</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span><b> such that </b><br/><a NAME="E3:3_1_1"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font>
 <b>and </b><br/><a NAME="E4:3_1_1"/>
 ex <font color="Olive">R</font> being  <a href="msualg_4.html#NM5">MSCongruence</a> of <font color="Maroon">F<sub>2</sub></font>() st <br/>( <font color="Olive">R</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font> &amp; <font color="Maroon">P<sub>1</sub></font>[ <a href="msualg_4.html#K14">QuotMSAlg</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Olive">R</font>] )
 ;<br/><b>thus </b><a NAME="E5:3_1_1"/>
<font color="Maroon">q</font> is   <a href="msualg_4.html#NM5">MSCongruence</a> of <font color="Maroon">F<sub>2</sub></font>()
 <b>by </b><i>, <a class="ref" href="msualg_5.html#D6">MSUALG_5:def 6</a></i>;<br/></div><b>end;</b></div>
<b>consider </b><font color="Maroon">EM</font> being  <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> , <font color="Maroon">EMF</font> being    <a href="pralg_2.html#M2">MSAlgebra-Family</a> of <font color="Maroon">EM</font>,<font color="Maroon">F<sub>1</sub></font>();<br/>
<a NAME="E4:3_1"/>
for <font color="Olive">i</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">EM</font> holds <br/> ex <font color="Olive">A</font> being   <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">F<sub>1</sub></font>() st <br/>( <font color="Olive">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">EMF</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">A</font>] )
 
;<br/>
<a NAME="E5:3_1"/><b>then </b><i><font color="Green">E67</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[ <a href="pralg_2.html#K15">product</a> <font color="Maroon">EMF</font>]
 
<b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">Xi</font> being    <a href="pboole.html#M1">ManySortedSet</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>()<b> such that </b><br/><a NAME="E7:3_1"/><i><font color="Green">E69</font></i>: 
<span class="p1"><a href="pzfmisc1.html#K1">{</a><span class="default"><font color="Maroon">Xi</font></span><a href="pzfmisc1.html#K1">}</a></span> <a href="pboole.html#R6">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="pre_circ.html#K2">--&gt;</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="msualg_9.html#T6">MSUALG_9:6</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:3_1_2"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>()
 ;<br/><b>then reconsider </b><font color="Maroon">s</font> = <font color="Maroon">i</font> as   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">F<sub>1</sub></font>() ;<br/><b>thus </b>the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">EMF</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> = 
 <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">EMF</font>,<font color="Maroon">s</font></span>)</span>
<b>by </b><i><a class="ref" href="pralg_2.html#D17">PRALG_2:def 17</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>
<b>by </b><i><a class="ref" href="card_3.html#T19">CARD_3:19</a>, <a class="ref" href="pralg_2.html#D16">PRALG_2:def 16</a></i>
<br/>.= 
<span class="p1">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="pre_circ.html#K2">--&gt;</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>
<b>by </b><i><a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>
<br/>.= 
<span class="p1"><a href="pzfmisc1.html#K1">{</a><span class="default"><font color="Maroon">Xi</font></span><a href="pzfmisc1.html#K1">}</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i/>
;<br/></div><b>end;</b></div>
<a NAME="E9:3_1"/><b>then </b>
the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">EMF</font></span>)</span> <a href="pboole.html#R6">=</a> <span class="p1"><a href="pzfmisc1.html#K1">{</a><span class="default"><font color="Maroon">Xi</font></span><a href="pzfmisc1.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="pboole.html#T3">PBOOLE:3</a></i>;<br/>
<a NAME="E10:3_1"/><b>then </b>
 <a href="pralg_2.html#K15">product</a> <font color="Maroon">EMF</font>, <a href="msafree2.html#K8">Trivial_Algebra</a> <font color="Maroon">F<sub>1</sub></font>() <a href="msualg_3.html#R6">are_isomorphic</a> 
 
<b>by </b><i><a class="ref" href="msualg_9.html#T27">MSUALG_9:27</a></i>;<br/>
<a NAME="E11:3_1"/><b>then </b><i><font color="Green">E70</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[ <a href="msafree2.html#K8">Trivial_Algebra</a> <font color="Maroon">F<sub>1</sub></font>()]
 
<b>by </b><i>, </i>;<br/>
<a NAME="E12:3_1"/>
<span class="p1"><a href="pboole.html#K10">[|</a><span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>(),the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>()</span><a href="pboole.html#K10">|]</a></span> is   <a href="msualg_4.html#NM5">MSCongruence</a> of <font color="Maroon">F<sub>2</sub></font>()
 
<b>by </b><i><a class="ref" href="msualg_5.html#T20">MSUALG_5:20</a></i>;<br/>
<a NAME="E13:3_1"/><b>then </b><i><font color="Green">E71</font></i>: 
<span class="p1"><a href="pboole.html#K10">[|</a><span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>(),the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>()</span><a href="pboole.html#K10">|]</a></span> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K6">CongrLatt</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span>
 
<b>by </b><i><a class="ref" href="msualg_5.html#D6">MSUALG_5:def 6</a></i>;<br/>
<b>reconsider </b><font color="Maroon">R</font> = <span class="p1"><a href="pboole.html#K10">[|</a><span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>(),the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>()</span><a href="pboole.html#K10">|]</a></span> as   <a href="msualg_4.html#NM5">MSCongruence</a> of <font color="Maroon">F<sub>2</sub></font>() <b>by </b><i><a class="ref" href="msualg_5.html#T20">MSUALG_5:20</a></i>;<br/>
<a NAME="E15:3_1"/>
the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msualg_4.html#K14">QuotMSAlg</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">R</font></span>)</span> <a href="pboole.html#R6">=</a> <span class="p1"><a href="pzfmisc1.html#K1">{</a><span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>()</span><a href="pzfmisc1.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="msualg_9.html#T30">MSUALG_9:30</a></i>;<br/>
<b>then consider </b><font color="Maroon">X</font> being  <a href="relat_1.html#V2">V5</a>  <a href="pboole.html#M1">ManySortedSet</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>()<b> such that </b><br/><a NAME="E17:3_1"/><i><font color="Green">E72</font></i>: 
the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msualg_4.html#K14">QuotMSAlg</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">R</font></span>)</span> <a href="pboole.html#R6">=</a> <span class="p1"><a href="pzfmisc1.html#K1">{</a><span class="default"><font color="Maroon">X</font></span><a href="pzfmisc1.html#K1">}</a></span>
 ;<br/>
<a NAME="E18:3_1"/>
 <a href="msualg_4.html#K14">QuotMSAlg</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">R</font>, <a href="msafree2.html#K8">Trivial_Algebra</a> <font color="Maroon">F<sub>1</sub></font>() <a href="msualg_3.html#R6">are_isomorphic</a> 
 
<b>by </b><i>, <a class="ref" href="msualg_9.html#T27">MSUALG_9:27</a></i>;<br/>
<a NAME="E19:3_1"/><b>then </b>
 <a href="msafree2.html#K8">Trivial_Algebra</a> <font color="Maroon">F<sub>1</sub></font>(), <a href="msualg_4.html#K14">QuotMSAlg</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">R</font> <a href="msualg_3.html#R6">are_isomorphic</a> 
 
<b>by </b><i><a class="ref" href="msualg_3.html#T17">MSUALG_3:17</a></i>;<br/>
<a NAME="E20:3_1"/><b>then </b>
<font color="Maroon">P<sub>1</sub></font>[ <a href="msualg_4.html#K14">QuotMSAlg</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">R</font>]
 
<b>by </b><i>, </i>;<br/>
<a NAME="E21:3_1"/><b>then </b><i><font color="Green">E73</font></i>: 
<span class="p1"><a href="pboole.html#K10">[|</a><span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>(),the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>()</span><a href="pboole.html#K10">|]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">C</font> where <font color="Olive">C</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K6">CongrLatt</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> :  ex <font color="Olive">R</font> being  <a href="msualg_4.html#NM5">MSCongruence</a> of <font color="Maroon">F<sub>2</sub></font>() st <br/>( <font color="Olive">R</font> <a href="hidden.html#R1">=</a> <font color="Olive">C</font> &amp; <font color="Maroon">P<sub>1</sub></font>[ <a href="msualg_4.html#K14">QuotMSAlg</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Olive">R</font>] ) </span>}</span> 
 
<b>by </b><i/>;<br/>
<a NAME="E22:3_1"/><i><font color="Green">E74</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">C</font> where <font color="Olive">C</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K6">CongrLatt</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> :  ex <font color="Olive">R</font> being  <a href="msualg_4.html#NM5">MSCongruence</a> of <font color="Maroon">F<sub>2</sub></font>() st <br/>( <font color="Olive">R</font> <a href="hidden.html#R1">=</a> <font color="Olive">C</font> &amp; <font color="Maroon">P<sub>1</sub></font>[ <a href="msualg_4.html#K14">QuotMSAlg</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Olive">R</font>] ) </span>}</span>  <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K5">EqRelLatt</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>()</span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">q</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:3_1_3"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">C</font> where <font color="Olive">C</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K6">CongrLatt</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> :  ex <font color="Olive">R</font> being  <a href="msualg_4.html#NM5">MSCongruence</a> of <font color="Maroon">F<sub>2</sub></font>() st <br/>( <font color="Olive">R</font> <a href="hidden.html#R1">=</a> <font color="Olive">C</font> &amp; <font color="Maroon">P<sub>1</sub></font>[ <a href="msualg_4.html#K14">QuotMSAlg</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Olive">R</font>] ) </span>}</span> 
 ;<br/>

<a NAME="E2:3_1_3"/><b>then </b>
<font color="Maroon">q</font> is   <a href="msualg_5.html#NM1">Equivalence_Relation</a> of the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>()
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:3_1_3"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K5">EqRelLatt</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>()</span>)</span>
 <b>by </b><i><a class="ref" href="msualg_5.html#D5">MSUALG_5:def 5</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">CC</font> = <span class="p1">{<span class="default"> <font color="Olive">C</font> where <font color="Olive">C</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K6">CongrLatt</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> :  ex <font color="Olive">R</font> being  <a href="msualg_4.html#NM5">MSCongruence</a> of <font color="Maroon">F<sub>2</sub></font>() st <br/>( <font color="Olive">R</font> <a href="hidden.html#R1">=</a> <font color="Olive">C</font> &amp; <font color="Maroon">P<sub>1</sub></font>[ <a href="msualg_4.html#K14">QuotMSAlg</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Olive">R</font>] ) </span>}</span>  as  non <a href="xboole_0.html#V1">empty</a> <a href="closure2.html#NM1">SubsetFamily</a> of <span class="p1"><a href="pboole.html#K10">[|</a><span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>(),the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>()</span><a href="pboole.html#K10">|]</a></span> <b>by </b><i>, <a class="ref" href="msualg_7.html#T6">MSUALG_7:6</a></i>;<br/>
<b>set </b><font color="Maroon">R0</font> =  <a href="mssubfam.html#K6">meet</a> <span class="p1"><a href="closure2.html#K5">|:</a><span class="default"><font color="Maroon">CC</font></span><a href="closure2.html#K5">:|</a></span>;<br/>
<a NAME="E24:3_1"/><i><font color="Green">E76</font></i>: 
 <a href="mssubfam.html#K6">meet</a> <span class="p1"><a href="closure2.html#K5">|:</a><span class="default"><font color="Maroon">CC</font></span><a href="closure2.html#K5">:|</a></span> is  <a href="msualg_4.html#V2">MSEquivalence_Relation-like</a> <a href="msualg_4.html#NM3">ManySortedRelation</a> of the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>()
 
<b>by </b><i>, <a class="ref" href="msualg_7.html#T9">MSUALG_7:9</a></i>;<br/>
<b>reconsider </b><font color="Maroon">R0</font> =  <a href="mssubfam.html#K6">meet</a> <span class="p1"><a href="closure2.html#K5">|:</a><span class="default"><font color="Maroon">CC</font></span><a href="closure2.html#K5">:|</a></span> as   <a href="msualg_4.html#NM4">ManySortedRelation</a> of <font color="Maroon">F<sub>2</sub></font>() <b>by </b><i>, <a class="ref" href="msualg_7.html#T9">MSUALG_7:9</a></i>;<br/>
<a NAME="E26:3_1"/>
<font color="Maroon">R0</font> is <a href="msualg_4.html#V3">MSEquivalence-like</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="msualg_4.html#D3">MSUALG_4:def 3</a>,<a class="ref" href="msualg_4.html#D5">MSUALG_4:def 5</a><br/><b>let </b><font color="Maroon">R</font> be   <a href="relset_1.html#NM3">Relation</a> of the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>;<br/>



<b>assume </b><a NAME="E1:3_1_4"/>
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon">R0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">R</font> )
 ;<br/>

<b>hence </b><a NAME="E2:3_1_4"/>
<font color="Maroon">R</font> is   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, <a class="ref" href="msualg_4.html#D3">MSUALG_4:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">R0</font> = <font color="Maroon">R0</font> as  <a href="msualg_4.html#V3">MSEquivalence-like</a> <a href="msualg_4.html#NM4">ManySortedRelation</a> of <font color="Maroon">F<sub>2</sub></font>() ;<br/>
<b>reconsider </b><font color="Maroon">R0</font> = <font color="Maroon">R0</font> as   <a href="msualg_4.html#NM5">MSCongruence</a> of <font color="Maroon">F<sub>2</sub></font>() <b>by </b><i>, <a class="ref" href="msualg_9.html#T29">MSUALG_9:29</a></i>;<br/>
<b>take </b><font color="Maroon">A</font> =  <a href="msualg_4.html#K14">QuotMSAlg</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">R0</font>;<br/>

<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">R</font> being  <a href="msualg_4.html#NM5">MSCongruence</a> of <font color="Maroon">F<sub>2</sub></font>() st <br/>( <font color="Olive">R</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="msualg_4.html#K14">QuotMSAlg</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Olive">R</font> );<br/>
<div><i><font color="Green">E77</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:3_1_5"/>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">CC</font>
 ;<br/><b>then reconsider </b><font color="Maroon">cc</font> = <font color="Maroon">c</font> as   <a href="msualg_4.html#NM5">MSCongruence</a> of <font color="Maroon">F<sub>2</sub></font>() <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">w</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:3_1_5"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">w</font> <a href="hidden.html#R1">=</a>  <a href="msualg_4.html#K14">QuotMSAlg</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">cc</font>
 ;<br/><b>take </b><font color="Maroon">w</font> = <font color="Maroon">w</font>;<br/><b>thus </b><a NAME="E5:3_1_5"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c</font>,<font color="Maroon">w</font>]
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>consider </b><font color="Maroon">FF</font> being    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">CC</font><b> such that </b><br/><a NAME="E31:3_1"/><i><font color="Green">E83</font></i>: 
for <font color="Olive">c</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">CC</font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">c</font>,<font color="Maroon">FF</font> <a href="funct_1.html#K1">.</a> <font color="Olive">c</font>]
 <b>from </b><i><a class="ref" href="pboole.html#S3">PBOOLE:sch 3</a>();<br/></i>
<a NAME="E32:3_1"/>
<font color="Maroon">FF</font> is    <a href="pralg_2.html#M2">MSAlgebra-Family</a> of <font color="Maroon">CC</font>,<font color="Maroon">F<sub>1</sub></font>()
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="pralg_2.html#D12">PRALG_2:def 12</a><br/>

<b>assume </b><a NAME="E1:3_1_6"/>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">CC</font>
 ;<br/>

<b>then consider </b><font color="Maroon">R</font> being   <a href="msualg_4.html#NM5">MSCongruence</a> of <font color="Maroon">F<sub>2</sub></font>()<b> such that </b><br/><a NAME="E3:3_1_6"/><i><font color="Green">E84</font></i>: 
( <font color="Maroon">R</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> &amp; <font color="Maroon">FF</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a>  <a href="msualg_4.html#K14">QuotMSAlg</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">R</font> )
 <b>by </b><i/>;<br/>
<b>thus </b><a NAME="E4:3_1_6"/>
<font color="Maroon">FF</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c</font> is  <a href="msualg_1.html#V5">non-empty</a>  <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">F<sub>1</sub></font>()
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">FF</font> = <font color="Maroon">FF</font> as    <a href="pralg_2.html#M2">MSAlgebra-Family</a> of <font color="Maroon">CC</font>,<font color="Maroon">F<sub>1</sub></font>() ;<br/>
<a NAME="E34:3_1"/>
for <font color="Olive">cc</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">cc</font> <a href="hidden.html#R2">in</a> <font color="Maroon">CC</font> holds <br/> ex <font color="Olive">A</font> being   <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">F<sub>1</sub></font>() st <br/>( <font color="Olive">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">FF</font> <a href="funct_1.html#K1">.</a> <font color="Olive">cc</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">A</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">cc</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:3_1_7"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">cc</font> <a href="hidden.html#R2">in</a> <font color="Maroon">CC</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c</font> = <font color="Maroon">cc</font> as    <a href="closure2.html#M1">Element</a> of <font color="Maroon">CC</font> ;<br/>
<b>take </b><font color="Maroon">V</font> = <font color="Maroon">FF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">c</font>;<br/>

<b>consider </b><font color="Maroon">C</font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K6">CongrLatt</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span><b> such that </b><br/><a NAME="E4:3_1_7"/><i><font color="Green">E87</font></i>: 
( <font color="Maroon">cc</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font> &amp;  ex <font color="Olive">R</font> being  <a href="msualg_4.html#NM5">MSCongruence</a> of <font color="Maroon">F<sub>2</sub></font>() st <br/>( <font color="Olive">R</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font> &amp; <font color="Maroon">P<sub>1</sub></font>[ <a href="msualg_4.html#K14">QuotMSAlg</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Olive">R</font>] ) )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">R</font> being   <a href="msualg_4.html#NM5">MSCongruence</a> of <font color="Maroon">F<sub>2</sub></font>()<b> such that </b><br/><a NAME="E6:3_1_7"/><i><font color="Green">E88</font></i>: 
( <font color="Maroon">R</font> <a href="hidden.html#R1">=</a> <font color="Maroon">cc</font> &amp; <font color="Maroon">FF</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">cc</font> <a href="hidden.html#R1">=</a>  <a href="msualg_4.html#K14">QuotMSAlg</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">R</font> )
 <b>by </b><i>, </i>;<br/>
<b>thus </b><a NAME="E7:3_1_7"/>
( <font color="Maroon">V</font> <a href="hidden.html#R1">=</a> <font color="Maroon">FF</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">cc</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">V</font>] )
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E35:3_1"/><b>then </b><i><font color="Green">E89</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[ <a href="pralg_2.html#K15">product</a> <font color="Maroon">FF</font>]
 
<b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">F</font> =  <a href="msualg_4.html#K16">MSNat_Hom</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">R0</font> as   <a href="msualg_3.html#NM1">ManySortedFunction</a> of <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">A</font> ;<br/>
<b>take </b>
<font color="Maroon">F</font>
;<br/>

<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="closure2.html#M1">Element</a> of <font color="Maroon">CC</font>,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">F1</font> being  <a href="msualg_3.html#NM1">ManySortedFunction</a> of <font color="Maroon">F<sub>2</sub></font>(),<span class="p1">(<span class="default"><font color="Maroon">FF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> st <br/>( <font color="Olive">F1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>2</sub></font> &amp; <font color="Olive">F1</font> <a href="msualg_3.html#R1">is_homomorphism</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">FF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">a<sub>1</sub></font> &amp;  ex <font color="Olive">R</font> being  <a href="msualg_4.html#NM5">MSCongruence</a> of <font color="Maroon">F<sub>2</sub></font>() st <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="pboole.html#R6">=</a> <font color="Olive">R</font> &amp; <font color="Olive">F1</font> <a href="pboole.html#R6">=</a>  <a href="msualg_4.html#K16">MSNat_Hom</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Olive">R</font> ) );<br/>
<a NAME="E37:3_1"/><i><font color="Green">E91</font></i>: 
for <font color="Olive">c</font> being   <a href="closure2.html#M1">Element</a> of <font color="Maroon">CC</font>  ex <font color="Olive">j</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">c</font>,<font color="Olive">j</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c</font> be    <a href="closure2.html#M1">Element</a> of <font color="Maroon">CC</font>;<br/>

<b>consider </b><font color="Maroon">R</font> being   <a href="msualg_4.html#NM5">MSCongruence</a> of <font color="Maroon">F<sub>2</sub></font>()<b> such that </b><br/><a NAME="E2:3_1_8"/><i><font color="Green">E93</font></i>: 
( <font color="Maroon">R</font> <a href="pboole.html#R6">=</a> <font color="Maroon">c</font> &amp; <font color="Maroon">FF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a>  <a href="msualg_4.html#K14">QuotMSAlg</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">R</font> )
 <b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">j</font> =  <a href="msualg_4.html#K16">MSNat_Hom</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">R</font>;<br/>
<b>take </b>
 <a href="msualg_4.html#K16">MSNat_Hom</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">R</font>
;<br/>

<b>reconsider </b><font color="Maroon">F1</font> =  <a href="msualg_4.html#K16">MSNat_Hom</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">R</font> as   <a href="msualg_3.html#NM1">ManySortedFunction</a> of <font color="Maroon">F<sub>2</sub></font>(),<span class="p1">(<span class="default"><font color="Maroon">FF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">c</font></span>)</span> <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">F1</font>
;<br/>

<b>thus </b><a NAME="E4:3_1_8"/>
<font color="Maroon">F1</font> <a href="pboole.html#R6">=</a>  <a href="msualg_4.html#K16">MSNat_Hom</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">R</font>
 ;<br/>

<a NAME="E5:3_1_8"/>
 <a href="msualg_4.html#K16">MSNat_Hom</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">R</font> <a href="msualg_3.html#R2">is_epimorphism</a> <font color="Maroon">F<sub>2</sub></font>(), <a href="msualg_4.html#K14">QuotMSAlg</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">R</font>
 
<b>by </b><i><a class="ref" href="msualg_4.html#T3">MSUALG_4:3</a></i>;<br/>
<b>hence </b><a NAME="E6:3_1_8"/>
<font color="Maroon">F1</font> <a href="msualg_3.html#R1">is_homomorphism</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">FF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">c</font>
 <b>by </b><i>, <a class="ref" href="msualg_3.html#D10">MSUALG_3:def 10</a></i>;<br/>

<b>take </b>
<font color="Maroon">R</font>
;<br/>

<b>thus </b><a NAME="E7:3_1_8"/>
( <font color="Maroon">c</font> <a href="pboole.html#R6">=</a> <font color="Maroon">R</font> &amp; <font color="Maroon">F1</font> <a href="pboole.html#R6">=</a>  <a href="msualg_4.html#K16">MSNat_Hom</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">R</font> )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">FofI1</font> being    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">CC</font><b> such that </b><br/><a NAME="E39:3_1"/><i><font color="Green">E95</font></i>: 
for <font color="Olive">c</font> being   <a href="closure2.html#M1">Element</a> of <font color="Maroon">CC</font> holds  <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">c</font>,<font color="Maroon">FofI1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">c</font>]
 <b>from </b><i><a class="ref" href="msualg_9.html#S1">MSUALG_9:sch 1</a>();<br/></i>
<a NAME="E40:3_1"/><i><font color="Green">E96</font></i>: 
for <font color="Olive">c</font> being   <a href="closure2.html#M1">Element</a> of <font color="Maroon">CC</font>  ex <font color="Olive">F1</font> being  <a href="msualg_3.html#NM1">ManySortedFunction</a> of <font color="Maroon">F<sub>2</sub></font>(),<span class="p1">(<span class="default"><font color="Maroon">FF</font> <a href="pralg_2.html#K6">.</a> <font color="Olive">c</font></span>)</span> st <br/>( <font color="Olive">F1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">FofI1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">c</font> &amp; <font color="Olive">F1</font> <a href="msualg_3.html#R1">is_homomorphism</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">FF</font> <a href="pralg_2.html#K6">.</a> <font color="Olive">c</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_9"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c</font> be    <a href="closure2.html#M1">Element</a> of <font color="Maroon">CC</font>;<br/>

<b>consider </b><font color="Maroon">F1</font> being   <a href="msualg_3.html#NM1">ManySortedFunction</a> of <font color="Maroon">F<sub>2</sub></font>(),<span class="p1">(<span class="default"><font color="Maroon">FF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">c</font></span>)</span><b> such that </b><br/><a NAME="E2:3_1_9"/><i><font color="Green">E97</font></i>: 
( <font color="Maroon">F1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">FofI1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c</font> &amp; <font color="Maroon">F1</font> <a href="msualg_3.html#R1">is_homomorphism</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">FF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">c</font> )
 <b>and </b><br/><a NAME="E3:3_1_9"/>
 ex <font color="Olive">R</font> being  <a href="msualg_4.html#NM5">MSCongruence</a> of <font color="Maroon">F<sub>2</sub></font>() st <br/>( <font color="Maroon">c</font> <a href="pboole.html#R6">=</a> <font color="Olive">R</font> &amp; <font color="Maroon">F1</font> <a href="pboole.html#R6">=</a>  <a href="msualg_4.html#K16">MSNat_Hom</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Olive">R</font> )
 <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">F1</font>
;<br/>

<b>thus </b><a NAME="E4:3_1_9"/>
( <font color="Maroon">F1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">FofI1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c</font> &amp; <font color="Maroon">F1</font> <a href="msualg_3.html#R1">is_homomorphism</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">FF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">c</font> )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E41:3_1"/>
<font color="Maroon">FofI1</font> is <a href="funcop_1.html#V1">Function-yielding</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_10"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="funcop_1.html#D6">FUNCOP_1:def 6</a><br/>

<b>assume </b><a NAME="E1:3_1_10"/>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">FofI1</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">cc</font> = <font color="Maroon">c</font> as    <a href="closure2.html#M1">Element</a> of <font color="Maroon">CC</font> <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<b>consider </b><font color="Maroon">F1</font> being   <a href="msualg_3.html#NM1">ManySortedFunction</a> of <font color="Maroon">F<sub>2</sub></font>(),<span class="p1">(<span class="default"><font color="Maroon">FF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">cc</font></span>)</span><b> such that </b><br/><a NAME="E4:3_1_10"/><i><font color="Green">E98</font></i>: 
( <font color="Maroon">F1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">FofI1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">cc</font> &amp; <font color="Maroon">F1</font> <a href="msualg_3.html#R1">is_homomorphism</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">FF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">cc</font> )
 <b>by </b><i/>;<br/>
<b>thus </b><a NAME="E5:3_1_10"/>
<font color="Maroon">FofI1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c</font> is   <a href="funct_1.html#NM1">Function</a>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">FofI1</font> = <font color="Maroon">FofI1</font> as   <a href="pboole.html#NM2">ManySortedFunction</a> of <font color="Maroon">CC</font> ;<br/>
<b>consider </b><font color="Maroon">H</font> being   <a href="msualg_3.html#NM1">ManySortedFunction</a> of <font color="Maroon">F<sub>2</sub></font>(),<span class="p1">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">FF</font></span>)</span><b> such that </b><br/><a NAME="E44:3_1"/><i><font color="Green">E99</font></i>: 
<font color="Maroon">H</font> <a href="msualg_3.html#R1">is_homomorphism</a> <font color="Maroon">F<sub>2</sub></font>(), <a href="pralg_2.html#K15">product</a> <font color="Maroon">FF</font>
 <b>and </b><br/><a NAME="E45:3_1"/><i><font color="Green">E100</font></i>: 
for <font color="Olive">c</font> being   <a href="closure2.html#M1">Element</a> of <font color="Maroon">CC</font> holds  <span class="p1">(<span class="default"><a href="pralg_3.html#K4">proj</a> <font color="Maroon">FF</font>,<font color="Olive">c</font></span>)</span> <a href="msualg_3.html#K3">**</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <font color="Maroon">FofI1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">c</font>
 <b>by </b><i>, <a class="ref" href="pralg_3.html#T30">PRALG_3:30</a></i>;<br/>
<a NAME="E46:3_1"/>
 <a href="msualg_4.html#K20">MSHomQuot</a> <font color="Maroon">H</font> <a href="msualg_3.html#R3">is_monomorphism</a>  <a href="msualg_4.html#K14">QuotMSAlg</a> <font color="Maroon">F<sub>2</sub></font>(),<span class="p1">(<span class="default"><a href="msualg_4.html#K18">MSCng</a> <font color="Maroon">H</font></span>)</span>, <a href="pralg_2.html#K15">product</a> <font color="Maroon">FF</font>
 
<b>by </b><i>, <a class="ref" href="msualg_4.html#T4">MSUALG_4:4</a></i>;<br/>
<a NAME="E47:3_1"/><b>then </b><i><font color="Green">E101</font></i>: 
 <a href="msualg_4.html#K14">QuotMSAlg</a> <font color="Maroon">F<sub>2</sub></font>(),<span class="p1">(<span class="default"><a href="msualg_4.html#K18">MSCng</a> <font color="Maroon">H</font></span>)</span>, <a href="msualg_3.html#K7">Image</a> <span class="p1">(<span class="default"><a href="msualg_4.html#K20">MSHomQuot</a> <font color="Maroon">H</font></span>)</span> <a href="msualg_3.html#R6">are_isomorphic</a> 
 
<b>by </b><i><a class="ref" href="msualg_9.html#T17">MSUALG_9:17</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_11"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:3_1_11"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>()
 ;<br/><b>then reconsider </b><font color="Maroon">s</font> = <font color="Maroon">i</font> as   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">F<sub>1</sub></font>() ;<br/><b>consider </b><font color="Maroon">Q</font> being   <a href="setfam_1.html#NM1">Subset-Family</a> of <span class="p1">(<span class="default"><span class="p2"><a href="pboole.html#K10">[|</a><span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>(),the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>()</span><a href="pboole.html#K10">|]</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font></span>)</span><b> such that </b><br/><a NAME="E4:3_1_11"/><i><font color="Green">E103</font></i>: 
<font color="Maroon">Q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="closure2.html#K5">|:</a><span class="default"><font color="Maroon">CC</font></span><a href="closure2.html#K5">:|</a></span> <a href="mssubfam.html#K4">.</a> <font color="Maroon">s</font>
 <b>and </b><br/><a NAME="E5:3_1_11"/><i><font color="Green">E104</font></i>: 
<span class="p1">(<span class="default"><a href="mssubfam.html#K6">meet</a> <span class="p2"><a href="closure2.html#K5">|:</a><span class="default"><font color="Maroon">CC</font></span><a href="closure2.html#K5">:|</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">Q</font>
 <b>by </b><i><a class="ref" href="mssubfam.html#D2">MSSUBFAM:def 2</a></i>;<br/><a NAME="E6:3_1_11"/><i><font color="Green">E105</font></i>: 
<span class="p1"><a href="closure2.html#K5">|:</a><span class="default"><font color="Maroon">CC</font></span><a href="closure2.html#K5">:|</a></span> <a href="mssubfam.html#K4">.</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">t</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font></span>)</span> where <font color="Olive">t</font> is    <a href="closure2.html#M1">Element</a> of  <a href="closure2.html#K6">Bool</a> <span class="p1"><a href="pboole.html#K10">[|</a><span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>(),the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>()</span><a href="pboole.html#K10">|]</a></span> : <font color="Olive">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">CC</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="closure2.html#T15">CLOSURE2:15</a></i>;<br/><a NAME="E7:3_1_11"/>
<span class="p1">(<span class="default"><a href="msualg_4.html#K18">MSCng</a> <font color="Maroon">H</font></span>)</span> <a href="msualg_4.html#K2">.</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">R0</font> <a href="msualg_4.html#K2">.</a> <font color="Maroon">s</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_11_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">b</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="relat_1.html#D2">RELAT_1:def 2</a><br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:3_1_11_1_1"/><i><font color="Green">E108</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="msualg_4.html#K18">MSCng</a> <font color="Maroon">H</font></span>)</span> <a href="msualg_4.html#K2">.</a> <font color="Maroon">s</font>
 ;<br/><a NAME="E2:3_1_11_1_1"/><b>then </b><i><font color="Green">E109</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E3:3_1_11_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font></span>)</span>,<span class="p2">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i/>;<br/><a NAME="E4:3_1_11_1_1"/><b>then </b><i><font color="Green">E110</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="pboole.html#K10">[|</a><span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>(),the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>()</span><a href="pboole.html#K10">|]</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>
 <b>by </b><i><a class="ref" href="pboole.html#D21">PBOOLE:def 21</a></i>;<br/><a NAME="E5:3_1_11_1_1"/><i><font color="Green">E111</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="msualg_4.html#K17">MSCng</a> <font color="Maroon">H</font>,<font color="Maroon">s</font>
 <b>by </b><i>, , <a class="ref" href="msualg_4.html#D20">MSUALG_4:def 20</a></i>;<br/><a NAME="E6:3_1_11_1_1"/>
for <font color="Olive">Y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> holds <br/><span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_11_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">Y</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:3_1_11_1_1_1"/>
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font>
 ;<br/>

<b>then consider </b><font color="Maroon">t</font> being    <a href="closure2.html#M1">Element</a> of  <a href="closure2.html#K6">Bool</a> <span class="p1"><a href="pboole.html#K10">[|</a><span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>(),the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>()</span><a href="pboole.html#K10">|]</a></span><b> such that </b><br/><a NAME="E3:3_1_11_1_1_1"/><i><font color="Green">E113</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>
 <b>and </b><br/><a NAME="E4:3_1_11_1_1_1"/><i><font color="Green">E114</font></i>: 
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">CC</font>
 <b>by </b><i>, </i>;<br/>
<b>reconsider </b><font color="Maroon">t1</font> = <font color="Maroon">t</font> as    <a href="closure2.html#M1">Element</a> of <font color="Maroon">CC</font> <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">F1</font> being   <a href="msualg_3.html#NM1">ManySortedFunction</a> of <font color="Maroon">F<sub>2</sub></font>(),<span class="p1">(<span class="default"><font color="Maroon">FF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">t1</font></span>)</span><b> such that </b><br/><a NAME="E7:3_1_11_1_1_1"/><i><font color="Green">E116</font></i>: 
( <font color="Maroon">F1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">FofI1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">t1</font> &amp; <font color="Maroon">F1</font> <a href="msualg_3.html#R1">is_homomorphism</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">FF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">t1</font> &amp;  ex <font color="Olive">R</font> being  <a href="msualg_4.html#NM5">MSCongruence</a> of <font color="Maroon">F<sub>2</sub></font>() st <br/>( <font color="Maroon">t1</font> <a href="pboole.html#R6">=</a> <font color="Olive">R</font> &amp; <font color="Maroon">F1</font> <a href="pboole.html#R6">=</a>  <a href="msualg_4.html#K16">MSNat_Hom</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Olive">R</font> ) )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">R</font> being   <a href="msualg_4.html#NM5">MSCongruence</a> of <font color="Maroon">F<sub>2</sub></font>()<b> such that </b><br/><a NAME="E9:3_1_11_1_1_1"/><i><font color="Green">E117</font></i>: 
( <font color="Maroon">t1</font> <a href="pboole.html#R6">=</a> <font color="Maroon">R</font> &amp; <font color="Maroon">F1</font> <a href="pboole.html#R6">=</a>  <a href="msualg_4.html#K16">MSNat_Hom</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">R</font> )
 <b>by </b><i/>;<br/>
<span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pralg_3.html#K4">proj</a> <font color="Maroon">FF</font>,<font color="Maroon">t1</font></span>)</span> <a href="msualg_3.html#K3">**</a> <font color="Maroon">H</font></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pralg_3.html#K4">proj</a> <font color="Maroon">FF</font>,<font color="Maroon">t1</font></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">H</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font>
<b>by </b><i><a class="ref" href="msualg_3.html#T2">MSUALG_3:2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_3.html#K4">proj</a> <font color="Maroon">FF</font>,<font color="Maroon">t1</font></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">H</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i>, <a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_3.html#K4">proj</a> <font color="Maroon">FF</font>,<font color="Maroon">t1</font></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">H</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="msualg_4.html#D19">MSUALG_4:def 19</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pralg_3.html#K4">proj</a> <font color="Maroon">FF</font>,<font color="Maroon">t1</font></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">H</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font>
<b>by </b><i>, <a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pralg_3.html#K4">proj</a> <font color="Maroon">FF</font>,<font color="Maroon">t1</font></span>)</span> <a href="msualg_3.html#K3">**</a> <font color="Maroon">H</font></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font>
<b>by </b><i><a class="ref" href="msualg_3.html#T2">MSUALG_3:2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font>
<b>by </b><i>, </i>
;<br/>
<b>hence </b><a NAME="E11:3_1_11_1_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>by </b><i>, , , <a class="ref" href="msualg_9.html#T34">MSUALG_9:34</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E7:3_1_11_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R0</font> <a href="msualg_4.html#K2">.</a> <font color="Maroon">s</font>
 <b>by </b><i>, , <a class="ref" href="setfam_1.html#T58">SETFAM_1:58</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:3_1_11_1"/><i><font color="Green">E118</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R0</font> <a href="msualg_4.html#K2">.</a> <font color="Maroon">s</font>
 ;<br/>

<a NAME="E3:3_1_11_1"/><b>then </b><i><font color="Green">E119</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E4:3_1_11_1"/>
<span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">FF</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>
 
<b>by </b><i>, <a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/>
<a NAME="E5:3_1_11_1"/><b>then </b><i><font color="Green">E120</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">FF</font>,<font color="Maroon">s</font></span>)</span>
 
<b>by </b><i><a class="ref" href="pralg_2.html#D17">PRALG_2:def 17</a></i>;<br/>
<a NAME="E6:3_1_11_1"/>
<span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">FF</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>
 
<b>by </b><i>, <a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/>
<a NAME="E7:3_1_11_1"/><b>then </b><i><font color="Green">E121</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">FF</font>,<font color="Maroon">s</font></span>)</span>
 
<b>by </b><i><a class="ref" href="pralg_2.html#D17">PRALG_2:def 17</a></i>;<br/>
<a NAME="E8:3_1_11_1"/>
for <font color="Olive">c</font> being   <a href="closure2.html#M1">Element</a> of <font color="Maroon">CC</font> holds  <span class="p1">(<span class="default"><a href="pralg_3.html#K3">proj</a> <span class="p2">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">FF</font>,<font color="Maroon">s</font></span>)</span>,<font color="Olive">c</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">H</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pralg_3.html#K3">proj</a> <span class="p2">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">FF</font>,<font color="Maroon">s</font></span>)</span>,<font color="Olive">c</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">H</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_11_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c</font> be    <a href="closure2.html#M1">Element</a> of <font color="Maroon">CC</font>;<br/>

<b>reconsider </b><font color="Maroon">t1</font> = <font color="Maroon">c</font> as   <a href="msualg_4.html#NM5">MSCongruence</a> of <font color="Maroon">F<sub>2</sub></font>() <b>by </b><i/>;<br/>
<a NAME="E2:3_1_11_1_2"/>
<font color="Maroon">t1</font> is    <a href="closure2.html#M1">Element</a> of  <a href="closure2.html#K6">Bool</a> <span class="p1"><a href="pboole.html#K10">[|</a><span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>(),the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>()</span><a href="pboole.html#K10">|]</a></span>
 
<b>by </b><i><a class="ref" href="closure2.html#D1">CLOSURE2:def 1</a></i>;<br/>
<a NAME="E3:3_1_11_1_2"/><b>then </b>
<font color="Maroon">t1</font> <a href="msualg_4.html#K2">.</a> <font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="closure2.html#K5">|:</a><span class="default"><font color="Maroon">CC</font></span><a href="closure2.html#K5">:|</a></span> <a href="mssubfam.html#K4">.</a> <font color="Maroon">s</font>
 
<b>by </b><i/>;<br/>
<a NAME="E4:3_1_11_1_2"/><b>then </b><i><font color="Green">E122</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">t1</font> <a href="msualg_4.html#K2">.</a> <font color="Maroon">s</font>
 
<b>by </b><i>, , , <a class="ref" href="setfam_1.html#T58">SETFAM_1:58</a></i>;<br/>
<b>consider </b><font color="Maroon">F1</font> being   <a href="msualg_3.html#NM1">ManySortedFunction</a> of <font color="Maroon">F<sub>2</sub></font>(),<span class="p1">(<span class="default"><font color="Maroon">FF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">c</font></span>)</span><b> such that </b><br/><a NAME="E6:3_1_11_1_2"/><i><font color="Green">E123</font></i>: 
( <font color="Maroon">F1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">FofI1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c</font> &amp; <font color="Maroon">F1</font> <a href="msualg_3.html#R1">is_homomorphism</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">FF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">c</font> &amp;  ex <font color="Olive">R</font> being  <a href="msualg_4.html#NM5">MSCongruence</a> of <font color="Maroon">F<sub>2</sub></font>() st <br/>( <font color="Maroon">c</font> <a href="pboole.html#R6">=</a> <font color="Olive">R</font> &amp; <font color="Maroon">F1</font> <a href="pboole.html#R6">=</a>  <a href="msualg_4.html#K16">MSNat_Hom</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Olive">R</font> ) )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">R</font> being   <a href="msualg_4.html#NM5">MSCongruence</a> of <font color="Maroon">F<sub>2</sub></font>()<b> such that </b><br/><a NAME="E8:3_1_11_1_2"/><i><font color="Green">E124</font></i>: 
( <font color="Maroon">t1</font> <a href="pboole.html#R6">=</a> <font color="Maroon">R</font> &amp; <font color="Maroon">F1</font> <a href="pboole.html#R6">=</a>  <a href="msualg_4.html#K16">MSNat_Hom</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">R</font> )
 <b>by </b><i/>;<br/>
<b>thus </b><span class="p1">(<span class="default"><a href="pralg_3.html#K3">proj</a> <span class="p2">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">FF</font>,<font color="Maroon">s</font></span>)</span>,<font color="Maroon">c</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">H</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_3.html#K4">proj</a> <font color="Maroon">FF</font>,<font color="Maroon">c</font></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">H</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i><a class="ref" href="pralg_3.html#D3">PRALG_3:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pralg_3.html#K4">proj</a> <font color="Maroon">FF</font>,<font color="Maroon">c</font></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">H</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font>
<b>by </b><i>, <a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pralg_3.html#K4">proj</a> <font color="Maroon">FF</font>,<font color="Maroon">c</font></span>)</span> <a href="msualg_3.html#K3">**</a> <font color="Maroon">H</font></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font>
<b>by </b><i><a class="ref" href="msualg_3.html#T2">MSUALG_3:2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font>
<b>by </b><i>, , , <a class="ref" href="msualg_9.html#T34">MSUALG_9:34</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pralg_3.html#K4">proj</a> <font color="Maroon">FF</font>,<font color="Maroon">c</font></span>)</span> <a href="msualg_3.html#K3">**</a> <font color="Maroon">H</font></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pralg_3.html#K4">proj</a> <font color="Maroon">FF</font>,<font color="Maroon">c</font></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">H</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font>
<b>by </b><i><a class="ref" href="msualg_3.html#T2">MSUALG_3:2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_3.html#K4">proj</a> <font color="Maroon">FF</font>,<font color="Maroon">c</font></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">H</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font></span>)</span>
<b>by </b><i>, <a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="pralg_3.html#K3">proj</a> <span class="p2">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">FF</font>,<font color="Maroon">s</font></span>)</span>,<font color="Maroon">c</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">H</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font></span>)</span>
<b>by </b><i><a class="ref" href="pralg_3.html#D3">PRALG_3:def 3</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E9:3_1_11_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font>
 
<b>by </b><i>, , <a class="ref" href="msualg_9.html#T15">MSUALG_9:15</a></i>;<br/>
<a NAME="E10:3_1_11_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="msualg_4.html#K17">MSCng</a> <font color="Maroon">H</font>,<font color="Maroon">s</font>
 
<b>by </b><i>, <a class="ref" href="msualg_4.html#D19">MSUALG_4:def 19</a></i>;<br/>
<b>hence </b><a NAME="E11:3_1_11_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="msualg_4.html#K18">MSCng</a> <font color="Maroon">H</font></span>)</span> <a href="msualg_4.html#K2">.</a> <font color="Maroon">s</font>
 <b>by </b><i>, <a class="ref" href="msualg_4.html#D20">MSUALG_4:def 20</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E8:3_1_11"/>
<span class="p1">(<span class="default"><a href="msualg_4.html#K18">MSCng</a> <font color="Maroon">H</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">R0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 ;<br/></div><b>end;</b></div>
<a NAME="E49:3_1"/><b>then </b>
 <a href="msualg_4.html#K18">MSCng</a> <font color="Maroon">H</font> <a href="pboole.html#R6">=</a> <font color="Maroon">R0</font>
 
<b>by </b><i><a class="ref" href="pboole.html#T3">PBOOLE:3</a></i>;<br/>
<b>then consider </b><font color="Maroon">XX</font> being  <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of  <a href="pralg_2.html#K15">product</a> <font color="Maroon">FF</font><b> such that </b><br/><a NAME="E51:3_1"/><i><font color="Green">E126</font></i>: 
<font color="Maroon">A</font>,<font color="Maroon">XX</font> <a href="msualg_3.html#R6">are_isomorphic</a> 
 <b>by </b><i/>;<br/>
<a NAME="E52:3_1"/><i><font color="Green">E127</font></i>: 
<font color="Maroon">XX</font>,<font color="Maroon">A</font> <a href="msualg_3.html#R6">are_isomorphic</a> 
 
<b>by </b><i>, <a class="ref" href="msualg_3.html#T17">MSUALG_3:17</a></i>;<br/>
<a NAME="E53:3_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">XX</font>]
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E54:3_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">A</font>]
 <b>by </b><i>, </i>;<br/>

<b>thus </b><a NAME="E55:3_1"/><i><font color="Green">E128</font></i>: 
<font color="Maroon">F</font> <a href="msualg_3.html#R2">is_epimorphism</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="msualg_4.html#T3">MSUALG_4:3</a></i>;<br/>

<b>let </b><font color="Maroon">B</font> be  <a href="msualg_1.html#V5">non-empty</a>  <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">F<sub>1</sub></font>();<br/><b>let </b><font color="Maroon">G</font> be   <a href="msualg_3.html#NM1">ManySortedFunction</a> of <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">B</font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E56:3_1"/><i><font color="Green">E129</font></i>: 
<font color="Maroon">G</font> <a href="msualg_3.html#R1">is_homomorphism</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">B</font>
 <b>and </b><br/><a NAME="E57:3_1"/><i><font color="Green">E130</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">B</font>]
 ;<br/>

<b>reconsider </b><font color="Maroon">C</font> =  <a href="msualg_3.html#K7">Image</a> <span class="p1">(<span class="default"><a href="msualg_4.html#K20">MSHomQuot</a> <font color="Maroon">G</font></span>)</span> as  <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">B</font> ;<br/>
<a NAME="E59:3_1"/>
 <a href="msualg_4.html#K20">MSHomQuot</a> <font color="Maroon">G</font> <a href="msualg_3.html#R3">is_monomorphism</a>  <a href="msualg_4.html#K14">QuotMSAlg</a> <font color="Maroon">F<sub>2</sub></font>(),<span class="p1">(<span class="default"><a href="msualg_4.html#K18">MSCng</a> <font color="Maroon">G</font></span>)</span>,<font color="Maroon">B</font>
 
<b>by </b><i>, <a class="ref" href="msualg_4.html#T4">MSUALG_4:4</a></i>;<br/>
<a NAME="E60:3_1"/><b>then </b>
 <a href="msualg_4.html#K14">QuotMSAlg</a> <font color="Maroon">F<sub>2</sub></font>(),<span class="p1">(<span class="default"><a href="msualg_4.html#K18">MSCng</a> <font color="Maroon">G</font></span>)</span>,<font color="Maroon">C</font> <a href="msualg_3.html#R6">are_isomorphic</a> 
 
<b>by </b><i><a class="ref" href="msualg_9.html#T17">MSUALG_9:17</a></i>;<br/>
<a NAME="E61:3_1"/><b>then </b><i><font color="Green">E131</font></i>: 
<font color="Maroon">C</font>, <a href="msualg_4.html#K14">QuotMSAlg</a> <font color="Maroon">F<sub>2</sub></font>(),<span class="p1">(<span class="default"><a href="msualg_4.html#K18">MSCng</a> <font color="Maroon">G</font></span>)</span> <a href="msualg_3.html#R6">are_isomorphic</a> 
 
<b>by </b><i><a class="ref" href="msualg_3.html#T17">MSUALG_3:17</a></i>;<br/>
<a NAME="E62:3_1"/><i><font color="Green">E132</font></i>: 
 <a href="msualg_4.html#K18">MSCng</a> <font color="Maroon">G</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K6">CongrLatt</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span>
 
<b>by </b><i><a class="ref" href="msualg_5.html#D6">MSUALG_5:def 6</a></i>;<br/>
<a NAME="E63:3_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">C</font>]
 
<b>by </b><i>, </i>;<br/>
<a NAME="E64:3_1"/><b>then </b>
<font color="Maroon">P<sub>1</sub></font>[ <a href="msualg_4.html#K14">QuotMSAlg</a> <font color="Maroon">F<sub>2</sub></font>(),<span class="p1">(<span class="default"><a href="msualg_4.html#K18">MSCng</a> <font color="Maroon">G</font></span>)</span>]
 
<b>by </b><i>, </i>;<br/>
<a NAME="E65:3_1"/><b>then </b>
 <a href="msualg_4.html#K18">MSCng</a> <font color="Maroon">G</font> <a href="hidden.html#R2">in</a> <font color="Maroon">CC</font>
 
<b>by </b><i/>;<br/>
<a NAME="E66:3_1"/><b>then </b>
 <a href="msualg_4.html#K18">MSCng</a> <font color="Maroon">G</font> <a href="pboole.html#R3">in</a> <span class="p1"><a href="closure2.html#K5">|:</a><span class="default"><font color="Maroon">CC</font></span><a href="closure2.html#K5">:|</a></span>
 
<b>by </b><i><a class="ref" href="closure2.html#T22">CLOSURE2:22</a></i>;<br/>
<a NAME="E67:3_1"/><b>then </b>
<font color="Maroon">R0</font> <a href="pboole.html#R2">c=</a>  <a href="msualg_4.html#K18">MSCng</a> <font color="Maroon">G</font>
 
<b>by </b><i><a class="ref" href="mssubfam.html#T43">MSSUBFAM:43</a></i>;<br/>
<b>then consider </b><font color="Maroon">H</font> being   <a href="msualg_3.html#NM1">ManySortedFunction</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font><b> such that </b><br/><a NAME="E69:3_1"/><i><font color="Green">E133</font></i>: 
( <font color="Maroon">H</font> <a href="msualg_3.html#R1">is_homomorphism</a> <font color="Maroon">A</font>,<font color="Maroon">B</font> &amp; <font color="Maroon">G</font> <a href="pboole.html#R6">=</a> <font color="Maroon">H</font> <a href="msualg_3.html#K3">**</a> <font color="Maroon">F</font> )
 <b>by </b><i>, <a class="ref" href="msualg_9.html#T37">MSUALG_9:37</a></i>;<br/>
<b>take </b>
<font color="Maroon">H</font>
;<br/>

<b>thus </b><a NAME="E70:3_1"/>
<font color="Maroon">H</font> <a href="msualg_3.html#R1">is_homomorphism</a> <font color="Maroon">A</font>,<font color="Maroon">B</font>
 <b>by </b><i/>;<br/>

<b>thus </b><a NAME="E71:3_1"/>
<font color="Maroon">G</font> <a href="pboole.html#R6">=</a> <font color="Maroon">H</font> <a href="msualg_3.html#K3">**</a> <font color="Maroon">F</font>
 <b>by </b><i/>;<br/>

<b>let </b><font color="Maroon">K</font> be   <a href="msualg_3.html#NM1">ManySortedFunction</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font>;<br/>

<b>assume </b><a NAME="E72:3_1"/><i><font color="Green">E134</font></i>: 
<font color="Maroon">K</font> <a href="msualg_3.html#K3">**</a> <font color="Maroon">F</font> <a href="pboole.html#R6">=</a> <font color="Maroon">G</font>
 ;<br/>

<a NAME="E73:3_1"/>
<font color="Maroon">F</font> is <a href="msualg_3.html#V2">"onto"</a>
 
<b>by </b><i>, <a class="ref" href="msualg_3.html#D10">MSUALG_3:def 10</a></i>;<br/>
<b>hence </b><a NAME="E74:3_1"/>
<font color="Maroon">H</font> <a href="pboole.html#R6">=</a> <font color="Maroon">K</font>
 <b>by </b><i>, , <a class="ref" href="extens_1.html#T16">EXTENS_1:16</a></i>;<br/>


</div>
