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

<b>set </b><font color="Maroon" title="c1">SF</font> = the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>();<br/>
<b>set </b><font color="Maroon" title="c2">I</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>();<br/>
<b>set </b><font color="Maroon" title="c3">CC</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">C</font> where <font color="Olive" title="b1">C</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K6" title="MSUALG_5:func.6">CongrLatt</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> :  ex <font color="Olive" title="b2">R</font> being  <a href="msualg_4.html#NM5" title="MSUALG_4:NM.5">MSCongruence</a> of <font color="Maroon">F<sub>2</sub></font>() st <br/>( <font color="Olive" title="b2">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">C</font> &amp; <font color="Maroon">P<sub>1</sub></font>[ <a href="msualg_4.html#K14" title="MSUALG_4:func.14">QuotMSAlg</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Olive" title="b2">R</font>] ) </span>}</span> ;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <font color="Olive" title="b1">R</font> being  <a href="msualg_4.html#NM5" title="MSUALG_4:NM.5">MSCongruence</a> of <font color="Maroon">F<sub>2</sub></font>() st <br/>( <font color="Olive" title="b1">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 &amp; <font color="Maroon">P<sub>1</sub></font>[ <a href="msualg_4.html#K14" title="MSUALG_4:func.14">QuotMSAlg</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Olive" title="b1">R</font>] );<br/>
<a NAME="E1:3_1"/><i><font color="Green" title="E6">A4</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive" title="b1">C</font> where <font color="Olive" title="b1">C</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K6" title="MSUALG_5:func.6">CongrLatt</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">C</font>] </span>}</span>  is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K6" title="MSUALG_5:func.6">CongrLatt</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span>
 
<b>from </b><i><a class="ref" href="domain_1.html#S7" title="DOMAIN_1:sch.7">DOMAIN_1:sch 7</a>();<br/></i>
<div><i><font color="Green" title="E7">A5</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" title="c4">q</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:3_1_1"/>
<font color="Maroon" title="c4">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">C</font> where <font color="Olive" title="b1">C</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K6" title="MSUALG_5:func.6">CongrLatt</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> :  ex <font color="Olive" title="b2">R</font> being  <a href="msualg_4.html#NM5" title="MSUALG_4:NM.5">MSCongruence</a> of <font color="Maroon">F<sub>2</sub></font>() st <br/>( <font color="Olive" title="b2">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">C</font> &amp; <font color="Maroon">P<sub>1</sub></font>[ <a href="msualg_4.html#K14" title="MSUALG_4:func.14">QuotMSAlg</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Olive" title="b2">R</font>] ) </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon" title="c5">C</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K6" title="MSUALG_5:func.6">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" title="E8">A6</font></i>: 
<font color="Maroon" title="c4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">C</font>
 <b>and </b><br/><a NAME="E4:3_1_1"/>
 ex <font color="Olive" title="b1">R</font> being  <a href="msualg_4.html#NM5" title="MSUALG_4:NM.5">MSCongruence</a> of <font color="Maroon">F<sub>2</sub></font>() st <br/>( <font color="Olive" title="b1">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">C</font> &amp; <font color="Maroon">P<sub>1</sub></font>[ <a href="msualg_4.html#K14" title="MSUALG_4:func.14">QuotMSAlg</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Olive" title="b1">R</font>] )
 ;<br/><b>thus </b><a NAME="E5:3_1_1"/>
<font color="Maroon" title="c4">q</font> is   <a href="msualg_4.html#NM5" title="MSUALG_4:NM.5">MSCongruence</a> of <font color="Maroon">F<sub>2</sub></font>()
 <b>by </b><i><a class="txt" href="birkhoff.html#E3:3_1_1"><i><font color="Green" title="E8">A6</font></i></a>, <a class="ref" href="msualg_5.html#D6" title="MSUALG_5:def.6">MSUALG_5:def 6</a></i>;<br/></div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c4">EM</font> being  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> , <font color="Maroon" title="c5">EMF</font> being    <a href="pralg_2.html#M2" title="PRALG_2:mode.2">MSAlgebra-Family</a> of <font color="Maroon" title="c4">EM</font>,<font color="Maroon">F<sub>1</sub></font>();<br/>
<a NAME="E4:3_1"/>
for <font color="Olive" title="b1">i</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">EM</font> holds <br/> ex <font color="Olive" title="b2">A</font> being   <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon">F<sub>1</sub></font>() st <br/>( <font color="Olive" title="b2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">EMF</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b2">A</font>] )
 
;<br/>
<a NAME="E5:3_1"/><b>then </b><i><font color="Green" title="E8">A7</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[ <a href="pralg_2.html#K15" title="PRALG_2:func.15">product</a> <font color="Maroon" title="c5">EMF</font>]
 
<b>by </b><i><a class="txt" href="birkhoff.html#E3:3"><i><font color="Green" title="E5">A3</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c6">Xi</font> being    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">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" title="E9">A8</font></i>: 
<span class="p1"><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">{</a><span class="default"><font color="Maroon" title="c6">Xi</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="msualg_9.html#T6" title="MSUALG_9:th.6">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" title="c7">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:3_1_2"/>
<font color="Maroon" title="c7">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>()
 ;<br/><b>then reconsider </b><font color="Maroon" title="c8">s</font> = <font color="Maroon" title="c7">i</font> as   <a href="msualg_1.html#NM1" title="MSUALG_1:NM.1">SortSymbol</a> of <font color="Maroon">F<sub>1</sub></font>() ;<br/><b>thus </b>the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K15" title="PRALG_2:func.15">product</a> <font color="Maroon" title="c5">EMF</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> = 
 <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10" title="PRALG_2:func.10">Carrier</a> <font color="Maroon" title="c5">EMF</font>,<font color="Maroon" title="c8">s</font></span>)</span>
<b>by </b><i><a class="ref" href="pralg_2.html#D17" title="PRALG_2:def.17">PRALG_2:def 17</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<b>by </b><i><a class="ref" href="card_3.html#T19" title="CARD_3:th.19">CARD_3:19</a>, <a class="ref" href="pralg_2.html#D16" title="PRALG_2:def.16">PRALG_2:def 16</a></i>
<br/>.= 
<span class="p1">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">s</font>
<b>by </b><i><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>
<br/>.= 
<span class="p1"><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">{</a><span class="default"><font color="Maroon" title="c6">Xi</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font>
<b>by </b><i><a class="txt" href="birkhoff.html#E7:3_1"><i><font color="Green" title="E9">A8</font></i></a></i>
;<br/></div><b>end;</b></div>
<a NAME="E9:3_1"/><b>then </b>
the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K15" title="PRALG_2:func.15">product</a> <font color="Maroon" title="c5">EMF</font></span>)</span> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <span class="p1"><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">{</a><span class="default"><font color="Maroon" title="c6">Xi</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="pboole.html#T3" title="PBOOLE:th.3">PBOOLE:3</a></i>;<br/>
<a NAME="E10:3_1"/><b>then </b>
 <a href="pralg_2.html#K15" title="PRALG_2:func.15">product</a> <font color="Maroon" title="c5">EMF</font>, <a href="msafree2.html#K8" title="MSAFREE2:func.8">Trivial_Algebra</a> <font color="Maroon">F<sub>1</sub></font>() <a href="msualg_3.html#R6" title="MSUALG_3:pred.6">are_isomorphic</a> 
 
<b>by </b><i><a class="ref" href="msualg_9.html#T27" title="MSUALG_9:th.27">MSUALG_9:27</a></i>;<br/>
<a NAME="E11:3_1"/><b>then </b><i><font color="Green" title="E10">A9</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[ <a href="msafree2.html#K8" title="MSAFREE2:func.8">Trivial_Algebra</a> <font color="Maroon">F<sub>1</sub></font>()]
 
<b>by </b><i><a class="txt" href="birkhoff.html#E1:3"><i><font color="Green" title="E3">A1</font></i></a>, <a class="txt" href="birkhoff.html#E5:3_1"><i><font color="Green" title="E8">A7</font></i></a></i>;<br/>
<a NAME="E12:3_1"/>
<span class="p1"><a href="pboole.html#K10" title="PBOOLE:func.10">[|</a><span class="default">the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>(),the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>()</span><a href="pboole.html#K10" title="PBOOLE:func.10">|]</a></span> is   <a href="msualg_4.html#NM5" title="MSUALG_4:NM.5">MSCongruence</a> of <font color="Maroon">F<sub>2</sub></font>()
 
<b>by </b><i><a class="ref" href="msualg_5.html#T20" title="MSUALG_5:th.20">MSUALG_5:20</a></i>;<br/>
<a NAME="E13:3_1"/><b>then </b><i><font color="Green" title="E11">A10</font></i>: 
<span class="p1"><a href="pboole.html#K10" title="PBOOLE:func.10">[|</a><span class="default">the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>(),the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>()</span><a href="pboole.html#K10" title="PBOOLE:func.10">|]</a></span> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K6" title="MSUALG_5:func.6">CongrLatt</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span>
 
<b>by </b><i><a class="ref" href="msualg_5.html#D6" title="MSUALG_5:def.6">MSUALG_5:def 6</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">R</font> = <span class="p1"><a href="pboole.html#K10" title="PBOOLE:func.10">[|</a><span class="default">the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>(),the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>()</span><a href="pboole.html#K10" title="PBOOLE:func.10">|]</a></span> as   <a href="msualg_4.html#NM5" title="MSUALG_4:NM.5">MSCongruence</a> of <font color="Maroon">F<sub>2</sub></font>() <b>by </b><i><a class="ref" href="msualg_5.html#T20" title="MSUALG_5:th.20">MSUALG_5:20</a></i>;<br/>
<a NAME="E15:3_1"/>
the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msualg_4.html#K14" title="MSUALG_4:func.14">QuotMSAlg</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon" title="c7">R</font></span>)</span> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <span class="p1"><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">{</a><span class="default">the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>()</span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="msualg_9.html#T30" title="MSUALG_9:th.30">MSUALG_9:30</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c8">X</font> being  <a href="relat_1.html#V2" title="RELAT_1:attr.2">V5</a>  <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">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" title="E12">A11</font></i>: 
the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msualg_4.html#K14" title="MSUALG_4:func.14">QuotMSAlg</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon" title="c7">R</font></span>)</span> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <span class="p1"><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">{</a><span class="default"><font color="Maroon" title="c8">X</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span>
 ;<br/>
<a NAME="E18:3_1"/>
 <a href="msualg_4.html#K14" title="MSUALG_4:func.14">QuotMSAlg</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon" title="c7">R</font>, <a href="msafree2.html#K8" title="MSAFREE2:func.8">Trivial_Algebra</a> <font color="Maroon">F<sub>1</sub></font>() <a href="msualg_3.html#R6" title="MSUALG_3:pred.6">are_isomorphic</a> 
 
<b>by </b><i><a class="txt" href="birkhoff.html#E17:3_1"><i><font color="Green" title="E12">A11</font></i></a>, <a class="ref" href="msualg_9.html#T27" title="MSUALG_9:th.27">MSUALG_9:27</a></i>;<br/>
<a NAME="E19:3_1"/><b>then </b>
 <a href="msafree2.html#K8" title="MSAFREE2:func.8">Trivial_Algebra</a> <font color="Maroon">F<sub>1</sub></font>(), <a href="msualg_4.html#K14" title="MSUALG_4:func.14">QuotMSAlg</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon" title="c7">R</font> <a href="msualg_3.html#R6" title="MSUALG_3:pred.6">are_isomorphic</a> 
 
<b>by </b><i><a class="ref" href="msualg_3.html#T17" title="MSUALG_3:th.17">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" title="MSUALG_4:func.14">QuotMSAlg</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon" title="c7">R</font>]
 
<b>by </b><i><a class="txt" href="birkhoff.html#E1:3"><i><font color="Green" title="E3">A1</font></i></a>, <a class="txt" href="birkhoff.html#E11:3_1"><i><font color="Green" title="E10">A9</font></i></a></i>;<br/>
<a NAME="E21:3_1"/><b>then </b><i><font color="Green" title="E13">A12</font></i>: 
<span class="p1"><a href="pboole.html#K10" title="PBOOLE:func.10">[|</a><span class="default">the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>(),the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>()</span><a href="pboole.html#K10" title="PBOOLE:func.10">|]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">C</font> where <font color="Olive" title="b1">C</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K6" title="MSUALG_5:func.6">CongrLatt</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> :  ex <font color="Olive" title="b2">R</font> being  <a href="msualg_4.html#NM5" title="MSUALG_4:NM.5">MSCongruence</a> of <font color="Maroon">F<sub>2</sub></font>() st <br/>( <font color="Olive" title="b2">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">C</font> &amp; <font color="Maroon">P<sub>1</sub></font>[ <a href="msualg_4.html#K14" title="MSUALG_4:func.14">QuotMSAlg</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Olive" title="b2">R</font>] ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="birkhoff.html#E13:3_1"><i><font color="Green" title="E11">A10</font></i></a></i>;<br/>
<a NAME="E22:3_1"/><i><font color="Green" title="E14">A13</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive" title="b1">C</font> where <font color="Olive" title="b1">C</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K6" title="MSUALG_5:func.6">CongrLatt</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> :  ex <font color="Olive" title="b2">R</font> being  <a href="msualg_4.html#NM5" title="MSUALG_4:NM.5">MSCongruence</a> of <font color="Maroon">F<sub>2</sub></font>() st <br/>( <font color="Olive" title="b2">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">C</font> &amp; <font color="Maroon">P<sub>1</sub></font>[ <a href="msualg_4.html#K14" title="MSUALG_4:func.14">QuotMSAlg</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Olive" title="b2">R</font>] ) </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K5" title="MSUALG_5:func.5">EqRelLatt</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">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" title="c9">q</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:3_1_3"/>
<font color="Maroon" title="c9">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">C</font> where <font color="Olive" title="b1">C</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K6" title="MSUALG_5:func.6">CongrLatt</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> :  ex <font color="Olive" title="b2">R</font> being  <a href="msualg_4.html#NM5" title="MSUALG_4:NM.5">MSCongruence</a> of <font color="Maroon">F<sub>2</sub></font>() st <br/>( <font color="Olive" title="b2">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">C</font> &amp; <font color="Maroon">P<sub>1</sub></font>[ <a href="msualg_4.html#K14" title="MSUALG_4:func.14">QuotMSAlg</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Olive" title="b2">R</font>] ) </span>}</span> 
 ;<br/>

<a NAME="E2:3_1_3"/><b>then </b>
<font color="Maroon" title="c9">q</font> is   <a href="msualg_5.html#NM1" title="MSUALG_5:NM.1">Equivalence_Relation</a> of the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>()
 
<b>by </b><i><a class="txt" href="birkhoff.html#E2:3_1"><i><font color="Green" title="E7">A5</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:3_1_3"/>
<font color="Maroon" title="c9">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K5" title="MSUALG_5:func.5">EqRelLatt</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">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" title="MSUALG_5:def.5">MSUALG_5:def 5</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c9">CC</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">C</font> where <font color="Olive" title="b1">C</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K6" title="MSUALG_5:func.6">CongrLatt</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> :  ex <font color="Olive" title="b2">R</font> being  <a href="msualg_4.html#NM5" title="MSUALG_4:NM.5">MSCongruence</a> of <font color="Maroon">F<sub>2</sub></font>() st <br/>( <font color="Olive" title="b2">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">C</font> &amp; <font color="Maroon">P<sub>1</sub></font>[ <a href="msualg_4.html#K14" title="MSUALG_4:func.14">QuotMSAlg</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Olive" title="b2">R</font>] ) </span>}</span>  as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="closure2.html#NM1" title="CLOSURE2:NM.1">SubsetFamily</a> of <span class="p1"><a href="pboole.html#K10" title="PBOOLE:func.10">[|</a><span class="default">the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>(),the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>()</span><a href="pboole.html#K10" title="PBOOLE:func.10">|]</a></span> <b>by </b><i><a class="txt" href="birkhoff.html#E21:3_1"><i><font color="Green" title="E13">A12</font></i></a>, <a class="ref" href="msualg_7.html#T6" title="MSUALG_7:th.6">MSUALG_7:6</a></i>;<br/>
<b>set </b><font color="Maroon" title="c10">R0</font> =  <a href="mssubfam.html#K6" title="MSSUBFAM:func.6">meet</a> <span class="p1"><a href="closure2.html#K5" title="CLOSURE2:func.5">|:</a><span class="default"><font color="Maroon" title="c9">CC</font></span><a href="closure2.html#K5" title="CLOSURE2:func.5">:|</a></span>;<br/>
<a NAME="E24:3_1"/><i><font color="Green" title="E15">A14</font></i>: 
 <a href="mssubfam.html#K6" title="MSSUBFAM:func.6">meet</a> <span class="p1"><a href="closure2.html#K5" title="CLOSURE2:func.5">|:</a><span class="default"><font color="Maroon" title="c9">CC</font></span><a href="closure2.html#K5" title="CLOSURE2:func.5">:|</a></span> is  <a href="msualg_4.html#V2" title="MSUALG_4:attr.2">MSEquivalence_Relation-like</a> <a href="msualg_4.html#NM3" title="MSUALG_4:NM.3">ManySortedRelation</a> of the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>()
 
<b>by </b><i><a class="txt" href="birkhoff.html#E22:3_1"><i><font color="Green" title="E14">A13</font></i></a>, <a class="ref" href="msualg_7.html#T9" title="MSUALG_7:th.9">MSUALG_7:9</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">R0</font> =  <a href="mssubfam.html#K6" title="MSSUBFAM:func.6">meet</a> <span class="p1"><a href="closure2.html#K5" title="CLOSURE2:func.5">|:</a><span class="default"><font color="Maroon" title="c9">CC</font></span><a href="closure2.html#K5" title="CLOSURE2:func.5">:|</a></span> as   <a href="msualg_4.html#NM4" title="MSUALG_4:NM.4">ManySortedRelation</a> of <font color="Maroon">F<sub>2</sub></font>() <b>by </b><i><a class="txt" href="birkhoff.html#E22:3_1"><i><font color="Green" title="E14">A13</font></i></a>, <a class="ref" href="msualg_7.html#T9" title="MSUALG_7:th.9">MSUALG_7:9</a></i>;<br/>
<a NAME="E26:3_1"/>
<font color="Maroon" title="c11">R0</font> is <a href="msualg_4.html#V3" title="MSUALG_4:attr.3">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" title="c12">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="msualg_4.html#D3" title="MSUALG_4:def.3">MSUALG_4:def 3</a>,<a class="ref" href="msualg_4.html#D5" title="MSUALG_4:def.5">MSUALG_4:def 5</a><br/><b>let </b><font color="Maroon" title="c13">R</font> be   <a href="relset_1.html#NM3" title="RELSET_1:NM.3">Relation</a> of the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font>;<br/>



<b>assume </b><a NAME="E1:3_1_4"/>
( <font color="Maroon" title="c12">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon" title="c11">R0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">R</font> )
 ;<br/>

<b>hence </b><a NAME="E2:3_1_4"/>
<font color="Maroon" title="c13">R</font> is   <a href="eqrel_1.html#NM2" title="EQREL_1:NM.2">Equivalence_Relation</a> of the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font>
 <b>by </b><i><a class="txt" href="birkhoff.html#E24:3_1"><i><font color="Green" title="E15">A14</font></i></a>, <a class="ref" href="msualg_4.html#D3" title="MSUALG_4:def.3">MSUALG_4:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c12">R0</font> = <font color="Maroon" title="c11">R0</font> as  <a href="msualg_4.html#V3" title="MSUALG_4:attr.3">MSEquivalence-like</a> <a href="msualg_4.html#NM4" title="MSUALG_4:NM.4">ManySortedRelation</a> of <font color="Maroon">F<sub>2</sub></font>() ;<br/>
<b>reconsider </b><font color="Maroon" title="c13">R0</font> = <font color="Maroon" title="c12">R0</font> as   <a href="msualg_4.html#NM5" title="MSUALG_4:NM.5">MSCongruence</a> of <font color="Maroon">F<sub>2</sub></font>() <b>by </b><i><a class="txt" href="birkhoff.html#E1:3_1"><i><font color="Green" title="E6">A4</font></i></a>, <a class="ref" href="msualg_9.html#T29" title="MSUALG_9:th.29">MSUALG_9:29</a></i>;<br/>
<b>take </b><font color="Maroon" title="c14">A</font> =  <a href="msualg_4.html#K14" title="MSUALG_4:func.14">QuotMSAlg</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon" title="c13">R0</font>;<br/>

<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <font color="Olive" title="b1">R</font> being  <a href="msualg_4.html#NM5" title="MSUALG_4:NM.5">MSCongruence</a> of <font color="Maroon">F<sub>2</sub></font>() st <br/>( <font color="Olive" title="b1">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msualg_4.html#K14" title="MSUALG_4:func.14">QuotMSAlg</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Olive" title="b1">R</font> );<br/>
<div><i><font color="Green" title="E16">A15</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" title="c15">c</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:3_1_5"/>
<font color="Maroon" title="c15">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">CC</font>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c16">cc</font> = <font color="Maroon" title="c15">c</font> as   <a href="msualg_4.html#NM5" title="MSUALG_4:NM.5">MSCongruence</a> of <font color="Maroon">F<sub>2</sub></font>() <b>by </b><i><a class="txt" href="birkhoff.html#E2:3_1"><i><font color="Green" title="E7">A5</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c17">w</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:3_1_5"/><i><font color="Green" title="E17">A16</font></i>: 
<font color="Maroon" title="c17">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msualg_4.html#K14" title="MSUALG_4:func.14">QuotMSAlg</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon" title="c16">cc</font>
 ;<br/><b>take </b><font color="Maroon" title="c18">w</font> = <font color="Maroon" title="c17">w</font>;<br/><b>thus </b><a NAME="E5:3_1_5"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c15">c</font>,<font color="Maroon" title="c18">w</font>]
 <b>by </b><i><a class="txt" href="birkhoff.html#E4:3_1_5"><i><font color="Green" title="E17">A16</font></i></a></i>;<br/></div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c15">FF</font> being    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c9">CC</font><b> such that </b><br/><a NAME="E31:3_1"/><i><font color="Green" title="E17">A17</font></i>: 
for <font color="Olive" title="b1">c</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">CC</font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">c</font>,<font color="Maroon" title="c15">FF</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">c</font>]
 <b>from </b><i><a class="ref" href="pboole.html#S3" title="PBOOLE:sch.3">PBOOLE:sch 3</a>(<a class="txt" href="birkhoff.html#E29:3_1"><i><font color="Green" title="E16">A15</font></i></a>);<br/></i>
<a NAME="E32:3_1"/>
<font color="Maroon" title="c15">FF</font> is    <a href="pralg_2.html#M2" title="PRALG_2:mode.2">MSAlgebra-Family</a> of <font color="Maroon" title="c9">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" title="c16">c</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="pralg_2.html#D12" title="PRALG_2:def.12">PRALG_2:def 12</a><br/>

<b>assume </b><a NAME="E1:3_1_6"/>
<font color="Maroon" title="c16">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">CC</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c17">R</font> being   <a href="msualg_4.html#NM5" title="MSUALG_4:NM.5">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" title="E18">A18</font></i>: 
( <font color="Maroon" title="c17">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">c</font> &amp; <font color="Maroon" title="c15">FF</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msualg_4.html#K14" title="MSUALG_4:func.14">QuotMSAlg</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon" title="c17">R</font> )
 <b>by </b><i><a class="txt" href="birkhoff.html#E31:3_1"><i><font color="Green" title="E17">A17</font></i></a></i>;<br/>
<b>thus </b><a NAME="E4:3_1_6"/>
<font color="Maroon" title="c15">FF</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">c</font> is  <a href="msualg_1.html#V5" title="MSUALG_1:attr.5">non-empty</a>  <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon">F<sub>1</sub></font>()
 <b>by </b><i><a class="txt" href="birkhoff.html#E3:3_1_6"><i><font color="Green" title="E18">A18</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c16">FF</font> = <font color="Maroon" title="c15">FF</font> as    <a href="pralg_2.html#M2" title="PRALG_2:mode.2">MSAlgebra-Family</a> of <font color="Maroon" title="c9">CC</font>,<font color="Maroon">F<sub>1</sub></font>() ;<br/>
<a NAME="E34:3_1"/>
for <font color="Olive" title="b1">cc</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">cc</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">CC</font> holds <br/> ex <font color="Olive" title="b2">A</font> being   <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon">F<sub>1</sub></font>() st <br/>( <font color="Olive" title="b2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">FF</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">cc</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b2">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" title="c17">cc</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:3_1_7"/><i><font color="Green" title="E18">A19</font></i>: 
<font color="Maroon" title="c17">cc</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">CC</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c18">c</font> = <font color="Maroon" title="c17">cc</font> as    <a href="closure2.html#M1" title="CLOSURE2:mode.1">Element</a> of <font color="Maroon" title="c9">CC</font> ;<br/>
<b>take </b><font color="Maroon" title="c19">V</font> = <font color="Maroon" title="c16">FF</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Maroon" title="c18">c</font>;<br/>

<b>consider </b><font color="Maroon" title="c20">C</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K6" title="MSUALG_5:func.6">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" title="E19">A20</font></i>: 
( <font color="Maroon" title="c17">cc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">C</font> &amp;  ex <font color="Olive" title="b1">R</font> being  <a href="msualg_4.html#NM5" title="MSUALG_4:NM.5">MSCongruence</a> of <font color="Maroon">F<sub>2</sub></font>() st <br/>( <font color="Olive" title="b1">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">C</font> &amp; <font color="Maroon">P<sub>1</sub></font>[ <a href="msualg_4.html#K14" title="MSUALG_4:func.14">QuotMSAlg</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Olive" title="b1">R</font>] ) )
 <b>by </b><i><a class="txt" href="birkhoff.html#E1:3_1_7"><i><font color="Green" title="E18">A19</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c21">R</font> being   <a href="msualg_4.html#NM5" title="MSUALG_4:NM.5">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" title="E20">A21</font></i>: 
( <font color="Maroon" title="c21">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">cc</font> &amp; <font color="Maroon" title="c16">FF</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">cc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msualg_4.html#K14" title="MSUALG_4:func.14">QuotMSAlg</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon" title="c21">R</font> )
 <b>by </b><i><a class="txt" href="birkhoff.html#E31:3_1"><i><font color="Green" title="E17">A17</font></i></a>, <a class="txt" href="birkhoff.html#E1:3_1_7"><i><font color="Green" title="E18">A19</font></i></a></i>;<br/>
<b>thus </b><a NAME="E7:3_1_7"/>
( <font color="Maroon" title="c19">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">FF</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">cc</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c19">V</font>] )
 <b>by </b><i><a class="txt" href="birkhoff.html#E4:3_1_7"><i><font color="Green" title="E19">A20</font></i></a>, <a class="txt" href="birkhoff.html#E6:3_1_7"><i><font color="Green" title="E20">A21</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E35:3_1"/><b>then </b><i><font color="Green" title="E18">A22</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[ <a href="pralg_2.html#K15" title="PRALG_2:func.15">product</a> <font color="Maroon" title="c16">FF</font>]
 
<b>by </b><i><a class="txt" href="birkhoff.html#E3:3"><i><font color="Green" title="E5">A3</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c17">F</font> =  <a href="msualg_4.html#K16" title="MSUALG_4:func.16">MSNat_Hom</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon" title="c13">R0</font> as   <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon" title="c14">A</font> ;<br/>
<b>take </b>
<font color="Maroon" title="c17">F</font>
;<br/>

<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="closure2.html#M1" title="CLOSURE2:mode.1">Element</a> of <font color="Maroon" title="c9">CC</font>,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <font color="Olive" title="b1">F1</font> being  <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon">F<sub>2</sub></font>(),<span class="p1">(<span class="default"><font color="Maroon" title="c16">FF</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> $1</span>)</span> st <br/>( <font color="Olive" title="b1">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $2 &amp; <font color="Olive" title="b1">F1</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon" title="c16">FF</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> $1 &amp;  ex <font color="Olive" title="b2">R</font> being  <a href="msualg_4.html#NM5" title="MSUALG_4:NM.5">MSCongruence</a> of <font color="Maroon">F<sub>2</sub></font>() st <br/>( $1 <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Olive" title="b2">R</font> &amp; <font color="Olive" title="b1">F1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="msualg_4.html#K16" title="MSUALG_4:func.16">MSNat_Hom</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Olive" title="b2">R</font> ) );<br/>
<a NAME="E37:3_1"/><i><font color="Green" title="E19">A23</font></i>: 
for <font color="Olive" title="b1">c</font> being   <a href="closure2.html#M1" title="CLOSURE2:mode.1">Element</a> of <font color="Maroon" title="c9">CC</font>  ex <font color="Olive" title="b2">j</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">c</font>,<font color="Olive" title="b2">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" title="c18">c</font> be    <a href="closure2.html#M1" title="CLOSURE2:mode.1">Element</a> of <font color="Maroon" title="c9">CC</font>;<br/>

<b>consider </b><font color="Maroon" title="c19">R</font> being   <a href="msualg_4.html#NM5" title="MSUALG_4:NM.5">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" title="E20">A24</font></i>: 
( <font color="Maroon" title="c19">R</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c18">c</font> &amp; <font color="Maroon" title="c16">FF</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Maroon" title="c18">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msualg_4.html#K14" title="MSUALG_4:func.14">QuotMSAlg</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon" title="c19">R</font> )
 <b>by </b><i><a class="txt" href="birkhoff.html#E31:3_1"><i><font color="Green" title="E17">A17</font></i></a></i>;<br/>
<b>set </b><font color="Maroon" title="c20">j</font> =  <a href="msualg_4.html#K16" title="MSUALG_4:func.16">MSNat_Hom</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon" title="c19">R</font>;<br/>
<b>take </b>
 <a href="msualg_4.html#K16" title="MSUALG_4:func.16">MSNat_Hom</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon" title="c19">R</font>
;<br/>

<b>reconsider </b><font color="Maroon" title="c21">F1</font> =  <a href="msualg_4.html#K16" title="MSUALG_4:func.16">MSNat_Hom</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon" title="c19">R</font> as   <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon">F<sub>2</sub></font>(),<span class="p1">(<span class="default"><font color="Maroon" title="c16">FF</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Maroon" title="c18">c</font></span>)</span> <b>by </b><i><a class="txt" href="birkhoff.html#E2:3_1_8"><i><font color="Green" title="E20">A24</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c21">F1</font>
;<br/>

<b>thus </b><a NAME="E4:3_1_8"/>
<font color="Maroon" title="c21">F1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="msualg_4.html#K16" title="MSUALG_4:func.16">MSNat_Hom</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon" title="c19">R</font>
 ;<br/>

<a NAME="E5:3_1_8"/>
 <a href="msualg_4.html#K16" title="MSUALG_4:func.16">MSNat_Hom</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon" title="c19">R</font> <a href="msualg_3.html#R2" title="MSUALG_3:pred.2">is_epimorphism</a> <font color="Maroon">F<sub>2</sub></font>(), <a href="msualg_4.html#K14" title="MSUALG_4:func.14">QuotMSAlg</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon" title="c19">R</font>
 
<b>by </b><i><a class="ref" href="msualg_4.html#T3" title="MSUALG_4:th.3">MSUALG_4:3</a></i>;<br/>
<b>hence </b><a NAME="E6:3_1_8"/>
<font color="Maroon" title="c21">F1</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon" title="c16">FF</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Maroon" title="c18">c</font>
 <b>by </b><i><a class="txt" href="birkhoff.html#E2:3_1_8"><i><font color="Green" title="E20">A24</font></i></a>, <a class="ref" href="msualg_3.html#D10" title="MSUALG_3:def.10">MSUALG_3:def 10</a></i>;<br/>

<b>take </b>
<font color="Maroon" title="c19">R</font>
;<br/>

<b>thus </b><a NAME="E7:3_1_8"/>
( <font color="Maroon" title="c18">c</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c19">R</font> &amp; <font color="Maroon" title="c21">F1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="msualg_4.html#K16" title="MSUALG_4:func.16">MSNat_Hom</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon" title="c19">R</font> )
 <b>by </b><i><a class="txt" href="birkhoff.html#E2:3_1_8"><i><font color="Green" title="E20">A24</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c18">FofI1</font> being    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c9">CC</font><b> such that </b><br/><a NAME="E39:3_1"/><i><font color="Green" title="E20">A25</font></i>: 
for <font color="Olive" title="b1">c</font> being   <a href="closure2.html#M1" title="CLOSURE2:mode.1">Element</a> of <font color="Maroon" title="c9">CC</font> holds  <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">c</font>,<font color="Maroon" title="c18">FofI1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">c</font>]
 <b>from </b><i><a class="ref" href="msualg_9.html#S1" title="MSUALG_9:sch.1">MSUALG_9:sch 1</a>(<a class="txt" href="birkhoff.html#E37:3_1"><i><font color="Green" title="E19">A23</font></i></a>);<br/></i>
<a NAME="E40:3_1"/><i><font color="Green" title="E21">A26</font></i>: 
for <font color="Olive" title="b1">c</font> being   <a href="closure2.html#M1" title="CLOSURE2:mode.1">Element</a> of <font color="Maroon" title="c9">CC</font>  ex <font color="Olive" title="b2">F1</font> being  <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon">F<sub>2</sub></font>(),<span class="p1">(<span class="default"><font color="Maroon" title="c16">FF</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Olive" title="b1">c</font></span>)</span> st <br/>( <font color="Olive" title="b2">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">FofI1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">c</font> &amp; <font color="Olive" title="b2">F1</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon" title="c16">FF</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Olive" title="b1">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" title="c19">c</font> be    <a href="closure2.html#M1" title="CLOSURE2:mode.1">Element</a> of <font color="Maroon" title="c9">CC</font>;<br/>

<b>consider </b><font color="Maroon" title="c20">F1</font> being   <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon">F<sub>2</sub></font>(),<span class="p1">(<span class="default"><font color="Maroon" title="c16">FF</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Maroon" title="c19">c</font></span>)</span><b> such that </b><br/><a NAME="E2:3_1_9"/><i><font color="Green" title="E22">A27</font></i>: 
( <font color="Maroon" title="c20">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">FofI1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">c</font> &amp; <font color="Maroon" title="c20">F1</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon" title="c16">FF</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Maroon" title="c19">c</font> )
 <b>and </b><br/><a NAME="E3:3_1_9"/>
 ex <font color="Olive" title="b1">R</font> being  <a href="msualg_4.html#NM5" title="MSUALG_4:NM.5">MSCongruence</a> of <font color="Maroon">F<sub>2</sub></font>() st <br/>( <font color="Maroon" title="c19">c</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Olive" title="b1">R</font> &amp; <font color="Maroon" title="c20">F1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="msualg_4.html#K16" title="MSUALG_4:func.16">MSNat_Hom</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Olive" title="b1">R</font> )
 <b>by </b><i><a class="txt" href="birkhoff.html#E39:3_1"><i><font color="Green" title="E20">A25</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c20">F1</font>
;<br/>

<b>thus </b><a NAME="E4:3_1_9"/>
( <font color="Maroon" title="c20">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">FofI1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">c</font> &amp; <font color="Maroon" title="c20">F1</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon" title="c16">FF</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Maroon" title="c19">c</font> )
 <b>by </b><i><a class="txt" href="birkhoff.html#E2:3_1_9"><i><font color="Green" title="E22">A27</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E41:3_1"/>
<font color="Maroon" title="c18">FofI1</font> is <a href="funcop_1.html#V1" title="FUNCOP_1:attr.1">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" title="c19">c</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="funcop_1.html#D6" title="FUNCOP_1:def.6">FUNCOP_1:def 6</a><br/>

<b>assume </b><a NAME="E1:3_1_10"/>
<font color="Maroon" title="c19">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c18">FofI1</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c20">cc</font> = <font color="Maroon" title="c19">c</font> as    <a href="closure2.html#M1" title="CLOSURE2:mode.1">Element</a> of <font color="Maroon" title="c9">CC</font> <b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c21">F1</font> being   <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon">F<sub>2</sub></font>(),<span class="p1">(<span class="default"><font color="Maroon" title="c16">FF</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Maroon" title="c20">cc</font></span>)</span><b> such that </b><br/><a NAME="E4:3_1_10"/><i><font color="Green" title="E22">A28</font></i>: 
( <font color="Maroon" title="c21">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">FofI1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">cc</font> &amp; <font color="Maroon" title="c21">F1</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon" title="c16">FF</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Maroon" title="c20">cc</font> )
 <b>by </b><i><a class="txt" href="birkhoff.html#E40:3_1"><i><font color="Green" title="E21">A26</font></i></a></i>;<br/>
<b>thus </b><a NAME="E5:3_1_10"/>
<font color="Maroon" title="c18">FofI1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">c</font> is   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>
 <b>by </b><i><a class="txt" href="birkhoff.html#E4:3_1_10"><i><font color="Green" title="E22">A28</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c19">FofI1</font> = <font color="Maroon" title="c18">FofI1</font> as   <a href="pboole.html#NM2" title="PBOOLE:NM.2">ManySortedFunction</a> of <font color="Maroon" title="c9">CC</font> ;<br/>
<b>consider </b><font color="Maroon" title="c20">H</font> being   <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon">F<sub>2</sub></font>(),<span class="p1">(<span class="default"><a href="pralg_2.html#K15" title="PRALG_2:func.15">product</a> <font color="Maroon" title="c16">FF</font></span>)</span><b> such that </b><br/><a NAME="E44:3_1"/><i><font color="Green" title="E22">A29</font></i>: 
<font color="Maroon" title="c20">H</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon">F<sub>2</sub></font>(), <a href="pralg_2.html#K15" title="PRALG_2:func.15">product</a> <font color="Maroon" title="c16">FF</font>
 <b>and </b><br/><a NAME="E45:3_1"/><i><font color="Green" title="E23">A30</font></i>: 
for <font color="Olive" title="b1">c</font> being   <a href="closure2.html#M1" title="CLOSURE2:mode.1">Element</a> of <font color="Maroon" title="c9">CC</font> holds  <span class="p1">(<span class="default"><a href="pralg_3.html#K4" title="PRALG_3:func.4">proj</a> <font color="Maroon" title="c16">FF</font>,<font color="Olive" title="b1">c</font></span>)</span> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c20">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">FofI1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">c</font>
 <b>by </b><i><a class="txt" href="birkhoff.html#E40:3_1"><i><font color="Green" title="E21">A26</font></i></a>, <a class="ref" href="pralg_3.html#T30" title="PRALG_3:th.30">PRALG_3:30</a></i>;<br/>
<a NAME="E46:3_1"/>
 <a href="msualg_4.html#K20" title="MSUALG_4:func.20">MSHomQuot</a> <font color="Maroon" title="c20">H</font> <a href="msualg_3.html#R3" title="MSUALG_3:pred.3">is_monomorphism</a>  <a href="msualg_4.html#K14" title="MSUALG_4:func.14">QuotMSAlg</a> <font color="Maroon">F<sub>2</sub></font>(),<span class="p1">(<span class="default"><a href="msualg_4.html#K18" title="MSUALG_4:func.18">MSCng</a> <font color="Maroon" title="c20">H</font></span>)</span>, <a href="pralg_2.html#K15" title="PRALG_2:func.15">product</a> <font color="Maroon" title="c16">FF</font>
 
<b>by </b><i><a class="txt" href="birkhoff.html#E44:3_1"><i><font color="Green" title="E22">A29</font></i></a>, <a class="ref" href="msualg_4.html#T4" title="MSUALG_4:th.4">MSUALG_4:4</a></i>;<br/>
<a NAME="E47:3_1"/><b>then </b><i><font color="Green" title="E24">A31</font></i>: 
 <a href="msualg_4.html#K14" title="MSUALG_4:func.14">QuotMSAlg</a> <font color="Maroon">F<sub>2</sub></font>(),<span class="p1">(<span class="default"><a href="msualg_4.html#K18" title="MSUALG_4:func.18">MSCng</a> <font color="Maroon" title="c20">H</font></span>)</span>, <a href="msualg_3.html#K7" title="MSUALG_3:func.7">Image</a> <span class="p1">(<span class="default"><a href="msualg_4.html#K20" title="MSUALG_4:func.20">MSHomQuot</a> <font color="Maroon" title="c20">H</font></span>)</span> <a href="msualg_3.html#R6" title="MSUALG_3:pred.6">are_isomorphic</a> 
 
<b>by </b><i><a class="ref" href="msualg_9.html#T17" title="MSUALG_9:th.17">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" title="c21">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:3_1_11"/>
<font color="Maroon" title="c21">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>()
 ;<br/><b>then reconsider </b><font color="Maroon" title="c22">s</font> = <font color="Maroon" title="c21">i</font> as   <a href="msualg_1.html#NM1" title="MSUALG_1:NM.1">SortSymbol</a> of <font color="Maroon">F<sub>1</sub></font>() ;<br/><b>consider </b><font color="Maroon" title="c23">Q</font> being   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <span class="p1">(<span class="default"><span class="p2"><a href="pboole.html#K10" title="PBOOLE:func.10">[|</a><span class="default">the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>(),the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>()</span><a href="pboole.html#K10" title="PBOOLE:func.10">|]</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">s</font></span>)</span><b> such that </b><br/><a NAME="E4:3_1_11"/><i><font color="Green" title="E25">A32</font></i>: 
<font color="Maroon" title="c23">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="closure2.html#K5" title="CLOSURE2:func.5">|:</a><span class="default"><font color="Maroon" title="c9">CC</font></span><a href="closure2.html#K5" title="CLOSURE2:func.5">:|</a></span> <a href="mssubfam.html#K4" title="MSSUBFAM:func.4">.</a> <font color="Maroon" title="c22">s</font>
 <b>and </b><br/><a NAME="E5:3_1_11"/><i><font color="Green" title="E26">A33</font></i>: 
<span class="p1">(<span class="default"><a href="mssubfam.html#K6" title="MSSUBFAM:func.6">meet</a> <span class="p2"><a href="closure2.html#K5" title="CLOSURE2:func.5">|:</a><span class="default"><font color="Maroon" title="c9">CC</font></span><a href="closure2.html#K5" title="CLOSURE2:func.5">:|</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c23">Q</font>
 <b>by </b><i><a class="ref" href="mssubfam.html#D2" title="MSSUBFAM:def.2">MSSUBFAM:def 2</a></i>;<br/><a NAME="E6:3_1_11"/><i><font color="Green" title="E27">A34</font></i>: 
<span class="p1"><a href="closure2.html#K5" title="CLOSURE2:func.5">|:</a><span class="default"><font color="Maroon" title="c9">CC</font></span><a href="closure2.html#K5" title="CLOSURE2:func.5">:|</a></span> <a href="mssubfam.html#K4" title="MSSUBFAM:func.4">.</a> <font color="Maroon" title="c22">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">s</font></span>)</span> where <font color="Olive" title="b1">t</font> is    <a href="closure2.html#M1" title="CLOSURE2:mode.1">Element</a> of  <a href="closure2.html#K6" title="CLOSURE2:func.6">Bool</a> <span class="p1"><a href="pboole.html#K10" title="PBOOLE:func.10">[|</a><span class="default">the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>(),the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>()</span><a href="pboole.html#K10" title="PBOOLE:func.10">|]</a></span> : <font color="Olive" title="b1">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">CC</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="closure2.html#T15" title="CLOSURE2:th.15">CLOSURE2:15</a></i>;<br/><a NAME="E7:3_1_11"/>
<span class="p1">(<span class="default"><a href="msualg_4.html#K18" title="MSUALG_4:func.18">MSCng</a> <font color="Maroon" title="c20">H</font></span>)</span> <a href="msualg_4.html#K2" title="MSUALG_4:func.2">.</a> <font color="Maroon" title="c22">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">R0</font> <a href="msualg_4.html#K2" title="MSUALG_4:func.2">.</a> <font color="Maroon" title="c22">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" title="c24">a</font>, <font color="Maroon" title="c25">b</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="relat_1.html#D2" title="RELAT_1:def.2">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" title="E28">A35</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c24">a</font>,<font color="Maroon" title="c25">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="msualg_4.html#K18" title="MSUALG_4:func.18">MSCng</a> <font color="Maroon" title="c20">H</font></span>)</span> <a href="msualg_4.html#K2" title="MSUALG_4:func.2">.</a> <font color="Maroon" title="c22">s</font>
 ;<br/><a NAME="E2:3_1_11_1_1"/><b>then </b><i><font color="Green" title="E29">A36</font></i>: 
( <font color="Maroon" title="c24">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">s</font> &amp; <font color="Maroon" title="c25">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">s</font> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><a NAME="E3:3_1_11_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c24">a</font>,<font color="Maroon" title="c25">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default">the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">s</font></span>)</span>,<span class="p2">(<span class="default">the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">s</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="txt" href="birkhoff.html#E1:3_1_11_1_1"><i><font color="Green" title="E28">A35</font></i></a></i>;<br/><a NAME="E4:3_1_11_1_1"/><b>then </b><i><font color="Green" title="E30">A37</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c24">a</font>,<font color="Maroon" title="c25">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="pboole.html#K10" title="PBOOLE:func.10">[|</a><span class="default">the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>(),the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>()</span><a href="pboole.html#K10" title="PBOOLE:func.10">|]</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">s</font>
 <b>by </b><i><a class="ref" href="pboole.html#D21" title="PBOOLE:def.21">PBOOLE:def 21</a></i>;<br/><a NAME="E5:3_1_11_1_1"/><i><font color="Green" title="E31">A38</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c24">a</font>,<font color="Maroon" title="c25">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msualg_4.html#K17" title="MSUALG_4:func.17">MSCng</a> <font color="Maroon" title="c20">H</font>,<font color="Maroon" title="c22">s</font>
 <b>by </b><i><a class="txt" href="birkhoff.html#E44:3_1"><i><font color="Green" title="E22">A29</font></i></a>, <a class="txt" href="birkhoff.html#E1:3_1_11_1_1"><i><font color="Green" title="E28">A35</font></i></a>, <a class="ref" href="msualg_4.html#D20" title="MSUALG_4:def.20">MSUALG_4:def 20</a></i>;<br/><a NAME="E6:3_1_11_1_1"/>
for <font color="Olive" title="b1">Y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">Q</font> holds <br/><span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c24">a</font>,<font color="Maroon" title="c25">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">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" title="c26">Y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:3_1_11_1_1_1"/>
<font color="Maroon" title="c26">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">Q</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c27">t</font> being    <a href="closure2.html#M1" title="CLOSURE2:mode.1">Element</a> of  <a href="closure2.html#K6" title="CLOSURE2:func.6">Bool</a> <span class="p1"><a href="pboole.html#K10" title="PBOOLE:func.10">[|</a><span class="default">the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>(),the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>()</span><a href="pboole.html#K10" title="PBOOLE:func.10">|]</a></span><b> such that </b><br/><a NAME="E3:3_1_11_1_1_1"/><i><font color="Green" title="E32">A39</font></i>: 
<font color="Maroon" title="c26">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">s</font>
 <b>and </b><br/><a NAME="E4:3_1_11_1_1_1"/><i><font color="Green" title="E33">A40</font></i>: 
<font color="Maroon" title="c27">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">CC</font>
 <b>by </b><i><a class="txt" href="birkhoff.html#E4:3_1_11"><i><font color="Green" title="E25">A32</font></i></a>, <a class="txt" href="birkhoff.html#E6:3_1_11"><i><font color="Green" title="E27">A34</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c28">t1</font> = <font color="Maroon" title="c27">t</font> as    <a href="closure2.html#M1" title="CLOSURE2:mode.1">Element</a> of <font color="Maroon" title="c9">CC</font> <b>by </b><i><a class="txt" href="birkhoff.html#E4:3_1_11_1_1_1"><i><font color="Green" title="E33">A40</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c29">F1</font> being   <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon">F<sub>2</sub></font>(),<span class="p1">(<span class="default"><font color="Maroon" title="c16">FF</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Maroon" title="c28">t1</font></span>)</span><b> such that </b><br/><a NAME="E7:3_1_11_1_1_1"/><i><font color="Green" title="E34">A41</font></i>: 
( <font color="Maroon" title="c29">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">FofI1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">t1</font> &amp; <font color="Maroon" title="c29">F1</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon" title="c16">FF</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Maroon" title="c28">t1</font> &amp;  ex <font color="Olive" title="b1">R</font> being  <a href="msualg_4.html#NM5" title="MSUALG_4:NM.5">MSCongruence</a> of <font color="Maroon">F<sub>2</sub></font>() st <br/>( <font color="Maroon" title="c28">t1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Olive" title="b1">R</font> &amp; <font color="Maroon" title="c29">F1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="msualg_4.html#K16" title="MSUALG_4:func.16">MSNat_Hom</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Olive" title="b1">R</font> ) )
 <b>by </b><i><a class="txt" href="birkhoff.html#E39:3_1"><i><font color="Green" title="E20">A25</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c30">R</font> being   <a href="msualg_4.html#NM5" title="MSUALG_4:NM.5">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" title="E35">A42</font></i>: 
( <font color="Maroon" title="c28">t1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c30">R</font> &amp; <font color="Maroon" title="c29">F1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="msualg_4.html#K16" title="MSUALG_4:func.16">MSNat_Hom</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon" title="c30">R</font> )
 <b>by </b><i><a class="txt" href="birkhoff.html#E7:3_1_11_1_1_1"><i><font color="Green" title="E34">A41</font></i></a></i>;<br/>
<span class="p1">(<span class="default"><font color="Maroon" title="c29">F1</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c22">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">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" title="PRALG_3:func.4">proj</a> <font color="Maroon" title="c16">FF</font>,<font color="Maroon" title="c28">t1</font></span>)</span> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c20">H</font></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c22">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">a</font>
<b>by </b><i><a class="txt" href="birkhoff.html#E45:3_1"><i><font color="Green" title="E23">A30</font></i></a>, <a class="txt" href="birkhoff.html#E7:3_1_11_1_1_1"><i><font color="Green" title="E34">A41</font></i></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" title="PRALG_3:func.4">proj</a> <font color="Maroon" title="c16">FF</font>,<font color="Maroon" title="c28">t1</font></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c22">s</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">H</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c22">s</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">a</font>
<b>by </b><i><a class="ref" href="msualg_3.html#T2" title="MSUALG_3:th.2">MSUALG_3:2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_3.html#K4" title="PRALG_3:func.4">proj</a> <font color="Maroon" title="c16">FF</font>,<font color="Maroon" title="c28">t1</font></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c22">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">H</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c22">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">a</font></span>)</span>
<b>by </b><i><a class="txt" href="birkhoff.html#E2:3_1_11_1_1"><i><font color="Green" title="E29">A36</font></i></a>, <a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_3.html#K4" title="PRALG_3:func.4">proj</a> <font color="Maroon" title="c16">FF</font>,<font color="Maroon" title="c28">t1</font></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c22">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">H</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c22">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">b</font></span>)</span>
<b>by </b><i><a class="txt" href="birkhoff.html#E2:3_1_11_1_1"><i><font color="Green" title="E29">A36</font></i></a>, <a class="txt" href="birkhoff.html#E5:3_1_11_1_1"><i><font color="Green" title="E31">A38</font></i></a>, <a class="ref" href="msualg_4.html#D19" title="MSUALG_4:def.19">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" title="PRALG_3:func.4">proj</a> <font color="Maroon" title="c16">FF</font>,<font color="Maroon" title="c28">t1</font></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c22">s</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">H</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c22">s</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">b</font>
<b>by </b><i><a class="txt" href="birkhoff.html#E2:3_1_11_1_1"><i><font color="Green" title="E29">A36</font></i></a>, <a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">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" title="PRALG_3:func.4">proj</a> <font color="Maroon" title="c16">FF</font>,<font color="Maroon" title="c28">t1</font></span>)</span> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c20">H</font></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c22">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">b</font>
<b>by </b><i><a class="ref" href="msualg_3.html#T2" title="MSUALG_3:th.2">MSUALG_3:2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c29">F1</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c22">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">b</font>
<b>by </b><i><a class="txt" href="birkhoff.html#E45:3_1"><i><font color="Green" title="E23">A30</font></i></a>, <a class="txt" href="birkhoff.html#E7:3_1_11_1_1_1"><i><font color="Green" title="E34">A41</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E11:3_1_11_1_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c24">a</font>,<font color="Maroon" title="c25">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c26">Y</font>
 <b>by </b><i><a class="txt" href="birkhoff.html#E2:3_1_11_1_1"><i><font color="Green" title="E29">A36</font></i></a>, <a class="txt" href="birkhoff.html#E3:3_1_11_1_1_1"><i><font color="Green" title="E32">A39</font></i></a>, <a class="txt" href="birkhoff.html#E9:3_1_11_1_1_1"><i><font color="Green" title="E35">A42</font></i></a>, <a class="ref" href="msualg_9.html#T34" title="MSUALG_9:th.34">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" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c24">a</font>,<font color="Maroon" title="c25">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">R0</font> <a href="msualg_4.html#K2" title="MSUALG_4:func.2">.</a> <font color="Maroon" title="c22">s</font>
 <b>by </b><i><a class="txt" href="birkhoff.html#E5:3_1_11"><i><font color="Green" title="E26">A33</font></i></a>, <a class="txt" href="birkhoff.html#E4:3_1_11_1_1"><i><font color="Green" title="E30">A37</font></i></a>, <a class="ref" href="setfam_1.html#T58" title="SETFAM_1:th.58">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" title="E28">A43</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c24">a</font>,<font color="Maroon" title="c25">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">R0</font> <a href="msualg_4.html#K2" title="MSUALG_4:func.2">.</a> <font color="Maroon" title="c22">s</font>
 ;<br/>

<a NAME="E3:3_1_11_1"/><b>then </b><i><font color="Green" title="E29">A44</font></i>: 
( <font color="Maroon" title="c24">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">s</font> &amp; <font color="Maroon" title="c25">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">s</font> )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E4:3_1_11_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c20">H</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c22">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K15" title="PRALG_2:func.15">product</a> <font color="Maroon" title="c16">FF</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">s</font>
 
<b>by </b><i><a class="txt" href="birkhoff.html#E3:3_1_11_1"><i><font color="Green" title="E29">A44</font></i></a>, <a class="ref" href="funct_2.html#T7" title="FUNCT_2:th.7">FUNCT_2:7</a></i>;<br/>
<a NAME="E5:3_1_11_1"/><b>then </b><i><font color="Green" title="E30">A45</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c20">H</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c22">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10" title="PRALG_2:func.10">Carrier</a> <font color="Maroon" title="c16">FF</font>,<font color="Maroon" title="c22">s</font></span>)</span>
 
<b>by </b><i><a class="ref" href="pralg_2.html#D17" title="PRALG_2:def.17">PRALG_2:def 17</a></i>;<br/>
<a NAME="E6:3_1_11_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c20">H</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c22">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K15" title="PRALG_2:func.15">product</a> <font color="Maroon" title="c16">FF</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">s</font>
 
<b>by </b><i><a class="txt" href="birkhoff.html#E3:3_1_11_1"><i><font color="Green" title="E29">A44</font></i></a>, <a class="ref" href="funct_2.html#T7" title="FUNCT_2:th.7">FUNCT_2:7</a></i>;<br/>
<a NAME="E7:3_1_11_1"/><b>then </b><i><font color="Green" title="E31">A46</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c20">H</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c22">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10" title="PRALG_2:func.10">Carrier</a> <font color="Maroon" title="c16">FF</font>,<font color="Maroon" title="c22">s</font></span>)</span>
 
<b>by </b><i><a class="ref" href="pralg_2.html#D17" title="PRALG_2:def.17">PRALG_2:def 17</a></i>;<br/>
<a NAME="E8:3_1_11_1"/>
for <font color="Olive" title="b1">c</font> being   <a href="closure2.html#M1" title="CLOSURE2:mode.1">Element</a> of <font color="Maroon" title="c9">CC</font> holds  <span class="p1">(<span class="default"><a href="pralg_3.html#K3" title="PRALG_3:func.3">proj</a> <span class="p2">(<span class="default"><a href="pralg_2.html#K10" title="PRALG_2:func.10">Carrier</a> <font color="Maroon" title="c16">FF</font>,<font color="Maroon" title="c22">s</font></span>)</span>,<font color="Olive" title="b1">c</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">H</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c22">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pralg_3.html#K3" title="PRALG_3:func.3">proj</a> <span class="p2">(<span class="default"><a href="pralg_2.html#K10" title="PRALG_2:func.10">Carrier</a> <font color="Maroon" title="c16">FF</font>,<font color="Maroon" title="c22">s</font></span>)</span>,<font color="Olive" title="b1">c</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">H</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c22">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">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" title="c26">c</font> be    <a href="closure2.html#M1" title="CLOSURE2:mode.1">Element</a> of <font color="Maroon" title="c9">CC</font>;<br/>

<b>reconsider </b><font color="Maroon" title="c27">t1</font> = <font color="Maroon" title="c26">c</font> as   <a href="msualg_4.html#NM5" title="MSUALG_4:NM.5">MSCongruence</a> of <font color="Maroon">F<sub>2</sub></font>() <b>by </b><i><a class="txt" href="birkhoff.html#E2:3_1"><i><font color="Green" title="E7">A5</font></i></a></i>;<br/>
<a NAME="E2:3_1_11_1_2"/>
<font color="Maroon" title="c27">t1</font> is    <a href="closure2.html#M1" title="CLOSURE2:mode.1">Element</a> of  <a href="closure2.html#K6" title="CLOSURE2:func.6">Bool</a> <span class="p1"><a href="pboole.html#K10" title="PBOOLE:func.10">[|</a><span class="default">the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>(),the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>()</span><a href="pboole.html#K10" title="PBOOLE:func.10">|]</a></span>
 
<b>by </b><i><a class="ref" href="closure2.html#D1" title="CLOSURE2:def.1">CLOSURE2:def 1</a></i>;<br/>
<a NAME="E3:3_1_11_1_2"/><b>then </b>
<font color="Maroon" title="c27">t1</font> <a href="msualg_4.html#K2" title="MSUALG_4:func.2">.</a> <font color="Maroon" title="c22">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="closure2.html#K5" title="CLOSURE2:func.5">|:</a><span class="default"><font color="Maroon" title="c9">CC</font></span><a href="closure2.html#K5" title="CLOSURE2:func.5">:|</a></span> <a href="mssubfam.html#K4" title="MSSUBFAM:func.4">.</a> <font color="Maroon" title="c22">s</font>
 
<b>by </b><i><a class="txt" href="birkhoff.html#E6:3_1_11"><i><font color="Green" title="E27">A34</font></i></a></i>;<br/>
<a NAME="E4:3_1_11_1_2"/><b>then </b><i><font color="Green" title="E32">A47</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c24">a</font>,<font color="Maroon" title="c25">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c27">t1</font> <a href="msualg_4.html#K2" title="MSUALG_4:func.2">.</a> <font color="Maroon" title="c22">s</font>
 
<b>by </b><i><a class="txt" href="birkhoff.html#E4:3_1_11"><i><font color="Green" title="E25">A32</font></i></a>, <a class="txt" href="birkhoff.html#E5:3_1_11"><i><font color="Green" title="E26">A33</font></i></a>, <a class="txt" href="birkhoff.html#E2:3_1_11_1"><i><font color="Green" title="E28">A43</font></i></a>, <a class="ref" href="setfam_1.html#T58" title="SETFAM_1:th.58">SETFAM_1:58</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c28">F1</font> being   <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon">F<sub>2</sub></font>(),<span class="p1">(<span class="default"><font color="Maroon" title="c16">FF</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Maroon" title="c26">c</font></span>)</span><b> such that </b><br/><a NAME="E6:3_1_11_1_2"/><i><font color="Green" title="E33">A48</font></i>: 
( <font color="Maroon" title="c28">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">FofI1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">c</font> &amp; <font color="Maroon" title="c28">F1</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon" title="c16">FF</font> <a href="pralg_2.html#K6" title="PRALG_2:func.6">.</a> <font color="Maroon" title="c26">c</font> &amp;  ex <font color="Olive" title="b1">R</font> being  <a href="msualg_4.html#NM5" title="MSUALG_4:NM.5">MSCongruence</a> of <font color="Maroon">F<sub>2</sub></font>() st <br/>( <font color="Maroon" title="c26">c</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Olive" title="b1">R</font> &amp; <font color="Maroon" title="c28">F1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="msualg_4.html#K16" title="MSUALG_4:func.16">MSNat_Hom</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Olive" title="b1">R</font> ) )
 <b>by </b><i><a class="txt" href="birkhoff.html#E39:3_1"><i><font color="Green" title="E20">A25</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c29">R</font> being   <a href="msualg_4.html#NM5" title="MSUALG_4:NM.5">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" title="E34">A49</font></i>: 
( <font color="Maroon" title="c27">t1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c29">R</font> &amp; <font color="Maroon" title="c28">F1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="msualg_4.html#K16" title="MSUALG_4:func.16">MSNat_Hom</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon" title="c29">R</font> )
 <b>by </b><i><a class="txt" href="birkhoff.html#E6:3_1_11_1_2"><i><font color="Green" title="E33">A48</font></i></a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><a href="pralg_3.html#K3" title="PRALG_3:func.3">proj</a> <span class="p2">(<span class="default"><a href="pralg_2.html#K10" title="PRALG_2:func.10">Carrier</a> <font color="Maroon" title="c16">FF</font>,<font color="Maroon" title="c22">s</font></span>)</span>,<font color="Maroon" title="c26">c</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">H</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c22">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">a</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_3.html#K4" title="PRALG_3:func.4">proj</a> <font color="Maroon" title="c16">FF</font>,<font color="Maroon" title="c26">c</font></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c22">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">H</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c22">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">a</font></span>)</span>
<b>by </b><i><a class="ref" href="pralg_3.html#D3" title="PRALG_3:def.3">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" title="PRALG_3:func.4">proj</a> <font color="Maroon" title="c16">FF</font>,<font color="Maroon" title="c26">c</font></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c22">s</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">H</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c22">s</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">a</font>
<b>by </b><i><a class="txt" href="birkhoff.html#E3:3_1_11_1"><i><font color="Green" title="E29">A44</font></i></a>, <a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">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" title="PRALG_3:func.4">proj</a> <font color="Maroon" title="c16">FF</font>,<font color="Maroon" title="c26">c</font></span>)</span> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c20">H</font></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c22">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">a</font>
<b>by </b><i><a class="ref" href="msualg_3.html#T2" title="MSUALG_3:th.2">MSUALG_3:2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c28">F1</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c22">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">a</font>
<b>by </b><i><a class="txt" href="birkhoff.html#E45:3_1"><i><font color="Green" title="E23">A30</font></i></a>, <a class="txt" href="birkhoff.html#E6:3_1_11_1_2"><i><font color="Green" title="E33">A48</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c28">F1</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c22">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">b</font>
<b>by </b><i><a class="txt" href="birkhoff.html#E3:3_1_11_1"><i><font color="Green" title="E29">A44</font></i></a>, <a class="txt" href="birkhoff.html#E4:3_1_11_1_2"><i><font color="Green" title="E32">A47</font></i></a>, <a class="txt" href="birkhoff.html#E8:3_1_11_1_2"><i><font color="Green" title="E34">A49</font></i></a>, <a class="ref" href="msualg_9.html#T34" title="MSUALG_9:th.34">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" title="PRALG_3:func.4">proj</a> <font color="Maroon" title="c16">FF</font>,<font color="Maroon" title="c26">c</font></span>)</span> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c20">H</font></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c22">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">b</font>
<b>by </b><i><a class="txt" href="birkhoff.html#E45:3_1"><i><font color="Green" title="E23">A30</font></i></a>, <a class="txt" href="birkhoff.html#E6:3_1_11_1_2"><i><font color="Green" title="E33">A48</font></i></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" title="PRALG_3:func.4">proj</a> <font color="Maroon" title="c16">FF</font>,<font color="Maroon" title="c26">c</font></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c22">s</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">H</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c22">s</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">b</font>
<b>by </b><i><a class="ref" href="msualg_3.html#T2" title="MSUALG_3:th.2">MSUALG_3:2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_3.html#K4" title="PRALG_3:func.4">proj</a> <font color="Maroon" title="c16">FF</font>,<font color="Maroon" title="c26">c</font></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c22">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">H</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c22">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">b</font></span>)</span>
<b>by </b><i><a class="txt" href="birkhoff.html#E3:3_1_11_1"><i><font color="Green" title="E29">A44</font></i></a>, <a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="pralg_3.html#K3" title="PRALG_3:func.3">proj</a> <span class="p2">(<span class="default"><a href="pralg_2.html#K10" title="PRALG_2:func.10">Carrier</a> <font color="Maroon" title="c16">FF</font>,<font color="Maroon" title="c22">s</font></span>)</span>,<font color="Maroon" title="c26">c</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">H</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c22">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">b</font></span>)</span>
<b>by </b><i><a class="ref" href="pralg_3.html#D3" title="PRALG_3:def.3">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" title="c20">H</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c22">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">H</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c22">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">b</font>
 
<b>by </b><i><a class="txt" href="birkhoff.html#E5:3_1_11_1"><i><font color="Green" title="E30">A45</font></i></a>, <a class="txt" href="birkhoff.html#E7:3_1_11_1"><i><font color="Green" title="E31">A46</font></i></a>, <a class="ref" href="msualg_9.html#T15" title="MSUALG_9:th.15">MSUALG_9:15</a></i>;<br/>
<a NAME="E10:3_1_11_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c24">a</font>,<font color="Maroon" title="c25">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msualg_4.html#K17" title="MSUALG_4:func.17">MSCng</a> <font color="Maroon" title="c20">H</font>,<font color="Maroon" title="c22">s</font>
 
<b>by </b><i><a class="txt" href="birkhoff.html#E3:3_1_11_1"><i><font color="Green" title="E29">A44</font></i></a>, <a class="ref" href="msualg_4.html#D19" title="MSUALG_4:def.19">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" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c24">a</font>,<font color="Maroon" title="c25">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="msualg_4.html#K18" title="MSUALG_4:func.18">MSCng</a> <font color="Maroon" title="c20">H</font></span>)</span> <a href="msualg_4.html#K2" title="MSUALG_4:func.2">.</a> <font color="Maroon" title="c22">s</font>
 <b>by </b><i><a class="txt" href="birkhoff.html#E44:3_1"><i><font color="Green" title="E22">A29</font></i></a>, <a class="ref" href="msualg_4.html#D20" title="MSUALG_4:def.20">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" title="MSUALG_4:func.18">MSCng</a> <font color="Maroon" title="c20">H</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">R0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font>
 ;<br/></div><b>end;</b></div>
<a NAME="E49:3_1"/><b>then </b>
 <a href="msualg_4.html#K18" title="MSUALG_4:func.18">MSCng</a> <font color="Maroon" title="c20">H</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c13">R0</font>
 
<b>by </b><i><a class="ref" href="pboole.html#T3" title="PBOOLE:th.3">PBOOLE:3</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c21">XX</font> being  <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">strict</a> <a href="msualg_1.html#V5" title="MSUALG_1:attr.5">non-empty</a>  <a href="msualg_2.html#M1" title="MSUALG_2:mode.1">MSSubAlgebra</a> of  <a href="pralg_2.html#K15" title="PRALG_2:func.15">product</a> <font color="Maroon" title="c16">FF</font><b> such that </b><br/><a NAME="E51:3_1"/><i><font color="Green" title="E25">A50</font></i>: 
<font color="Maroon" title="c14">A</font>,<font color="Maroon" title="c21">XX</font> <a href="msualg_3.html#R6" title="MSUALG_3:pred.6">are_isomorphic</a> 
 <b>by </b><i><a class="txt" href="birkhoff.html#E47:3_1"><i><font color="Green" title="E24">A31</font></i></a></i>;<br/>
<a NAME="E52:3_1"/><i><font color="Green" title="E26">A51</font></i>: 
<font color="Maroon" title="c21">XX</font>,<font color="Maroon" title="c14">A</font> <a href="msualg_3.html#R6" title="MSUALG_3:pred.6">are_isomorphic</a> 
 
<b>by </b><i><a class="txt" href="birkhoff.html#E51:3_1"><i><font color="Green" title="E25">A50</font></i></a>, <a class="ref" href="msualg_3.html#T17" title="MSUALG_3:th.17">MSUALG_3:17</a></i>;<br/>
<a NAME="E53:3_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c21">XX</font>]
 
<b>by </b><i><a class="txt" href="birkhoff.html#E2:3"><i><font color="Green" title="E4">A2</font></i></a>, <a class="txt" href="birkhoff.html#E35:3_1"><i><font color="Green" title="E18">A22</font></i></a></i>;<br/>
<b>hence </b><a NAME="E54:3_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c14">A</font>]
 <b>by </b><i><a class="txt" href="birkhoff.html#E1:3"><i><font color="Green" title="E3">A1</font></i></a>, <a class="txt" href="birkhoff.html#E52:3_1"><i><font color="Green" title="E26">A51</font></i></a></i>;<br/>

<b>thus </b><a NAME="E55:3_1"/><i><font color="Green" title="E27">A52</font></i>: 
<font color="Maroon" title="c17">F</font> <a href="msualg_3.html#R2" title="MSUALG_3:pred.2">is_epimorphism</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon" title="c14">A</font>
 <b>by </b><i><a class="ref" href="msualg_4.html#T3" title="MSUALG_4:th.3">MSUALG_4:3</a></i>;<br/>

<b>let </b><font color="Maroon" title="c22">B</font> be  <a href="msualg_1.html#V5" title="MSUALG_1:attr.5">non-empty</a>  <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon">F<sub>1</sub></font>();<br/><b>let </b><font color="Maroon" title="c23">G</font> be   <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon" title="c22">B</font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E56:3_1"/><i><font color="Green" title="E28">A53</font></i>: 
<font color="Maroon" title="c23">G</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon" title="c22">B</font>
 <b>and </b><br/><a NAME="E57:3_1"/><i><font color="Green" title="E29">A54</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c22">B</font>]
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c24">C</font> =  <a href="msualg_3.html#K7" title="MSUALG_3:func.7">Image</a> <span class="p1">(<span class="default"><a href="msualg_4.html#K20" title="MSUALG_4:func.20">MSHomQuot</a> <font color="Maroon" title="c23">G</font></span>)</span> as  <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">strict</a> <a href="msualg_1.html#V5" title="MSUALG_1:attr.5">non-empty</a>  <a href="msualg_2.html#M1" title="MSUALG_2:mode.1">MSSubAlgebra</a> of <font color="Maroon" title="c22">B</font> ;<br/>
<a NAME="E59:3_1"/>
 <a href="msualg_4.html#K20" title="MSUALG_4:func.20">MSHomQuot</a> <font color="Maroon" title="c23">G</font> <a href="msualg_3.html#R3" title="MSUALG_3:pred.3">is_monomorphism</a>  <a href="msualg_4.html#K14" title="MSUALG_4:func.14">QuotMSAlg</a> <font color="Maroon">F<sub>2</sub></font>(),<span class="p1">(<span class="default"><a href="msualg_4.html#K18" title="MSUALG_4:func.18">MSCng</a> <font color="Maroon" title="c23">G</font></span>)</span>,<font color="Maroon" title="c22">B</font>
 
<b>by </b><i><a class="txt" href="birkhoff.html#E56:3_1"><i><font color="Green" title="E28">A53</font></i></a>, <a class="ref" href="msualg_4.html#T4" title="MSUALG_4:th.4">MSUALG_4:4</a></i>;<br/>
<a NAME="E60:3_1"/><b>then </b>
 <a href="msualg_4.html#K14" title="MSUALG_4:func.14">QuotMSAlg</a> <font color="Maroon">F<sub>2</sub></font>(),<span class="p1">(<span class="default"><a href="msualg_4.html#K18" title="MSUALG_4:func.18">MSCng</a> <font color="Maroon" title="c23">G</font></span>)</span>,<font color="Maroon" title="c24">C</font> <a href="msualg_3.html#R6" title="MSUALG_3:pred.6">are_isomorphic</a> 
 
<b>by </b><i><a class="ref" href="msualg_9.html#T17" title="MSUALG_9:th.17">MSUALG_9:17</a></i>;<br/>
<a NAME="E61:3_1"/><b>then </b><i><font color="Green" title="E30">A55</font></i>: 
<font color="Maroon" title="c24">C</font>, <a href="msualg_4.html#K14" title="MSUALG_4:func.14">QuotMSAlg</a> <font color="Maroon">F<sub>2</sub></font>(),<span class="p1">(<span class="default"><a href="msualg_4.html#K18" title="MSUALG_4:func.18">MSCng</a> <font color="Maroon" title="c23">G</font></span>)</span> <a href="msualg_3.html#R6" title="MSUALG_3:pred.6">are_isomorphic</a> 
 
<b>by </b><i><a class="ref" href="msualg_3.html#T17" title="MSUALG_3:th.17">MSUALG_3:17</a></i>;<br/>
<a NAME="E62:3_1"/><i><font color="Green" title="E31">A56</font></i>: 
 <a href="msualg_4.html#K18" title="MSUALG_4:func.18">MSCng</a> <font color="Maroon" title="c23">G</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K6" title="MSUALG_5:func.6">CongrLatt</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span>
 
<b>by </b><i><a class="ref" href="msualg_5.html#D6" title="MSUALG_5:def.6">MSUALG_5:def 6</a></i>;<br/>
<a NAME="E63:3_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c24">C</font>]
 
<b>by </b><i><a class="txt" href="birkhoff.html#E2:3"><i><font color="Green" title="E4">A2</font></i></a>, <a class="txt" href="birkhoff.html#E57:3_1"><i><font color="Green" title="E29">A54</font></i></a></i>;<br/>
<a NAME="E64:3_1"/><b>then </b>
<font color="Maroon">P<sub>1</sub></font>[ <a href="msualg_4.html#K14" title="MSUALG_4:func.14">QuotMSAlg</a> <font color="Maroon">F<sub>2</sub></font>(),<span class="p1">(<span class="default"><a href="msualg_4.html#K18" title="MSUALG_4:func.18">MSCng</a> <font color="Maroon" title="c23">G</font></span>)</span>]
 
<b>by </b><i><a class="txt" href="birkhoff.html#E1:3"><i><font color="Green" title="E3">A1</font></i></a>, <a class="txt" href="birkhoff.html#E61:3_1"><i><font color="Green" title="E30">A55</font></i></a></i>;<br/>
<a NAME="E65:3_1"/><b>then </b>
 <a href="msualg_4.html#K18" title="MSUALG_4:func.18">MSCng</a> <font color="Maroon" title="c23">G</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">CC</font>
 
<b>by </b><i><a class="txt" href="birkhoff.html#E62:3_1"><i><font color="Green" title="E31">A56</font></i></a></i>;<br/>
<a NAME="E66:3_1"/><b>then </b>
 <a href="msualg_4.html#K18" title="MSUALG_4:func.18">MSCng</a> <font color="Maroon" title="c23">G</font> <a href="pboole.html#R3" title="PBOOLE:pred.3">in</a> <span class="p1"><a href="closure2.html#K5" title="CLOSURE2:func.5">|:</a><span class="default"><font color="Maroon" title="c9">CC</font></span><a href="closure2.html#K5" title="CLOSURE2:func.5">:|</a></span>
 
<b>by </b><i><a class="ref" href="closure2.html#T22" title="CLOSURE2:th.22">CLOSURE2:22</a></i>;<br/>
<a NAME="E67:3_1"/><b>then </b>
<font color="Maroon" title="c13">R0</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a>  <a href="msualg_4.html#K18" title="MSUALG_4:func.18">MSCng</a> <font color="Maroon" title="c23">G</font>
 
<b>by </b><i><a class="ref" href="mssubfam.html#T43" title="MSSUBFAM:th.43">MSSUBFAM:43</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c25">H</font> being   <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon" title="c14">A</font>,<font color="Maroon" title="c22">B</font><b> such that </b><br/><a NAME="E69:3_1"/><i><font color="Green" title="E32">A57</font></i>: 
( <font color="Maroon" title="c25">H</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon" title="c14">A</font>,<font color="Maroon" title="c22">B</font> &amp; <font color="Maroon" title="c23">G</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c25">H</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c17">F</font> )
 <b>by </b><i><a class="txt" href="birkhoff.html#E56:3_1"><i><font color="Green" title="E28">A53</font></i></a>, <a class="ref" href="msualg_9.html#T37" title="MSUALG_9:th.37">MSUALG_9:37</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c25">H</font>
;<br/>

<b>thus </b><a NAME="E70:3_1"/>
<font color="Maroon" title="c25">H</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon" title="c14">A</font>,<font color="Maroon" title="c22">B</font>
 <b>by </b><i><a class="txt" href="birkhoff.html#E69:3_1"><i><font color="Green" title="E32">A57</font></i></a></i>;<br/>

<b>thus </b><a NAME="E71:3_1"/>
<font color="Maroon" title="c23">G</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c25">H</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c17">F</font>
 <b>by </b><i><a class="txt" href="birkhoff.html#E69:3_1"><i><font color="Green" title="E32">A57</font></i></a></i>;<br/>

<b>let </b><font color="Maroon" title="c26">K</font> be   <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon" title="c14">A</font>,<font color="Maroon" title="c22">B</font>;<br/>

<b>assume </b><a NAME="E72:3_1"/><i><font color="Green" title="E33">A58</font></i>: 
<font color="Maroon" title="c26">K</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c17">F</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c23">G</font>
 ;<br/>

<a NAME="E73:3_1"/>
<font color="Maroon" title="c17">F</font> is <a href="msualg_3.html#V2" title="MSUALG_3:attr.2">"onto"</a>
 
<b>by </b><i><a class="txt" href="birkhoff.html#E55:3_1"><i><font color="Green" title="E27">A52</font></i></a>, <a class="ref" href="msualg_3.html#D10" title="MSUALG_3:def.10">MSUALG_3:def 10</a></i>;<br/>
<b>hence </b><a NAME="E74:3_1"/>
<font color="Maroon" title="c25">H</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c26">K</font>
 <b>by </b><i><a class="txt" href="birkhoff.html#E69:3_1"><i><font color="Green" title="E32">A57</font></i></a>, <a class="txt" href="birkhoff.html#E72:3_1"><i><font color="Green" title="E33">A58</font></i></a>, <a class="ref" href="extens_1.html#T16" title="EXTENS_1:th.16">EXTENS_1:16</a></i>;<br/>


</div>
