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

<b>set </b><font color="Maroon">c<sub>1</sub></font> = the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>();<br/>
<b>set </b><font color="Maroon">c<sub>2</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>();<br/>
<b>set </b><font color="Maroon">c<sub>3</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K6">CongrLatt</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> : ex <font color="Olive">b<sub>1</sub></font> being  <a href="msualg_4.html#NM5">MSCongruence</a> of <font color="Maroon">F<sub>2</sub></font>() st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">P<sub>1</sub></font>[ <a href="msualg_4.html#K14">QuotMSAlg</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Olive">b<sub>2</sub></font>] ) </span>}</span> ;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] means ex <font color="Olive">b<sub>1</sub></font> being  <a href="msualg_4.html#NM5">MSCongruence</a> of <font color="Maroon">F<sub>2</sub></font>() st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Maroon">P<sub>1</sub></font>[ <a href="msualg_4.html#K14">QuotMSAlg</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Olive">b<sub>1</sub></font>] );<br/>
<a NAME="E1:3_1"/><i><font color="Green">E6</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K6">CongrLatt</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span>  is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K6">CongrLatt</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span>
 
<b>from </b><i><a class="ref" href="domain_1.html#S7">DOMAIN_1:sch 7</a>();<br/></i>
<div><i><font color="Green">E7</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">c<sub>4</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:3_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K6">CongrLatt</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> : ex <font color="Olive">b<sub>1</sub></font> being  <a href="msualg_4.html#NM5">MSCongruence</a> of <font color="Maroon">F<sub>2</sub></font>() st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">P<sub>1</sub></font>[ <a href="msualg_4.html#K14">QuotMSAlg</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Olive">b<sub>2</sub></font>] ) </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K6">CongrLatt</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span><b> such that </b><br/><a NAME="E3:3_1_1"/><i><font color="Green">E8</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>and </b><br/><a NAME="E4:3_1_1"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="msualg_4.html#NM5">MSCongruence</a> of <font color="Maroon">F<sub>2</sub></font>() st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">P<sub>1</sub></font>[ <a href="msualg_4.html#K14">QuotMSAlg</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Olive">b<sub>1</sub></font>] )
 ;<br/><b>thus </b><a NAME="E5:3_1_1"/>
<font color="Maroon">c<sub>4</sub></font> is   <a href="msualg_4.html#NM5">MSCongruence</a> of <font color="Maroon">F<sub>2</sub></font>()
 <b>by </b><i><a class="txt" href="birkhoff.html#E3:3_1_1"><i><font color="Green">E8</font></i></a>, <a class="ref" href="msualg_5.html#D6">MSUALG_5:def 6</a></i>;<br/></div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>4</sub></font> being  <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> , <font color="Maroon">c<sub>5</sub></font> being    <a href="pralg_2.html#M2">MSAlgebra-Family</a> of <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">F<sub>1</sub></font>();<br/>
<a NAME="E4:3_1"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> holds <br/>ex <font color="Olive">b<sub>2</sub></font> being   <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">F<sub>1</sub></font>() st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">b<sub>2</sub></font>] )
 
;<br/>
<a NAME="E5:3_1"/><b>then </b><i><font color="Green">E8</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[ <a href="pralg_2.html#K15">product</a> <font color="Maroon">c<sub>5</sub></font>]
 
<b>by </b><i><a class="txt" href="birkhoff.html#E3:3"><i><font color="Green">E5</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>6</sub></font> being    <a href="pboole.html#M1">ManySortedSet</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>()<b> such that </b><br/><a NAME="E7:3_1"/><i><font color="Green">E9</font></i>: 
<span class="p1"><a href="pzfmisc1.html#K1">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="pzfmisc1.html#K1">}</a></span> <a href="pboole.html#R6">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="pre_circ.html#K2">--&gt;</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="msualg_9.html#T6">MSUALG_9:6</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>7</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:3_1_2"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>()
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</sub></font> as   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">F<sub>1</sub></font>() ;<br/><b>thus </b>the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> = 
<span class="p1">(<span class="default"><a href="pralg_2.html#K11">SORTS</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font>

<br/>.= 
 <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="pralg_2.html#D17">PRALG_2:def 17</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>
<b>by </b><i><a class="ref" href="card_3.html#T19">CARD_3:19</a>, <a class="ref" href="pralg_2.html#D16">PRALG_2:def 16</a></i>
<br/>.= 
<span class="p1">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="pre_circ.html#K2">--&gt;</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>
<br/>.= 
<span class="p1"><a href="pzfmisc1.html#K1">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="pzfmisc1.html#K1">}</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="txt" href="birkhoff.html#E7:3_1"><i><font color="Green">E9</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">Sorts</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="pboole.html#R6">=</a> <span class="p1"><a href="pzfmisc1.html#K1">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="pzfmisc1.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="pboole.html#T3">PBOOLE:3</a></i>;<br/>
<a NAME="E10:3_1"/><b>then </b>
 <a href="pralg_2.html#K15">product</a> <font color="Maroon">c<sub>5</sub></font>, <a href="msafree2.html#K8">Trivial_Algebra</a> <font color="Maroon">F<sub>1</sub></font>() <a href="msualg_3.html#R6">are_isomorphic</a> 
 
<b>by </b><i><a class="ref" href="msualg_9.html#T27">MSUALG_9:27</a></i>;<br/>
<a NAME="E11:3_1"/><b>then </b><i><font color="Green">E10</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[ <a href="msafree2.html#K8">Trivial_Algebra</a> <font color="Maroon">F<sub>1</sub></font>()]
 
<b>by </b><i><a class="txt" href="birkhoff.html#E1:3"><i><font color="Green">E3</font></i></a>, <a class="txt" href="birkhoff.html#E5:3_1"><i><font color="Green">E8</font></i></a></i>;<br/>
<a NAME="E12:3_1"/>
<span class="p1"><a href="pboole.html#K10">[|</a><span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>(),the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>()</span><a href="pboole.html#K10">|]</a></span> is   <a href="msualg_4.html#NM5">MSCongruence</a> of <font color="Maroon">F<sub>2</sub></font>()
 
<b>by </b><i><a class="ref" href="msualg_5.html#T20">MSUALG_5:20</a></i>;<br/>
<a NAME="E13:3_1"/><b>then </b><i><font color="Green">E11</font></i>: 
<span class="p1"><a href="pboole.html#K10">[|</a><span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>(),the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>()</span><a href="pboole.html#K10">|]</a></span> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K6">CongrLatt</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span>
 
<b>by </b><i><a class="ref" href="msualg_5.html#D6">MSUALG_5:def 6</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <span class="p1"><a href="pboole.html#K10">[|</a><span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>(),the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>()</span><a href="pboole.html#K10">|]</a></span> as   <a href="msualg_4.html#NM5">MSCongruence</a> of <font color="Maroon">F<sub>2</sub></font>() <b>by </b><i><a class="ref" href="msualg_5.html#T20">MSUALG_5:20</a></i>;<br/>
<a NAME="E15:3_1"/>
the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msualg_4.html#K14">QuotMSAlg</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="pboole.html#R6">=</a> <span class="p1"><a href="pzfmisc1.html#K1">{</a><span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>()</span><a href="pzfmisc1.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="msualg_9.html#T30">MSUALG_9:30</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>8</sub></font> being  <a href="relat_1.html#V2">V5</a>  <a href="pboole.html#M1">ManySortedSet</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>()<b> such that </b><br/><a NAME="E17:3_1"/><i><font color="Green">E12</font></i>: 
the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msualg_4.html#K14">QuotMSAlg</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="pboole.html#R6">=</a> <span class="p1"><a href="pzfmisc1.html#K1">{</a><span class="default"><font color="Maroon">c<sub>8</sub></font></span><a href="pzfmisc1.html#K1">}</a></span>
 ;<br/>
<a NAME="E18:3_1"/>
 <a href="msualg_4.html#K14">QuotMSAlg</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">c<sub>7</sub></font>, <a href="msafree2.html#K8">Trivial_Algebra</a> <font color="Maroon">F<sub>1</sub></font>() <a href="msualg_3.html#R6">are_isomorphic</a> 
 
<b>by </b><i><a class="txt" href="birkhoff.html#E17:3_1"><i><font color="Green">E12</font></i></a>, <a class="ref" href="msualg_9.html#T27">MSUALG_9:27</a></i>;<br/>
<a NAME="E19:3_1"/><b>then </b>
 <a href="msafree2.html#K8">Trivial_Algebra</a> <font color="Maroon">F<sub>1</sub></font>(), <a href="msualg_4.html#K14">QuotMSAlg</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">c<sub>7</sub></font> <a href="msualg_3.html#R6">are_isomorphic</a> 
 
<b>by </b><i><a class="ref" href="msualg_3.html#T17">MSUALG_3:17</a></i>;<br/>
<a NAME="E20:3_1"/><b>then </b>
<font color="Maroon">P<sub>1</sub></font>[ <a href="msualg_4.html#K14">QuotMSAlg</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">c<sub>7</sub></font>]
 
<b>by </b><i><a class="txt" href="birkhoff.html#E1:3"><i><font color="Green">E3</font></i></a>, <a class="txt" href="birkhoff.html#E11:3_1"><i><font color="Green">E10</font></i></a></i>;<br/>
<a NAME="E21:3_1"/><b>then </b><i><font color="Green">E13</font></i>: 
<span class="p1"><a href="pboole.html#K10">[|</a><span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>(),the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>()</span><a href="pboole.html#K10">|]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K6">CongrLatt</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> : ex <font color="Olive">b<sub>1</sub></font> being  <a href="msualg_4.html#NM5">MSCongruence</a> of <font color="Maroon">F<sub>2</sub></font>() st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">P<sub>1</sub></font>[ <a href="msualg_4.html#K14">QuotMSAlg</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Olive">b<sub>2</sub></font>] ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="birkhoff.html#E13:3_1"><i><font color="Green">E11</font></i></a></i>;<br/>
<a NAME="E22:3_1"/><i><font color="Green">E14</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K6">CongrLatt</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> : ex <font color="Olive">b<sub>1</sub></font> being  <a href="msualg_4.html#NM5">MSCongruence</a> of <font color="Maroon">F<sub>2</sub></font>() st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">P<sub>1</sub></font>[ <a href="msualg_4.html#K14">QuotMSAlg</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Olive">b<sub>2</sub></font>] ) </span>}</span>  <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K5">EqRelLatt</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>()</span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:3_1_3"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K6">CongrLatt</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> : ex <font color="Olive">b<sub>1</sub></font> being  <a href="msualg_4.html#NM5">MSCongruence</a> of <font color="Maroon">F<sub>2</sub></font>() st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">P<sub>1</sub></font>[ <a href="msualg_4.html#K14">QuotMSAlg</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Olive">b<sub>2</sub></font>] ) </span>}</span> 
 ;<br/>

<a NAME="E2:3_1_3"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> is   <a href="msualg_5.html#NM1">Equivalence_Relation</a> of the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>()
 
<b>by </b><i><a class="txt" href="birkhoff.html#E2:3_1"><i><font color="Green">E7</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:3_1_3"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K5">EqRelLatt</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>()</span>)</span>
 <b>by </b><i><a class="ref" href="msualg_5.html#D5">MSUALG_5:def 5</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>9</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K6">CongrLatt</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> : ex <font color="Olive">b<sub>1</sub></font> being  <a href="msualg_4.html#NM5">MSCongruence</a> of <font color="Maroon">F<sub>2</sub></font>() st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">P<sub>1</sub></font>[ <a href="msualg_4.html#K14">QuotMSAlg</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Olive">b<sub>2</sub></font>] ) </span>}</span>  as  non <a href="xboole_0.html#V1">empty</a> <a href="closure2.html#NM1">SubsetFamily</a> of <span class="p1"><a href="pboole.html#K10">[|</a><span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>(),the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>()</span><a href="pboole.html#K10">|]</a></span> <b>by </b><i><a class="txt" href="birkhoff.html#E21:3_1"><i><font color="Green">E13</font></i></a>, <a class="ref" href="msualg_7.html#T6">MSUALG_7:6</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>10</sub></font> =  <a href="mssubfam.html#K6">meet</a> <span class="p1"><a href="closure2.html#K5">|:</a><span class="default"><font color="Maroon">c<sub>9</sub></font></span><a href="closure2.html#K5">:|</a></span>;<br/>
<a NAME="E24:3_1"/><i><font color="Green">E15</font></i>: 
 <a href="mssubfam.html#K6">meet</a> <span class="p1"><a href="closure2.html#K5">|:</a><span class="default"><font color="Maroon">c<sub>9</sub></font></span><a href="closure2.html#K5">:|</a></span> is  <a href="msualg_4.html#V2">MSEquivalence_Relation-like</a> <a href="msualg_4.html#NM3">ManySortedRelation</a> of the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>()
 
<b>by </b><i><a class="txt" href="birkhoff.html#E22:3_1"><i><font color="Green">E14</font></i></a>, <a class="ref" href="msualg_7.html#T9">MSUALG_7:9</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>11</sub></font> =  <a href="mssubfam.html#K6">meet</a> <span class="p1"><a href="closure2.html#K5">|:</a><span class="default"><font color="Maroon">c<sub>9</sub></font></span><a href="closure2.html#K5">:|</a></span> as   <a href="msualg_4.html#NM4">ManySortedRelation</a> of <font color="Maroon">F<sub>2</sub></font>() <b>by </b><i><a class="txt" href="birkhoff.html#E22:3_1"><i><font color="Green">E14</font></i></a>, <a class="ref" href="msualg_7.html#T9">MSUALG_7:9</a></i>;<br/>
<a NAME="E26:3_1"/>
<font color="Maroon">c<sub>11</sub></font> is <a href="msualg_4.html#V3">MSEquivalence-like</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>12</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="msualg_4.html#D3">MSUALG_4:def 3</a>,<a class="ref" href="msualg_4.html#D5">MSUALG_4:def 5</a><br/><b>let </b><font color="Maroon">c<sub>13</sub></font> be   <a href="relset_1.html#NM3">Relation</a> of the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font>;<br/>



<b>assume </b><a NAME="E1:3_1_4"/>
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> )
 ;<br/>

<b>hence </b><a NAME="E2:3_1_4"/>
<font color="Maroon">c<sub>13</sub></font> is   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="birkhoff.html#E24:3_1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="msualg_4.html#D3">MSUALG_4:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>11</sub></font> as  <a href="msualg_4.html#V3">MSEquivalence-like</a> <a href="msualg_4.html#NM4">ManySortedRelation</a> of <font color="Maroon">F<sub>2</sub></font>() ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>12</sub></font> as   <a href="msualg_4.html#NM5">MSCongruence</a> of <font color="Maroon">F<sub>2</sub></font>() <b>by </b><i><a class="txt" href="birkhoff.html#E1:3_1"><i><font color="Green">E6</font></i></a>, <a class="ref" href="msualg_9.html#T29">MSUALG_9:29</a></i>;<br/>
<b>take </b><font color="Maroon">c<sub>14</sub></font> =  <a href="msualg_4.html#K14">QuotMSAlg</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">c<sub>13</sub></font>;<br/>

<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] means ex <font color="Olive">b<sub>1</sub></font> being  <a href="msualg_4.html#NM5">MSCongruence</a> of <font color="Maroon">F<sub>2</sub></font>() st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="msualg_4.html#K14">QuotMSAlg</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Olive">b<sub>1</sub></font> );<br/>
<div><i><font color="Green">E16</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>15</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:3_1_5"/>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>15</sub></font> as   <a href="msualg_4.html#NM5">MSCongruence</a> of <font color="Maroon">F<sub>2</sub></font>() <b>by </b><i><a class="txt" href="birkhoff.html#E2:3_1"><i><font color="Green">E7</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>17</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:3_1_5"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a>  <a href="msualg_4.html#K14">QuotMSAlg</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">c<sub>16</sub></font>
 ;<br/><b>take </b><font color="Maroon">c<sub>18</sub></font> = <font color="Maroon">c<sub>17</sub></font>;<br/><b>thus </b><a NAME="E5:3_1_5"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>18</sub></font>]
 <b>by </b><i><a class="txt" href="birkhoff.html#E4:3_1_5"><i><font color="Green">E17</font></i></a></i>;<br/></div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>15</sub></font> being    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">c<sub>9</sub></font><b> such that </b><br/><a NAME="E31:3_1"/><i><font color="Green">E17</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>15</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="pboole.html#S3">PBOOLE:sch 3</a>(<a class="txt" href="birkhoff.html#E29:3_1"><i><font color="Green">E16</font></i></a>);<br/></i>
<a NAME="E32:3_1"/>
<font color="Maroon">c<sub>15</sub></font> is    <a href="pralg_2.html#M2">MSAlgebra-Family</a> of <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">F<sub>1</sub></font>()
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>16</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="pralg_2.html#D12">PRALG_2:def 12</a><br/>

<b>assume </b><a NAME="E1:3_1_6"/>
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>17</sub></font> being   <a href="msualg_4.html#NM5">MSCongruence</a> of <font color="Maroon">F<sub>2</sub></font>()<b> such that </b><br/><a NAME="E3:3_1_6"/><i><font color="Green">E18</font></i>: 
( <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a>  <a href="msualg_4.html#K14">QuotMSAlg</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">c<sub>17</sub></font> )
 <b>by </b><i><a class="txt" href="birkhoff.html#E31:3_1"><i><font color="Green">E17</font></i></a></i>;<br/>
<b>thus </b><a NAME="E4:3_1_6"/>
<font color="Maroon">c<sub>15</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font> is  <a href="msualg_1.html#V5">non-empty</a>  <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">F<sub>1</sub></font>()
 <b>by </b><i><a class="txt" href="birkhoff.html#E3:3_1_6"><i><font color="Green">E18</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>15</sub></font> as    <a href="pralg_2.html#M2">MSAlgebra-Family</a> of <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">F<sub>1</sub></font>() ;<br/>
<a NAME="E34:3_1"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> holds <br/>ex <font color="Olive">b<sub>2</sub></font> being   <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">F<sub>1</sub></font>() st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">b<sub>2</sub></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">c<sub>17</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:3_1_7"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>18</sub></font> = <font color="Maroon">c<sub>17</sub></font> as    <a href="closure2.html#M1">Element</a> of <font color="Maroon">c<sub>9</sub></font> ;<br/>
<b>take </b><font color="Maroon">c<sub>19</sub></font> = <font color="Maroon">c<sub>16</sub></font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">c<sub>18</sub></font>;<br/>

<b>consider </b><font color="Maroon">c<sub>20</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K6">CongrLatt</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span><b> such that </b><br/><a NAME="E4:3_1_7"/><i><font color="Green">E19</font></i>: 
( <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> &amp; ex <font color="Olive">b<sub>1</sub></font> being  <a href="msualg_4.html#NM5">MSCongruence</a> of <font color="Maroon">F<sub>2</sub></font>() st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> &amp; <font color="Maroon">P<sub>1</sub></font>[ <a href="msualg_4.html#K14">QuotMSAlg</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Olive">b<sub>1</sub></font>] ) )
 <b>by </b><i><a class="txt" href="birkhoff.html#E1:3_1_7"><i><font color="Green">E18</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>21</sub></font> being   <a href="msualg_4.html#NM5">MSCongruence</a> of <font color="Maroon">F<sub>2</sub></font>()<b> such that </b><br/><a NAME="E6:3_1_7"/><i><font color="Green">E20</font></i>: 
( <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a>  <a href="msualg_4.html#K14">QuotMSAlg</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">c<sub>21</sub></font> )
 <b>by </b><i><a class="txt" href="birkhoff.html#E31:3_1"><i><font color="Green">E17</font></i></a>, <a class="txt" href="birkhoff.html#E1:3_1_7"><i><font color="Green">E18</font></i></a></i>;<br/>
<b>thus </b><a NAME="E7:3_1_7"/>
( <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>19</sub></font>] )
 <b>by </b><i><a class="txt" href="birkhoff.html#E4:3_1_7"><i><font color="Green">E19</font></i></a>, <a class="txt" href="birkhoff.html#E6:3_1_7"><i><font color="Green">E20</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E35:3_1"/><b>then </b><i><font color="Green">E18</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[ <a href="pralg_2.html#K15">product</a> <font color="Maroon">c<sub>16</sub></font>]
 
<b>by </b><i><a class="txt" href="birkhoff.html#E3:3"><i><font color="Green">E5</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>17</sub></font> =  <a href="msualg_4.html#K16">MSNat_Hom</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">c<sub>13</sub></font> as   <a href="msualg_3.html#NM1">ManySortedFunction</a> of <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">c<sub>14</sub></font> ;<br/>
<b>take </b>
<font color="Maroon">c<sub>17</sub></font>
;<br/>

<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="closure2.html#M1">Element</a> of <font color="Maroon">c<sub>9</sub></font>,   <a href="hidden.html#M1">set</a> ] means ex <font color="Olive">b<sub>1</sub></font> being  <a href="msualg_3.html#NM1">ManySortedFunction</a> of <font color="Maroon">F<sub>2</sub></font>(),<span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>2</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="msualg_3.html#R1">is_homomorphism</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">c<sub>16</sub></font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">a<sub>1</sub></font> &amp; ex <font color="Olive">b<sub>2</sub></font> being  <a href="msualg_4.html#NM5">MSCongruence</a> of <font color="Maroon">F<sub>2</sub></font>() st <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="pboole.html#R6">=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="pboole.html#R6">=</a>  <a href="msualg_4.html#K16">MSNat_Hom</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Olive">b<sub>2</sub></font> ) );<br/>
<a NAME="E37:3_1"/><i><font color="Green">E19</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="closure2.html#M1">Element</a> of <font color="Maroon">c<sub>9</sub></font>ex <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>18</sub></font> be    <a href="closure2.html#M1">Element</a> of <font color="Maroon">c<sub>9</sub></font>;<br/>

<b>consider </b><font color="Maroon">c<sub>19</sub></font> being   <a href="msualg_4.html#NM5">MSCongruence</a> of <font color="Maroon">F<sub>2</sub></font>()<b> such that </b><br/><a NAME="E2:3_1_8"/><i><font color="Green">E20</font></i>: 
( <font color="Maroon">c<sub>19</sub></font> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a>  <a href="msualg_4.html#K14">QuotMSAlg</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">c<sub>19</sub></font> )
 <b>by </b><i><a class="txt" href="birkhoff.html#E31:3_1"><i><font color="Green">E17</font></i></a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>20</sub></font> =  <a href="msualg_4.html#K16">MSNat_Hom</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">c<sub>19</sub></font>;<br/>
<b>take </b>
 <a href="msualg_4.html#K16">MSNat_Hom</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">c<sub>19</sub></font>
;<br/>

<b>reconsider </b><font color="Maroon">c<sub>21</sub></font> =  <a href="msualg_4.html#K16">MSNat_Hom</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">c<sub>19</sub></font> as   <a href="msualg_3.html#NM1">ManySortedFunction</a> of <font color="Maroon">F<sub>2</sub></font>(),<span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <b>by </b><i><a class="txt" href="birkhoff.html#E2:3_1_8"><i><font color="Green">E20</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>21</sub></font>
;<br/>

<b>thus </b><a NAME="E4:3_1_8"/>
<font color="Maroon">c<sub>21</sub></font> <a href="pboole.html#R6">=</a>  <a href="msualg_4.html#K16">MSNat_Hom</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">c<sub>19</sub></font>
 ;<br/>

<a NAME="E5:3_1_8"/>
 <a href="msualg_4.html#K16">MSNat_Hom</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">c<sub>19</sub></font> <a href="msualg_3.html#R2">is_epimorphism</a> <font color="Maroon">F<sub>2</sub></font>(), <a href="msualg_4.html#K14">QuotMSAlg</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">c<sub>19</sub></font>
 
<b>by </b><i><a class="ref" href="msualg_4.html#T3">MSUALG_4:3</a></i>;<br/>
<b>hence </b><a NAME="E6:3_1_8"/>
<font color="Maroon">c<sub>21</sub></font> <a href="msualg_3.html#R1">is_homomorphism</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">c<sub>16</sub></font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">c<sub>18</sub></font>
 <b>by </b><i><a class="txt" href="birkhoff.html#E2:3_1_8"><i><font color="Green">E20</font></i></a>, <a class="ref" href="msualg_3.html#D10">MSUALG_3:def 10</a></i>;<br/>

<b>take </b>
<font color="Maroon">c<sub>19</sub></font>
;<br/>

<b>thus </b><a NAME="E7:3_1_8"/>
( <font color="Maroon">c<sub>18</sub></font> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>19</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="pboole.html#R6">=</a>  <a href="msualg_4.html#K16">MSNat_Hom</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">c<sub>19</sub></font> )
 <b>by </b><i><a class="txt" href="birkhoff.html#E2:3_1_8"><i><font color="Green">E20</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>18</sub></font> being    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">c<sub>9</sub></font><b> such that </b><br/><a NAME="E39:3_1"/><i><font color="Green">E20</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="closure2.html#M1">Element</a> of <font color="Maroon">c<sub>9</sub></font> holds <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>18</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="msualg_9.html#S1">MSUALG_9:sch 1</a>(<a class="txt" href="birkhoff.html#E37:3_1"><i><font color="Green">E19</font></i></a>);<br/></i>
<a NAME="E40:3_1"/><i><font color="Green">E21</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="closure2.html#M1">Element</a> of <font color="Maroon">c<sub>9</sub></font>ex <font color="Olive">b<sub>2</sub></font> being  <a href="msualg_3.html#NM1">ManySortedFunction</a> of <font color="Maroon">F<sub>2</sub></font>(),<span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="pralg_2.html#K6">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="msualg_3.html#R1">is_homomorphism</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">c<sub>16</sub></font> <a href="pralg_2.html#K6">.</a> <font color="Olive">b<sub>1</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_9"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>19</sub></font> be    <a href="closure2.html#M1">Element</a> of <font color="Maroon">c<sub>9</sub></font>;<br/>

<b>consider </b><font color="Maroon">c<sub>20</sub></font> being   <a href="msualg_3.html#NM1">ManySortedFunction</a> of <font color="Maroon">F<sub>2</sub></font>(),<span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span><b> such that </b><br/><a NAME="E2:3_1_9"/><i><font color="Green">E22</font></i>: 
( <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="msualg_3.html#R1">is_homomorphism</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">c<sub>16</sub></font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">c<sub>19</sub></font> )
 <b>and </b><br/><a NAME="E3:3_1_9"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="msualg_4.html#NM5">MSCongruence</a> of <font color="Maroon">F<sub>2</sub></font>() st <br/>( <font color="Maroon">c<sub>19</sub></font> <a href="pboole.html#R6">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="pboole.html#R6">=</a>  <a href="msualg_4.html#K16">MSNat_Hom</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Olive">b<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="birkhoff.html#E39:3_1"><i><font color="Green">E20</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>20</sub></font>
;<br/>

<b>thus </b><a NAME="E4:3_1_9"/>
( <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="msualg_3.html#R1">is_homomorphism</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">c<sub>16</sub></font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">c<sub>19</sub></font> )
 <b>by </b><i><a class="txt" href="birkhoff.html#E2:3_1_9"><i><font color="Green">E22</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E41:3_1"/>
<font color="Maroon">c<sub>18</sub></font> is <a href="funcop_1.html#V1">Function-yielding</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_10"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>19</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="funcop_1.html#D6">FUNCOP_1:def 6</a><br/>

<b>assume </b><a NAME="E1:3_1_10"/>
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>18</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>20</sub></font> = <font color="Maroon">c<sub>19</sub></font> as    <a href="closure2.html#M1">Element</a> of <font color="Maroon">c<sub>9</sub></font> <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>21</sub></font> being   <a href="msualg_3.html#NM1">ManySortedFunction</a> of <font color="Maroon">F<sub>2</sub></font>(),<span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span><b> such that </b><br/><a NAME="E4:3_1_10"/><i><font color="Green">E22</font></i>: 
( <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="msualg_3.html#R1">is_homomorphism</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">c<sub>16</sub></font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">c<sub>20</sub></font> )
 <b>by </b><i><a class="txt" href="birkhoff.html#E40:3_1"><i><font color="Green">E21</font></i></a></i>;<br/>
<b>thus </b><a NAME="E5:3_1_10"/>
<font color="Maroon">c<sub>18</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font> is   <a href="funct_1.html#NM1">Function</a>
 <b>by </b><i><a class="txt" href="birkhoff.html#E4:3_1_10"><i><font color="Green">E22</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>19</sub></font> = <font color="Maroon">c<sub>18</sub></font> as   <a href="pboole.html#NM2">ManySortedFunction</a> of <font color="Maroon">c<sub>9</sub></font> ;<br/>
<b>consider </b><font color="Maroon">c<sub>20</sub></font> being   <a href="msualg_3.html#NM1">ManySortedFunction</a> of <font color="Maroon">F<sub>2</sub></font>(),<span class="p1">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">c<sub>16</sub></font></span>)</span><b> such that </b><br/><a NAME="E44:3_1"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="msualg_3.html#R1">is_homomorphism</a> <font color="Maroon">F<sub>2</sub></font>(), <a href="pralg_2.html#K15">product</a> <font color="Maroon">c<sub>16</sub></font>
 <b>and </b><br/><a NAME="E45:3_1"/><i><font color="Green">E23</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="closure2.html#M1">Element</a> of <font color="Maroon">c<sub>9</sub></font> holds <span class="p1">(<span class="default"><a href="pralg_3.html#K4">proj</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Olive">b<sub>1</sub></font></span>)</span> <a href="msualg_3.html#K3">**</a> <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="birkhoff.html#E40:3_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="pralg_3.html#T30">PRALG_3:30</a></i>;<br/>
<a NAME="E46:3_1"/>
 <a href="msualg_4.html#K20">MSHomQuot</a> <font color="Maroon">c<sub>20</sub></font> <a href="msualg_3.html#R3">is_monomorphism</a>  <a href="msualg_4.html#K14">QuotMSAlg</a> <font color="Maroon">F<sub>2</sub></font>(),<span class="p1">(<span class="default"><a href="msualg_4.html#K18">MSCng</a> <font color="Maroon">c<sub>20</sub></font></span>)</span>, <a href="pralg_2.html#K15">product</a> <font color="Maroon">c<sub>16</sub></font>
 
<b>by </b><i><a class="txt" href="birkhoff.html#E44:3_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="msualg_4.html#T4">MSUALG_4:4</a></i>;<br/>
<a NAME="E47:3_1"/><b>then </b><i><font color="Green">E24</font></i>: 
 <a href="msualg_4.html#K14">QuotMSAlg</a> <font color="Maroon">F<sub>2</sub></font>(),<span class="p1">(<span class="default"><a href="msualg_4.html#K18">MSCng</a> <font color="Maroon">c<sub>20</sub></font></span>)</span>, <a href="msualg_3.html#K7">Image</a> <span class="p1">(<span class="default"><a href="msualg_4.html#K20">MSHomQuot</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="msualg_3.html#R6">are_isomorphic</a> 
 
<b>by </b><i><a class="ref" href="msualg_9.html#T17">MSUALG_9:17</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_11"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>21</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:3_1_11"/>
<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>()
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>22</sub></font> = <font color="Maroon">c<sub>21</sub></font> as   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">F<sub>1</sub></font>() ;<br/><b>consider </b><font color="Maroon">c<sub>23</sub></font> being   <a href="setfam_1.html#NM1">Subset-Family</a> of <span class="p1">(<span class="default"><span class="p2"><a href="pboole.html#K10">[|</a><span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>(),the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>()</span><a href="pboole.html#K10">|]</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font></span>)</span><b> such that </b><br/><a NAME="E4:3_1_11"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="closure2.html#K5">|:</a><span class="default"><font color="Maroon">c<sub>9</sub></font></span><a href="closure2.html#K5">:|</a></span> <a href="mssubfam.html#K4">.</a> <font color="Maroon">c<sub>22</sub></font>
 <b>and </b><br/><a NAME="E5:3_1_11"/><i><font color="Green">E26</font></i>: 
<span class="p1">(<span class="default"><a href="mssubfam.html#K6">meet</a> <span class="p2"><a href="closure2.html#K5">|:</a><span class="default"><font color="Maroon">c<sub>9</sub></font></span><a href="closure2.html#K5">:|</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">c<sub>23</sub></font>
 <b>by </b><i><a class="ref" href="mssubfam.html#D2">MSSUBFAM:def 2</a></i>;<br/><a NAME="E6:3_1_11"/><i><font color="Green">E27</font></i>: 
<span class="p1"><a href="closure2.html#K5">|:</a><span class="default"><font color="Maroon">c<sub>9</sub></font></span><a href="closure2.html#K5">:|</a></span> <a href="mssubfam.html#K4">.</a> <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> where B is    <a href="closure2.html#M1">Element</a> of  <a href="closure2.html#K6">Bool</a> <span class="p1"><a href="pboole.html#K10">[|</a><span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>(),the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>()</span><a href="pboole.html#K10">|]</a></span> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> </span>}</span> 
 <b>by </b><i><a class="ref" href="closure2.html#T15">CLOSURE2:15</a></i>;<br/><a NAME="E7:3_1_11"/>
<span class="p1">(<span class="default"><a href="msualg_4.html#K18">MSCng</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="msualg_4.html#K2">.</a> <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> <a href="msualg_4.html#K2">.</a> <font color="Maroon">c<sub>22</sub></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">c<sub>24</sub></font>, <font color="Maroon">c<sub>25</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="relat_1.html#D2">RELAT_1:def 2</a><br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:3_1_11_1_1"/><i><font color="Green">E28</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>24</sub></font>,<font color="Maroon">c<sub>25</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="msualg_4.html#K18">MSCng</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="msualg_4.html#K2">.</a> <font color="Maroon">c<sub>22</sub></font>
 ;<br/><a NAME="E2:3_1_11_1_1"/><b>then </b><i><font color="Green">E29</font></i>: 
( <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E3:3_1_11_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>24</sub></font>,<font color="Maroon">c<sub>25</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font></span>)</span>,<span class="p2">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="txt" href="birkhoff.html#E1:3_1_11_1_1"><i><font color="Green">E28</font></i></a></i>;<br/><a NAME="E4:3_1_11_1_1"/><b>then </b><i><font color="Green">E30</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>24</sub></font>,<font color="Maroon">c<sub>25</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="pboole.html#K10">[|</a><span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>(),the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>()</span><a href="pboole.html#K10">|]</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font>
 <b>by </b><i><a class="ref" href="pboole.html#D21">PBOOLE:def 21</a></i>;<br/><a NAME="E5:3_1_11_1_1"/><i><font color="Green">E31</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>24</sub></font>,<font color="Maroon">c<sub>25</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="msualg_4.html#K17">MSCng</a> <font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>22</sub></font>
 <b>by </b><i><a class="txt" href="birkhoff.html#E44:3_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="birkhoff.html#E1:3_1_11_1_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="msualg_4.html#D20">MSUALG_4:def 20</a></i>;<br/><a NAME="E6:3_1_11_1_1"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>23</sub></font> holds <br/><span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>24</sub></font>,<font color="Maroon">c<sub>25</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></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">c<sub>26</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:3_1_11_1_1_1"/>
<font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>23</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>27</sub></font> being    <a href="closure2.html#M1">Element</a> of  <a href="closure2.html#K6">Bool</a> <span class="p1"><a href="pboole.html#K10">[|</a><span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>(),the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>()</span><a href="pboole.html#K10">|]</a></span><b> such that </b><br/><a NAME="E3:3_1_11_1_1_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>27</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font>
 <b>and </b><br/><a NAME="E4:3_1_11_1_1_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="birkhoff.html#E4:3_1_11"><i><font color="Green">E25</font></i></a>, <a class="txt" href="birkhoff.html#E6:3_1_11"><i><font color="Green">E27</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>28</sub></font> = <font color="Maroon">c<sub>27</sub></font> as    <a href="closure2.html#M1">Element</a> of <font color="Maroon">c<sub>9</sub></font> <b>by </b><i><a class="txt" href="birkhoff.html#E4:3_1_11_1_1_1"><i><font color="Green">E33</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>29</sub></font> being   <a href="msualg_3.html#NM1">ManySortedFunction</a> of <font color="Maroon">F<sub>2</sub></font>(),<span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">c<sub>28</sub></font></span>)</span><b> such that </b><br/><a NAME="E7:3_1_11_1_1_1"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">c<sub>29</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>28</sub></font> &amp; <font color="Maroon">c<sub>29</sub></font> <a href="msualg_3.html#R1">is_homomorphism</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">c<sub>16</sub></font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">c<sub>28</sub></font> &amp; ex <font color="Olive">b<sub>1</sub></font> being  <a href="msualg_4.html#NM5">MSCongruence</a> of <font color="Maroon">F<sub>2</sub></font>() st <br/>( <font color="Maroon">c<sub>28</sub></font> <a href="pboole.html#R6">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>29</sub></font> <a href="pboole.html#R6">=</a>  <a href="msualg_4.html#K16">MSNat_Hom</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Olive">b<sub>1</sub></font> ) )
 <b>by </b><i><a class="txt" href="birkhoff.html#E39:3_1"><i><font color="Green">E20</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>30</sub></font> being   <a href="msualg_4.html#NM5">MSCongruence</a> of <font color="Maroon">F<sub>2</sub></font>()<b> such that </b><br/><a NAME="E9:3_1_11_1_1_1"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">c<sub>28</sub></font> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>30</sub></font> &amp; <font color="Maroon">c<sub>29</sub></font> <a href="pboole.html#R6">=</a>  <a href="msualg_4.html#K16">MSNat_Hom</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">c<sub>30</sub></font> )
 <b>by </b><i><a class="txt" href="birkhoff.html#E7:3_1_11_1_1_1"><i><font color="Green">E34</font></i></a></i>;<br/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>29</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pralg_3.html#K4">proj</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>28</sub></font></span>)</span> <a href="msualg_3.html#K3">**</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font>
<b>by </b><i><a class="txt" href="birkhoff.html#E45:3_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="birkhoff.html#E7:3_1_11_1_1_1"><i><font color="Green">E34</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">proj</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>28</sub></font></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font>
<b>by </b><i><a class="ref" href="msualg_3.html#T2">MSUALG_3:2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_3.html#K4">proj</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>28</sub></font></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="birkhoff.html#E2:3_1_11_1_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_3.html#K4">proj</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>28</sub></font></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>25</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="birkhoff.html#E2:3_1_11_1_1"><i><font color="Green">E29</font></i></a>, <a class="txt" href="birkhoff.html#E5:3_1_11_1_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="msualg_4.html#D19">MSUALG_4:def 19</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pralg_3.html#K4">proj</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>28</sub></font></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>25</sub></font>
<b>by </b><i><a class="txt" href="birkhoff.html#E2:3_1_11_1_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pralg_3.html#K4">proj</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>28</sub></font></span>)</span> <a href="msualg_3.html#K3">**</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>25</sub></font>
<b>by </b><i><a class="ref" href="msualg_3.html#T2">MSUALG_3:2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>29</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>25</sub></font>
<b>by </b><i><a class="txt" href="birkhoff.html#E45:3_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="birkhoff.html#E7:3_1_11_1_1_1"><i><font color="Green">E34</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">[</a><span class="default"><font color="Maroon">c<sub>24</sub></font>,<font color="Maroon">c<sub>25</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>26</sub></font>
 <b>by </b><i><a class="txt" href="birkhoff.html#E2:3_1_11_1_1"><i><font color="Green">E29</font></i></a>, <a class="txt" href="birkhoff.html#E3:3_1_11_1_1_1"><i><font color="Green">E32</font></i></a>, <a class="txt" href="birkhoff.html#E9:3_1_11_1_1_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="msualg_9.html#T34">MSUALG_9:34</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E7:3_1_11_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>24</sub></font>,<font color="Maroon">c<sub>25</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>13</sub></font> <a href="msualg_4.html#K2">.</a> <font color="Maroon">c<sub>22</sub></font>
 <b>by </b><i><a class="txt" href="birkhoff.html#E5:3_1_11"><i><font color="Green">E26</font></i></a>, <a class="txt" href="birkhoff.html#E4:3_1_11_1_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="setfam_1.html#T58">SETFAM_1:58</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:3_1_11_1"/><i><font color="Green">E28</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>24</sub></font>,<font color="Maroon">c<sub>25</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>13</sub></font> <a href="msualg_4.html#K2">.</a> <font color="Maroon">c<sub>22</sub></font>
 ;<br/>

<a NAME="E3:3_1_11_1"/><b>then </b><i><font color="Green">E29</font></i>: 
( <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font> )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E4:3_1_11_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font>
 
<b>by </b><i><a class="txt" href="birkhoff.html#E3:3_1_11_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/>
<a NAME="E5:3_1_11_1"/><b>then </b><i><font color="Green">E30</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>22</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="pralg_2.html#D17">PRALG_2:def 17</a></i>;<br/>
<a NAME="E6:3_1_11_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K15">product</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font>
 
<b>by </b><i><a class="txt" href="birkhoff.html#E3:3_1_11_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/>
<a NAME="E7:3_1_11_1"/><b>then </b><i><font color="Green">E31</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>22</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="pralg_2.html#D17">PRALG_2:def 17</a></i>;<br/>
<a NAME="E8:3_1_11_1"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="closure2.html#M1">Element</a> of <font color="Maroon">c<sub>9</sub></font> holds <span class="p1">(<span class="default"><a href="pralg_3.html#K3">proj</a> <span class="p2">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>22</sub></font></span>)</span>,<font color="Olive">b<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pralg_3.html#K3">proj</a> <span class="p2">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>22</sub></font></span>)</span>,<font color="Olive">b<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>25</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_11_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>26</sub></font> be    <a href="closure2.html#M1">Element</a> of <font color="Maroon">c<sub>9</sub></font>;<br/>

<b>reconsider </b><font color="Maroon">c<sub>27</sub></font> = <font color="Maroon">c<sub>26</sub></font> as   <a href="msualg_4.html#NM5">MSCongruence</a> of <font color="Maroon">F<sub>2</sub></font>() <b>by </b><i><a class="txt" href="birkhoff.html#E2:3_1"><i><font color="Green">E7</font></i></a></i>;<br/>
<a NAME="E2:3_1_11_1_2"/>
<font color="Maroon">c<sub>27</sub></font> is    <a href="closure2.html#M1">Element</a> of  <a href="closure2.html#K6">Bool</a> <span class="p1"><a href="pboole.html#K10">[|</a><span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>(),the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>()</span><a href="pboole.html#K10">|]</a></span>
 
<b>by </b><i><a class="ref" href="closure2.html#D1">CLOSURE2:def 1</a></i>;<br/>
<a NAME="E3:3_1_11_1_2"/><b>then </b>
<font color="Maroon">c<sub>27</sub></font> <a href="msualg_4.html#K2">.</a> <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="closure2.html#K5">|:</a><span class="default"><font color="Maroon">c<sub>9</sub></font></span><a href="closure2.html#K5">:|</a></span> <a href="mssubfam.html#K4">.</a> <font color="Maroon">c<sub>22</sub></font>
 
<b>by </b><i><a class="txt" href="birkhoff.html#E6:3_1_11"><i><font color="Green">E27</font></i></a></i>;<br/>
<a NAME="E4:3_1_11_1_2"/><b>then </b><i><font color="Green">E32</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>24</sub></font>,<font color="Maroon">c<sub>25</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>27</sub></font> <a href="msualg_4.html#K2">.</a> <font color="Maroon">c<sub>22</sub></font>
 
<b>by </b><i><a class="txt" href="birkhoff.html#E4:3_1_11"><i><font color="Green">E25</font></i></a>, <a class="txt" href="birkhoff.html#E5:3_1_11"><i><font color="Green">E26</font></i></a>, <a class="txt" href="birkhoff.html#E2:3_1_11_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="setfam_1.html#T58">SETFAM_1:58</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>28</sub></font> being   <a href="msualg_3.html#NM1">ManySortedFunction</a> of <font color="Maroon">F<sub>2</sub></font>(),<span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">c<sub>26</sub></font></span>)</span><b> such that </b><br/><a NAME="E6:3_1_11_1_2"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>26</sub></font> &amp; <font color="Maroon">c<sub>28</sub></font> <a href="msualg_3.html#R1">is_homomorphism</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">c<sub>16</sub></font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">c<sub>26</sub></font> &amp; ex <font color="Olive">b<sub>1</sub></font> being  <a href="msualg_4.html#NM5">MSCongruence</a> of <font color="Maroon">F<sub>2</sub></font>() st <br/>( <font color="Maroon">c<sub>26</sub></font> <a href="pboole.html#R6">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>28</sub></font> <a href="pboole.html#R6">=</a>  <a href="msualg_4.html#K16">MSNat_Hom</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Olive">b<sub>1</sub></font> ) )
 <b>by </b><i><a class="txt" href="birkhoff.html#E39:3_1"><i><font color="Green">E20</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>29</sub></font> being   <a href="msualg_4.html#NM5">MSCongruence</a> of <font color="Maroon">F<sub>2</sub></font>()<b> such that </b><br/><a NAME="E8:3_1_11_1_2"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">c<sub>27</sub></font> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>29</sub></font> &amp; <font color="Maroon">c<sub>28</sub></font> <a href="pboole.html#R6">=</a>  <a href="msualg_4.html#K16">MSNat_Hom</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">c<sub>29</sub></font> )
 <b>by </b><i><a class="txt" href="birkhoff.html#E6:3_1_11_1_2"><i><font color="Green">E33</font></i></a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><a href="pralg_3.html#K3">proj</a> <span class="p2">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>22</sub></font></span>)</span>,<font color="Maroon">c<sub>26</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_3.html#K4">proj</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>26</sub></font></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="pralg_3.html#D3">PRALG_3:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pralg_3.html#K4">proj</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>26</sub></font></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font>
<b>by </b><i><a class="txt" href="birkhoff.html#E3:3_1_11_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pralg_3.html#K4">proj</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>26</sub></font></span>)</span> <a href="msualg_3.html#K3">**</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font>
<b>by </b><i><a class="ref" href="msualg_3.html#T2">MSUALG_3:2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>28</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font>
<b>by </b><i><a class="txt" href="birkhoff.html#E45:3_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="birkhoff.html#E6:3_1_11_1_2"><i><font color="Green">E33</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>28</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>25</sub></font>
<b>by </b><i><a class="txt" href="birkhoff.html#E3:3_1_11_1"><i><font color="Green">E29</font></i></a>, <a class="txt" href="birkhoff.html#E4:3_1_11_1_2"><i><font color="Green">E32</font></i></a>, <a class="txt" href="birkhoff.html#E8:3_1_11_1_2"><i><font color="Green">E34</font></i></a>, <a class="ref" href="msualg_9.html#T34">MSUALG_9:34</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pralg_3.html#K4">proj</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>26</sub></font></span>)</span> <a href="msualg_3.html#K3">**</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>25</sub></font>
<b>by </b><i><a class="txt" href="birkhoff.html#E45:3_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="birkhoff.html#E6:3_1_11_1_2"><i><font color="Green">E33</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">proj</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>26</sub></font></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>25</sub></font>
<b>by </b><i><a class="ref" href="msualg_3.html#T2">MSUALG_3:2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_3.html#K4">proj</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>26</sub></font></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>25</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="birkhoff.html#E3:3_1_11_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="pralg_3.html#K3">proj</a> <span class="p2">(<span class="default"><a href="pralg_2.html#K10">Carrier</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>22</sub></font></span>)</span>,<font color="Maroon">c<sub>26</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>25</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="pralg_3.html#D3">PRALG_3:def 3</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E9:3_1_11_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>25</sub></font>
 
<b>by </b><i><a class="txt" href="birkhoff.html#E5:3_1_11_1"><i><font color="Green">E30</font></i></a>, <a class="txt" href="birkhoff.html#E7:3_1_11_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="msualg_9.html#T15">MSUALG_9:15</a></i>;<br/>
<a NAME="E10:3_1_11_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>24</sub></font>,<font color="Maroon">c<sub>25</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="msualg_4.html#K17">MSCng</a> <font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>22</sub></font>
 
<b>by </b><i><a class="txt" href="birkhoff.html#E3:3_1_11_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="msualg_4.html#D19">MSUALG_4:def 19</a></i>;<br/>
<b>hence </b><a NAME="E11:3_1_11_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>24</sub></font>,<font color="Maroon">c<sub>25</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="msualg_4.html#K18">MSCng</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="msualg_4.html#K2">.</a> <font color="Maroon">c<sub>22</sub></font>
 <b>by </b><i><a class="txt" href="birkhoff.html#E44:3_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="msualg_4.html#D20">MSUALG_4:def 20</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E8:3_1_11"/>
<span class="p1">(<span class="default"><a href="msualg_4.html#K18">MSCng</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font>
 ;<br/></div><b>end;</b></div>
<a NAME="E49:3_1"/><b>then </b>
 <a href="msualg_4.html#K18">MSCng</a> <font color="Maroon">c<sub>20</sub></font> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>13</sub></font>
 
<b>by </b><i><a class="ref" href="pboole.html#T3">PBOOLE:3</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>21</sub></font> being  <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of  <a href="pralg_2.html#K15">product</a> <font color="Maroon">c<sub>16</sub></font><b> such that </b><br/><a NAME="E51:3_1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>21</sub></font> <a href="msualg_3.html#R6">are_isomorphic</a> 
 <b>by </b><i><a class="txt" href="birkhoff.html#E47:3_1"><i><font color="Green">E24</font></i></a></i>;<br/>
<a NAME="E52:3_1"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>14</sub></font> <a href="msualg_3.html#R6">are_isomorphic</a> 
 
<b>by </b><i><a class="txt" href="birkhoff.html#E51:3_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="msualg_3.html#T17">MSUALG_3:17</a></i>;<br/>
<a NAME="E53:3_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>21</sub></font>]
 
<b>by </b><i><a class="txt" href="birkhoff.html#E2:3"><i><font color="Green">E4</font></i></a>, <a class="txt" href="birkhoff.html#E35:3_1"><i><font color="Green">E18</font></i></a></i>;<br/>
<b>hence </b><a NAME="E54:3_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>14</sub></font>]
 <b>by </b><i><a class="txt" href="birkhoff.html#E1:3"><i><font color="Green">E3</font></i></a>, <a class="txt" href="birkhoff.html#E52:3_1"><i><font color="Green">E26</font></i></a></i>;<br/>

<b>thus </b><a NAME="E55:3_1"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="msualg_3.html#R2">is_epimorphism</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="ref" href="msualg_4.html#T3">MSUALG_4:3</a></i>;<br/>

<b>let </b><font color="Maroon">c<sub>22</sub></font> be  <a href="msualg_1.html#V5">non-empty</a>  <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">F<sub>1</sub></font>();<br/><b>let </b><font color="Maroon">c<sub>23</sub></font> be   <a href="msualg_3.html#NM1">ManySortedFunction</a> of <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">c<sub>22</sub></font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E56:3_1"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>23</sub></font> <a href="msualg_3.html#R1">is_homomorphism</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">c<sub>22</sub></font>
 <b>and </b><br/><a NAME="E57:3_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>22</sub></font>]
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>24</sub></font> =  <a href="msualg_3.html#K7">Image</a> <span class="p1">(<span class="default"><a href="msualg_4.html#K20">MSHomQuot</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> as  <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">c<sub>22</sub></font> ;<br/>
<a NAME="E59:3_1"/>
 <a href="msualg_4.html#K20">MSHomQuot</a> <font color="Maroon">c<sub>23</sub></font> <a href="msualg_3.html#R3">is_monomorphism</a>  <a href="msualg_4.html#K14">QuotMSAlg</a> <font color="Maroon">F<sub>2</sub></font>(),<span class="p1">(<span class="default"><a href="msualg_4.html#K18">MSCng</a> <font color="Maroon">c<sub>23</sub></font></span>)</span>,<font color="Maroon">c<sub>22</sub></font>
 
<b>by </b><i><a class="txt" href="birkhoff.html#E56:3_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="msualg_4.html#T4">MSUALG_4:4</a></i>;<br/>
<a NAME="E60:3_1"/><b>then </b>
 <a href="msualg_4.html#K14">QuotMSAlg</a> <font color="Maroon">F<sub>2</sub></font>(),<span class="p1">(<span class="default"><a href="msualg_4.html#K18">MSCng</a> <font color="Maroon">c<sub>23</sub></font></span>)</span>,<font color="Maroon">c<sub>24</sub></font> <a href="msualg_3.html#R6">are_isomorphic</a> 
 
<b>by </b><i><a class="ref" href="msualg_9.html#T17">MSUALG_9:17</a></i>;<br/>
<a NAME="E61:3_1"/><b>then </b><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>24</sub></font>, <a href="msualg_4.html#K14">QuotMSAlg</a> <font color="Maroon">F<sub>2</sub></font>(),<span class="p1">(<span class="default"><a href="msualg_4.html#K18">MSCng</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> <a href="msualg_3.html#R6">are_isomorphic</a> 
 
<b>by </b><i><a class="ref" href="msualg_3.html#T17">MSUALG_3:17</a></i>;<br/>
<a NAME="E62:3_1"/><i><font color="Green">E31</font></i>: 
 <a href="msualg_4.html#K18">MSCng</a> <font color="Maroon">c<sub>23</sub></font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K6">CongrLatt</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span>
 
<b>by </b><i><a class="ref" href="msualg_5.html#D6">MSUALG_5:def 6</a></i>;<br/>
<a NAME="E63:3_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>24</sub></font>]
 
<b>by </b><i><a class="txt" href="birkhoff.html#E2:3"><i><font color="Green">E4</font></i></a>, <a class="txt" href="birkhoff.html#E57:3_1"><i><font color="Green">E29</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">QuotMSAlg</a> <font color="Maroon">F<sub>2</sub></font>(),<span class="p1">(<span class="default"><a href="msualg_4.html#K18">MSCng</a> <font color="Maroon">c<sub>23</sub></font></span>)</span>]
 
<b>by </b><i><a class="txt" href="birkhoff.html#E1:3"><i><font color="Green">E3</font></i></a>, <a class="txt" href="birkhoff.html#E61:3_1"><i><font color="Green">E30</font></i></a></i>;<br/>
<a NAME="E65:3_1"/><b>then </b>
 <a href="msualg_4.html#K18">MSCng</a> <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="birkhoff.html#E62:3_1"><i><font color="Green">E31</font></i></a></i>;<br/>
<a NAME="E66:3_1"/><b>then </b>
 <a href="msualg_4.html#K18">MSCng</a> <font color="Maroon">c<sub>23</sub></font> <a href="pboole.html#R3">in</a> <span class="p1"><a href="closure2.html#K5">|:</a><span class="default"><font color="Maroon">c<sub>9</sub></font></span><a href="closure2.html#K5">:|</a></span>
 
<b>by </b><i><a class="ref" href="closure2.html#T22">CLOSURE2:22</a></i>;<br/>
<a NAME="E67:3_1"/><b>then </b>
<font color="Maroon">c<sub>13</sub></font> <a href="pboole.html#R2">c=</a>  <a href="msualg_4.html#K18">MSCng</a> <font color="Maroon">c<sub>23</sub></font>
 
<b>by </b><i><a class="ref" href="mssubfam.html#T43">MSSUBFAM:43</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>25</sub></font> being   <a href="msualg_3.html#NM1">ManySortedFunction</a> of <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>22</sub></font><b> such that </b><br/><a NAME="E69:3_1"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">c<sub>25</sub></font> <a href="msualg_3.html#R1">is_homomorphism</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>23</sub></font> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>25</sub></font> <a href="msualg_3.html#K3">**</a> <font color="Maroon">c<sub>17</sub></font> )
 <b>by </b><i><a class="txt" href="birkhoff.html#E56:3_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="msualg_9.html#T37">MSUALG_9:37</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>25</sub></font>
;<br/>

<b>thus </b><a NAME="E70:3_1"/>
<font color="Maroon">c<sub>25</sub></font> <a href="msualg_3.html#R1">is_homomorphism</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>22</sub></font>
 <b>by </b><i><a class="txt" href="birkhoff.html#E69:3_1"><i><font color="Green">E32</font></i></a></i>;<br/>

<b>thus </b><a NAME="E71:3_1"/>
<font color="Maroon">c<sub>23</sub></font> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>25</sub></font> <a href="msualg_3.html#K3">**</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="txt" href="birkhoff.html#E69:3_1"><i><font color="Green">E32</font></i></a></i>;<br/>

<b>let </b><font color="Maroon">c<sub>26</sub></font> be   <a href="msualg_3.html#NM1">ManySortedFunction</a> of <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>22</sub></font>;<br/>

<b>assume </b><a NAME="E72:3_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>26</sub></font> <a href="msualg_3.html#K3">**</a> <font color="Maroon">c<sub>17</sub></font> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>23</sub></font>
 ;<br/>

<a NAME="E73:3_1"/>
<font color="Maroon">c<sub>17</sub></font> is <a href="msualg_3.html#V2">"onto"</a>
 
<b>by </b><i><a class="txt" href="birkhoff.html#E55:3_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="msualg_3.html#D10">MSUALG_3:def 10</a></i>;<br/>
<b>hence </b><a NAME="E74:3_1"/>
<font color="Maroon">c<sub>25</sub></font> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>26</sub></font>
 <b>by </b><i><a class="txt" href="birkhoff.html#E69:3_1"><i><font color="Green">E32</font></i></a>, <a class="txt" href="birkhoff.html#E72:3_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="extens_1.html#T16">EXTENS_1:16</a></i>;<br/>


</div>
