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

<span class="kw">consider </span><font color="Maroon" title="c1">EM</font> being   <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> , <font color="Maroon" title="c2">EMF</font> being    <a href="pralg_2.html#M2" title="PRALG_2:mode.2">MSAlgebra-Family</a> of <font color="Maroon" title="c1">EM</font>,<font color="Maroon">F<sub>1</sub></font>();<br/>
<span class="kw">set </span><font color="Maroon" title="c3">CC</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">C</font> where <font color="Olive" title="b1">C</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K6" title="MSUALG_5:func.6">CongrLatt</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> :  ex <font color="Olive" title="b2">R</font> being  <a href="msualg_4.html#NM5" title="MSUALG_4:NM.5">MSCongruence</a> of <font color="Maroon">F<sub>2</sub></font>() st <br/>( <font color="Olive" title="b2">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">C</font> &amp; <font color="Maroon">P<sub>1</sub></font>[ <a href="msualg_4.html#K13" title="MSUALG_4:func.13">QuotMSAlg</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Olive" title="b2">R</font>] ) </span>}</span> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c4">SF</font> = the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>();<br/>
<span class="kw">set </span><font color="Maroon" title="c5">I</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>();<br/>
<span class="kw">consider </span><font color="Maroon" title="c6">Xi</font> being   <a href="pboole.html#NM1" title="PBOOLE:NM.1">ManySortedSet</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>()<span class="kw"> such that </span><br/><a NAME="E3:3_1"/><span class="lab"><font color="Green" title="E6">A4</font></span>: 
<span class="p1"><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">{</a><span class="default"><font color="Maroon" title="c6">Xi</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_9.html#T6" title="MSUALG_9:th.6">MSUALG_9:6</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c7">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() implies the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <font color="Maroon" title="c2">EMF</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">{</a><span class="default"><font color="Maroon" title="c6">Xi</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> )</span><br/><span class="kw">assume </span><a NAME="E1:3_1_1"/>
<font color="Maroon" title="c7">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>()
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <font color="Maroon" title="c2">EMF</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">{</a><span class="default"><font color="Maroon" title="c6">Xi</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font></span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">s</font> = <font color="Maroon" title="c7">i</font> as   <a href="msualg_1.html#NM1" title="MSUALG_1:NM.1">SortSymbol</a> of <font color="Maroon">F<sub>1</sub></font>() ;<br/><span class="kw">thus </span>the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <font color="Maroon" title="c2">EMF</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> = 
 <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K9" title="PRALG_2:func.9">Carrier</a> <font color="Maroon" title="c2">EMF</font>,<font color="Maroon" title="c8">s</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="pralg_2.html#D17" title="PRALG_2:def.17">PRALG_2:def 17</a></span>
<br/>.= 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_3.html#T19" title="CARD_3:th.19">CARD_3:19</a>, <a class="ref" href="pralg_2.html#D16" title="PRALG_2:def.16">PRALG_2:def 16</a></span>
<br/>.= 
<span class="p1">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">s</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>
<br/>.= 
<span class="p1"><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">{</a><span class="default"><font color="Maroon" title="c6">Xi</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:3_1"><span class="lab"><font color="Green" title="E6">A4</font></span></a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E5:3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E7">A5</font></span>: 
the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <font color="Maroon" title="c2">EMF</font></span>)</span> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <span class="p1"><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">{</a><span class="default"><font color="Maroon" title="c6">Xi</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pboole.html#T3" title="PBOOLE:th.3">PBOOLE:3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">R</font> = <span class="p1"><a href="pboole.html#K8" title="PBOOLE:func.8">[|</a><span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>(),the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>()</span><a href="pboole.html#K8" title="PBOOLE:func.8">|]</a></span> as   <a href="msualg_4.html#NM5" title="MSUALG_4:NM.5">MSCongruence</a> of <font color="Maroon">F<sub>2</sub></font>() <span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_5.html#T20" title="MSUALG_5:th.20">MSUALG_5:20</a></span>;<br/>
<a NAME="E7:3_1"/>
<span class="p1"><a href="pboole.html#K8" title="PBOOLE:func.8">[|</a><span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>(),the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>()</span><a href="pboole.html#K8" title="PBOOLE:func.8">|]</a></span> is   <a href="msualg_4.html#NM5" title="MSUALG_4:NM.5">MSCongruence</a> of <font color="Maroon">F<sub>2</sub></font>()
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_5.html#T20" title="MSUALG_5:th.20">MSUALG_5:20</a></span>;<br/>
<a NAME="E8:3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E8">A6</font></span>: 
<span class="p1"><a href="pboole.html#K8" title="PBOOLE:func.8">[|</a><span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>(),the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>()</span><a href="pboole.html#K8" title="PBOOLE:func.8">|]</a></span> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K6" title="MSUALG_5:func.6">CongrLatt</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_5.html#D6" title="MSUALG_5:def.6">MSUALG_5:def 6</a></span>;<br/>
<a NAME="E9:3_1"/>
the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="msualg_4.html#K13" title="MSUALG_4:func.13">QuotMSAlg</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon" title="c7">R</font></span>)</span> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <span class="p1"><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">{</a><span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>()</span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_9.html#T30" title="MSUALG_9:th.30">MSUALG_9:30</a></span>;<br/>
<a NAME="E10:3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E9">A7</font></span>: 
 <a href="msualg_4.html#K13" title="MSUALG_4:func.13">QuotMSAlg</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon" title="c7">R</font>, <a href="msafree2.html#K8" title="MSAFREE2:func.8">Trivial_Algebra</a> <font color="Maroon">F<sub>1</sub></font>() <a href="msualg_3.html#R6" title="MSUALG_3:pred.6">are_isomorphic</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_9.html#T27" title="MSUALG_9:th.27">MSUALG_9:27</a></span>;<br/>
<a NAME="E11:3_1"/>
for <font color="Olive" title="b1">i</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">EM</font> holds <br/> ex <font color="Olive" title="b2">A</font> being   <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon">F<sub>1</sub></font>() st <br/>( <font color="Olive" title="b2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">EMF</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b2">A</font>] )
 
;<br/>
<a NAME="E12:3_1"/><span class="kw">then </span>
<font color="Maroon">P<sub>1</sub></font>[ <a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <font color="Maroon" title="c2">EMF</font>]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:3"><span class="lab"><font color="Green" title="E5">A3</font></span></a></span>;<br/>
<a NAME="E13:3_1"/><span class="kw">then </span>
<font color="Maroon">P<sub>1</sub></font>[ <a href="msafree2.html#K8" title="MSAFREE2:func.8">Trivial_Algebra</a> <font color="Maroon">F<sub>1</sub></font>()]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:3"><span class="lab"><font color="Green" title="E3">A1</font></span></a>, <a class="txt" href="#E5:3_1"><span class="lab"><font color="Green" title="E7">A5</font></span></a>, <a class="ref" href="msualg_9.html#T27" title="MSUALG_9:th.27">MSUALG_9:27</a></span>;<br/>
<a NAME="E14:3_1"/><span class="kw">then </span>
<font color="Maroon">P<sub>1</sub></font>[ <a href="msualg_4.html#K13" title="MSUALG_4:func.13">QuotMSAlg</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon" title="c7">R</font>]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:3"><span class="lab"><font color="Green" title="E3">A1</font></span></a>, <a class="txt" href="#E10:3_1"><span class="lab"><font color="Green" title="E9">A7</font></span></a>, <a class="ref" href="msualg_3.html#T17" title="MSUALG_3:th.17">MSUALG_3:17</a></span>;<br/>
<a NAME="E15:3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E10">A8</font></span>: 
<span class="p1"><a href="pboole.html#K8" title="PBOOLE:func.8">[|</a><span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>(),the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>()</span><a href="pboole.html#K8" title="PBOOLE:func.8">|]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">C</font> where <font color="Olive" title="b1">C</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K6" title="MSUALG_5:func.6">CongrLatt</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> :  ex <font color="Olive" title="b2">R</font> being  <a href="msualg_4.html#NM5" title="MSUALG_4:NM.5">MSCongruence</a> of <font color="Maroon">F<sub>2</sub></font>() st <br/>( <font color="Olive" title="b2">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">C</font> &amp; <font color="Maroon">P<sub>1</sub></font>[ <a href="msualg_4.html#K13" title="MSUALG_4:func.13">QuotMSAlg</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Olive" title="b2">R</font>] ) </span>}</span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:3_1"><span class="lab"><font color="Green" title="E8">A6</font></span></a></span>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span> ex <font color="Olive" title="b1">R</font> being  <a href="msualg_4.html#NM5" title="MSUALG_4:NM.5">MSCongruence</a> of <font color="Maroon">F<sub>2</sub></font>() st <br/>( <font color="Olive" title="b1">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msualg_4.html#K13" title="MSUALG_4:func.13">QuotMSAlg</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Olive" title="b1">R</font> );<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span> ex <font color="Olive" title="b1">R</font> being  <a href="msualg_4.html#NM5" title="MSUALG_4:NM.5">MSCongruence</a> of <font color="Maroon">F<sub>2</sub></font>() st <br/>( <font color="Olive" title="b1">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 &amp; <font color="Maroon">P<sub>1</sub></font>[ <a href="msualg_4.html#K13" title="MSUALG_4:func.13">QuotMSAlg</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Olive" title="b1">R</font>] );<br/>
<div><span class="lab"><font color="Green" title="E11">A9</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c8">q</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">C</font> where <font color="Olive" title="b1">C</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K6" title="MSUALG_5:func.6">CongrLatt</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> :  ex <font color="Olive" title="b2">R</font> being  <a href="msualg_4.html#NM5" title="MSUALG_4:NM.5">MSCongruence</a> of <font color="Maroon">F<sub>2</sub></font>() st <br/>( <font color="Olive" title="b2">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">C</font> &amp; <font color="Maroon">P<sub>1</sub></font>[ <a href="msualg_4.html#K13" title="MSUALG_4:func.13">QuotMSAlg</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Olive" title="b2">R</font>] ) </span>}</span>  implies <font color="Maroon" title="c8">q</font> is   <a href="msualg_4.html#NM5" title="MSUALG_4:NM.5">MSCongruence</a> of <font color="Maroon">F<sub>2</sub></font>() )</span><br/><span class="kw">assume </span><a NAME="E1:3_1_2"/>
<font color="Maroon" title="c8">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">C</font> where <font color="Olive" title="b1">C</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K6" title="MSUALG_5:func.6">CongrLatt</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> :  ex <font color="Olive" title="b2">R</font> being  <a href="msualg_4.html#NM5" title="MSUALG_4:NM.5">MSCongruence</a> of <font color="Maroon">F<sub>2</sub></font>() st <br/>( <font color="Olive" title="b2">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">C</font> &amp; <font color="Maroon">P<sub>1</sub></font>[ <a href="msualg_4.html#K13" title="MSUALG_4:func.13">QuotMSAlg</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Olive" title="b2">R</font>] ) </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">q</font> is   <a href="msualg_4.html#NM5" title="MSUALG_4:NM.5">MSCongruence</a> of <font color="Maroon">F<sub>2</sub></font>()</span><br/><a NAME="E2:3_1_2"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">C</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K6" title="MSUALG_5:func.6">CongrLatt</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> st <br/>( <font color="Maroon" title="c8">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">C</font> &amp;  ex <font color="Olive" title="b2">R</font> being  <a href="msualg_4.html#NM5" title="MSUALG_4:NM.5">MSCongruence</a> of <font color="Maroon">F<sub>2</sub></font>() st <br/>( <font color="Olive" title="b2">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">C</font> &amp; <font color="Maroon">P<sub>1</sub></font>[ <a href="msualg_4.html#K13" title="MSUALG_4:func.13">QuotMSAlg</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Olive" title="b2">R</font>] ) )
 ;<br/><span class="kw">hence </span><a NAME="E3:3_1_2"/>
<font color="Maroon" title="c8">q</font> is   <a href="msualg_4.html#NM5" title="MSUALG_4:NM.5">MSCongruence</a> of <font color="Maroon">F<sub>2</sub></font>()
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E17:3_1"/><span class="lab"><font color="Green" title="E12">A10</font></span>: 
<span class="p1">{<span class="default"> <font color="Olive" title="b1">C</font> where <font color="Olive" title="b1">C</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K6" title="MSUALG_5:func.6">CongrLatt</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> :  ex <font color="Olive" title="b2">R</font> being  <a href="msualg_4.html#NM5" title="MSUALG_4:NM.5">MSCongruence</a> of <font color="Maroon">F<sub>2</sub></font>() st <br/>( <font color="Olive" title="b2">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">C</font> &amp; <font color="Maroon">P<sub>1</sub></font>[ <a href="msualg_4.html#K13" title="MSUALG_4:func.13">QuotMSAlg</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Olive" title="b2">R</font>] ) </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K5" title="MSUALG_5:func.5">EqRelLatt</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">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"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">q</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c8">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">C</font> where <font color="Olive" title="b1">C</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K6" title="MSUALG_5:func.6">CongrLatt</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> :  ex <font color="Olive" title="b2">R</font> being  <a href="msualg_4.html#NM5" title="MSUALG_4:NM.5">MSCongruence</a> of <font color="Maroon">F<sub>2</sub></font>() st <br/>( <font color="Olive" title="b2">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">C</font> &amp; <font color="Maroon">P<sub>1</sub></font>[ <a href="msualg_4.html#K13" title="MSUALG_4:func.13">QuotMSAlg</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Olive" title="b2">R</font>] ) </span>}</span>  or <font color="Maroon" title="c8">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K5" title="MSUALG_5:func.5">EqRelLatt</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>()</span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:3_1_3"/>
<font color="Maroon" title="c8">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">C</font> where <font color="Olive" title="b1">C</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K6" title="MSUALG_5:func.6">CongrLatt</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> :  ex <font color="Olive" title="b2">R</font> being  <a href="msualg_4.html#NM5" title="MSUALG_4:NM.5">MSCongruence</a> of <font color="Maroon">F<sub>2</sub></font>() st <br/>( <font color="Olive" title="b2">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">C</font> &amp; <font color="Maroon">P<sub>1</sub></font>[ <a href="msualg_4.html#K13" title="MSUALG_4:func.13">QuotMSAlg</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Olive" title="b2">R</font>] ) </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K5" title="MSUALG_5:func.5">EqRelLatt</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>()</span>)</span></span><br/>

<a NAME="E2:3_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c8">q</font> is   <a href="msualg_5.html#NM1" title="MSUALG_5:NM.1">Equivalence_Relation</a> of the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>()
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:3_1"><span class="lab"><font color="Green" title="E11">A9</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:3_1_3"/>
<font color="Maroon" title="c8">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K5" title="MSUALG_5:func.5">EqRelLatt</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>()</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_5.html#D5" title="MSUALG_5:def.5">MSUALG_5:def 5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">CC</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">C</font> where <font color="Olive" title="b1">C</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K6" title="MSUALG_5:func.6">CongrLatt</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> :  ex <font color="Olive" title="b2">R</font> being  <a href="msualg_4.html#NM5" title="MSUALG_4:NM.5">MSCongruence</a> of <font color="Maroon">F<sub>2</sub></font>() st <br/>( <font color="Olive" title="b2">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">C</font> &amp; <font color="Maroon">P<sub>1</sub></font>[ <a href="msualg_4.html#K13" title="MSUALG_4:func.13">QuotMSAlg</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Olive" title="b2">R</font>] ) </span>}</span>  as  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="closure2.html#NM1" title="CLOSURE2:NM.1">SubsetFamily</a> of <span class="p1"><a href="pboole.html#K8" title="PBOOLE:func.8">[|</a><span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>(),the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>()</span><a href="pboole.html#K8" title="PBOOLE:func.8">|]</a></span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:3_1"><span class="lab"><font color="Green" title="E10">A8</font></span></a>, <a class="ref" href="msualg_7.html#T6" title="MSUALG_7:th.6">MSUALG_7:6</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c9">R0</font> =  <a href="mssubfam.html#K4" title="MSSUBFAM:func.4">meet</a> <span class="p1"><a href="closure2.html#K5" title="CLOSURE2:func.5">|:</a><span class="default"><font color="Maroon" title="c8">CC</font></span><a href="closure2.html#K5" title="CLOSURE2:func.5">:|</a></span>;<br/>
<a NAME="E19:3_1"/><span class="lab"><font color="Green" title="E13">A11</font></span>: 
 <a href="mssubfam.html#K4" title="MSSUBFAM:func.4">meet</a> <span class="p1"><a href="closure2.html#K5" title="CLOSURE2:func.5">|:</a><span class="default"><font color="Maroon" title="c8">CC</font></span><a href="closure2.html#K5" title="CLOSURE2:func.5">:|</a></span> is   <a href="msualg_4.html#V2" title="MSUALG_4:attr.2">MSEquivalence_Relation-like</a>  <a href="msualg_4.html#NM3" title="MSUALG_4:NM.3">ManySortedRelation</a> of the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>()
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:3_1"><span class="lab"><font color="Green" title="E12">A10</font></span></a>, <a class="ref" href="msualg_7.html#T9" title="MSUALG_7:th.9">MSUALG_7:9</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">R0</font> =  <a href="mssubfam.html#K4" title="MSSUBFAM:func.4">meet</a> <span class="p1"><a href="closure2.html#K5" title="CLOSURE2:func.5">|:</a><span class="default"><font color="Maroon" title="c8">CC</font></span><a href="closure2.html#K5" title="CLOSURE2:func.5">:|</a></span> as   <a href="msualg_4.html#NM4" title="MSUALG_4:NM.4">ManySortedRelation</a> of <font color="Maroon">F<sub>2</sub></font>() <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:3_1"><span class="lab"><font color="Green" title="E12">A10</font></span></a>, <a class="ref" href="msualg_7.html#T9" title="MSUALG_7:th.9">MSUALG_7:9</a></span>;<br/>
<a NAME="E21:3_1"/>
<font color="Maroon" title="c10">R0</font> is  <a href="msualg_4.html#V3" title="MSUALG_4:attr.3">MSEquivalence-like</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="msualg_4.html#D3" title="MSUALG_4:def.3">MSUALG_4:def 3</a>,<a class="ref" href="msualg_4.html#D5" title="MSUALG_4:def.5">MSUALG_4:def 5</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font></span>)</span>,<span class="p2">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>  holds <br/>(  not <font color="Maroon" title="c11">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() or  not <font color="Maroon" title="c10">R0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>1</sub></font> or <font color="Olive">b<sub>1</sub></font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font></span>)</span>,<span class="p2">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c12">R</font> be   <a href="relset_1.html#NM2" title="RELSET_1:NM.2">Relation</a> of <span class="p1">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c11">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() or  not <font color="Maroon" title="c10">R0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">R</font> or <font color="Maroon" title="c12">R</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font></span>)</span>,<span class="p2">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:3_1_4"/><span class="lab"><font color="Green" title="E14">A12</font></span>: 
<font color="Maroon" title="c11">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>()
 <span class="kw">and </span><br/><a NAME="E2:3_1_4"/><span class="lab"><font color="Green" title="E15">A13</font></span>: 
<font color="Maroon" title="c10">R0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">R</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c12">R</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font></span>)</span>,<span class="p2">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span><br/>

<span class="kw">thus </span><a NAME="E3:3_1_4"/>
<font color="Maroon" title="c12">R</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font></span>)</span>,<span class="p2">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:3_1"><span class="lab"><font color="Green" title="E13">A11</font></span></a>, <a class="txt" href="#E1:3_1_4"><span class="lab"><font color="Green" title="E14">A12</font></span></a>, <a class="txt" href="#E2:3_1_4"><span class="lab"><font color="Green" title="E15">A13</font></span></a>, <a class="ref" href="msualg_4.html#D3" title="MSUALG_4:def.3">MSUALG_4:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c11">R0</font> = <font color="Maroon" title="c10">R0</font> as   <a href="msualg_4.html#V3" title="MSUALG_4:attr.3">MSEquivalence-like</a>  <a href="msualg_4.html#NM4" title="MSUALG_4:NM.4">ManySortedRelation</a> of <font color="Maroon">F<sub>2</sub></font>() ;<br/>
<a NAME="E23:3_1"/>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">C</font> where <font color="Olive" title="b1">C</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K6" title="MSUALG_5:func.6">CongrLatt</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">C</font>] </span>}</span>  is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K6" title="MSUALG_5:func.6">CongrLatt</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span>
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="domain_1.html#S7" title="DOMAIN_1:sch.7">DOMAIN_1:sch 7</a>();<br/></span>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c12">R0</font> = <font color="Maroon" title="c11">R0</font> as   <a href="msualg_4.html#NM5" title="MSUALG_4:NM.5">MSCongruence</a> of <font color="Maroon">F<sub>2</sub></font>() <span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_9.html#T29" title="MSUALG_9:th.29">MSUALG_9:29</a></span>;<br/>
<span class="kw">take </span><font color="Maroon" title="c13">A</font> =  <a href="msualg_4.html#K13" title="MSUALG_4:func.13">QuotMSAlg</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon" title="c12">R0</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">F</font> being  <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon" title="c13">A</font> st <br/>( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c13">A</font>] &amp; <font color="Olive" title="b1">F</font> <a href="msualg_3.html#R2" title="MSUALG_3:pred.2">is_epimorphism</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon" title="c13">A</font> &amp; ( for <font color="Olive" title="b2">B</font> being  <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">non-empty</a>   <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon">F<sub>1</sub></font>()<br/> for <font color="Olive" title="b3">G</font> being  <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon">F<sub>2</sub></font>(),<font color="Olive" title="b2">B</font>  st <font color="Olive" title="b3">G</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Olive" title="b2">B</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b2">B</font>] holds <br/> ex <font color="Olive" title="b4">H</font> being  <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon" title="c13">A</font>,<font color="Olive" title="b2">B</font> st <br/>( <font color="Olive" title="b4">H</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon" title="c13">A</font>,<font color="Olive" title="b2">B</font> &amp; <font color="Olive" title="b4">H</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Olive" title="b1">F</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Olive" title="b3">G</font> &amp; ( for <font color="Olive" title="b5">K</font> being  <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon" title="c13">A</font>,<font color="Olive" title="b2">B</font>  st <font color="Olive" title="b5">K</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Olive" title="b1">F</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Olive" title="b3">G</font> holds <br/><font color="Olive" title="b4">H</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Olive" title="b5">K</font> ) ) ) )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c14">F</font> =  <a href="msualg_4.html#K15" title="MSUALG_4:func.15">MSNat_Hom</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon" title="c12">R0</font> as   <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon" title="c13">A</font> ;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c14">F</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c13">A</font>] &amp; <font color="Maroon" title="c14">F</font> <a href="msualg_3.html#R2" title="MSUALG_3:pred.2">is_epimorphism</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon" title="c13">A</font> &amp; ( for <font color="Olive" title="b1">B</font> being  <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">non-empty</a>   <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon">F<sub>1</sub></font>()<br/> for <font color="Olive" title="b2">G</font> being  <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon">F<sub>2</sub></font>(),<font color="Olive" title="b1">B</font>  st <font color="Olive" title="b2">G</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Olive" title="b1">B</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">B</font>] holds <br/> ex <font color="Olive" title="b3">H</font> being  <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon" title="c13">A</font>,<font color="Olive" title="b1">B</font> st <br/>( <font color="Olive" title="b3">H</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon" title="c13">A</font>,<font color="Olive" title="b1">B</font> &amp; <font color="Olive" title="b3">H</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c14">F</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Olive" title="b2">G</font> &amp; ( for <font color="Olive" title="b4">K</font> being  <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon" title="c13">A</font>,<font color="Olive" title="b1">B</font>  st <font color="Olive" title="b4">K</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c14">F</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Olive" title="b2">G</font> holds <br/><font color="Olive" title="b3">H</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Olive" title="b4">K</font> ) ) ) )</span><br/>

<div><span class="lab"><font color="Green" title="E14">A14</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_5"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c15">c</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c15">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">CC</font> implies  ex <font color="Olive" title="b1">w</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c15">c</font>,<font color="Olive" title="b1">w</font>] )</span><br/><span class="kw">assume </span><a NAME="E1:3_1_5"/>
<font color="Maroon" title="c15">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">CC</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">w</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c15">c</font>,<font color="Olive" title="b1">w</font>]</span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c16">cc</font> = <font color="Maroon" title="c15">c</font> as   <a href="msualg_4.html#NM5" title="MSUALG_4:NM.5">MSCongruence</a> of <font color="Maroon">F<sub>2</sub></font>() <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:3_1"><span class="lab"><font color="Green" title="E11">A9</font></span></a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c17">w</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E4:3_1_5"/><span class="lab"><font color="Green" title="E15">A15</font></span>: 
<font color="Maroon" title="c17">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msualg_4.html#K13" title="MSUALG_4:func.13">QuotMSAlg</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon" title="c16">cc</font>
 ;<br/><span class="kw">take </span><font color="Maroon" title="c18">w</font> = <font color="Maroon" title="c17">w</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c15">c</font>,<font color="Maroon" title="c18">w</font>]</span><br/><span class="kw">thus </span><a NAME="E5:3_1_5"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c15">c</font>,<font color="Maroon" title="c18">w</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:3_1_5"><span class="lab"><font color="Green" title="E15">A15</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">consider </span><font color="Maroon" title="c15">FF</font> being   <a href="pboole.html#NM1" title="PBOOLE:NM.1">ManySortedSet</a> of <font color="Maroon" title="c8">CC</font><span class="kw"> such that </span><br/><a NAME="E28:3_1"/><span class="lab"><font color="Green" title="E15">A16</font></span>: 
for <font color="Olive" title="b1">c</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">CC</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">c</font>,<font color="Maroon" title="c15">FF</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">c</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="pboole.html#S3" title="PBOOLE:sch.3">PBOOLE:sch 3</a>(<a class="txt" href="#E26:3_1"><span class="lab"><font color="Green" title="E14">A14</font></span></a>);<br/></span>
<a NAME="E29:3_1"/>
<font color="Maroon" title="c15">FF</font> is    <a href="pralg_2.html#M2" title="PRALG_2:mode.2">MSAlgebra-Family</a> of <font color="Maroon" title="c8">CC</font>,<font color="Maroon">F<sub>1</sub></font>()
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c16">c</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="pralg_2.html#D12" title="PRALG_2:def.12">PRALG_2:def 12</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c16">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">CC</font> or <font color="Maroon" title="c15">FF</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">c</font> is    <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon">F<sub>1</sub></font>() )</span><br/>

<span class="kw">assume </span><a NAME="E1:3_1_6"/>
<font color="Maroon" title="c16">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">CC</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c15">FF</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">c</font> is    <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon">F<sub>1</sub></font>()</span><br/>

<a NAME="E2:3_1_6"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">R</font> being  <a href="msualg_4.html#NM5" title="MSUALG_4:NM.5">MSCongruence</a> of <font color="Maroon">F<sub>2</sub></font>() st <br/>( <font color="Olive" title="b1">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">c</font> &amp; <font color="Maroon" title="c15">FF</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msualg_4.html#K13" title="MSUALG_4:func.13">QuotMSAlg</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Olive" title="b1">R</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:3_1"><span class="lab"><font color="Green" title="E15">A16</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:3_1_6"/>
<font color="Maroon" title="c15">FF</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">c</font> is    <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon">F<sub>1</sub></font>()
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c16">FF</font> = <font color="Maroon" title="c15">FF</font> as    <a href="pralg_2.html#M2" title="PRALG_2:mode.2">MSAlgebra-Family</a> of <font color="Maroon" title="c8">CC</font>,<font color="Maroon">F<sub>1</sub></font>() ;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>3</sub></font>[   <a href="closure2.html#M1" title="CLOSURE2:mode.1">Element</a> of <font color="Maroon" title="c8">CC</font>,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span> ex <font color="Olive" title="b1">F1</font> being  <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon">F<sub>2</sub></font>(),<span class="p1">(<span class="default"><font color="Maroon" title="c16">FF</font> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> $1</span>)</span> st <br/>( <font color="Olive" title="b1">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $2 &amp; <font color="Olive" title="b1">F1</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon" title="c16">FF</font> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> $1 &amp;  ex <font color="Olive" title="b2">R</font> being  <a href="msualg_4.html#NM5" title="MSUALG_4:NM.5">MSCongruence</a> of <font color="Maroon">F<sub>2</sub></font>() st <br/>( $1 <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Olive" title="b2">R</font> &amp; <font color="Olive" title="b1">F1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="msualg_4.html#K15" title="MSUALG_4:func.15">MSNat_Hom</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Olive" title="b2">R</font> ) );<br/>
<a NAME="E31:3_1"/><span class="lab"><font color="Green" title="E16">A17</font></span>: 
for <font color="Olive" title="b1">c</font> being   <a href="closure2.html#M1" title="CLOSURE2:mode.1">Element</a> of <font color="Maroon" title="c8">CC</font>  ex <font color="Olive" title="b2">j</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">c</font>,<font color="Olive" title="b2">j</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_7"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c17">c</font> be    <a href="closure2.html#M1" title="CLOSURE2:mode.1">Element</a> of <font color="Maroon" title="c8">CC</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">j</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c17">c</font>,<font color="Olive" title="b1">j</font>]</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c18">R</font> being   <a href="msualg_4.html#NM5" title="MSUALG_4:NM.5">MSCongruence</a> of <font color="Maroon">F<sub>2</sub></font>()<span class="kw"> such that </span><br/><a NAME="E2:3_1_7"/><span class="lab"><font color="Green" title="E17">A18</font></span>: 
<font color="Maroon" title="c18">R</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c17">c</font>
 <span class="kw">and </span><br/><a NAME="E3:3_1_7"/><span class="lab"><font color="Green" title="E18">A19</font></span>: 
<font color="Maroon" title="c16">FF</font> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Maroon" title="c17">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msualg_4.html#K13" title="MSUALG_4:func.13">QuotMSAlg</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon" title="c18">R</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:3_1"><span class="lab"><font color="Green" title="E15">A16</font></span></a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c19">j</font> =  <a href="msualg_4.html#K15" title="MSUALG_4:func.15">MSNat_Hom</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon" title="c18">R</font>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c20">F1</font> =  <a href="msualg_4.html#K15" title="MSUALG_4:func.15">MSNat_Hom</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon" title="c18">R</font> as   <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon">F<sub>2</sub></font>(),<span class="p1">(<span class="default"><font color="Maroon" title="c16">FF</font> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Maroon" title="c17">c</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:3_1_7"><span class="lab"><font color="Green" title="E18">A19</font></span></a></span>;<br/>
<span class="kw">take </span>
 <a href="msualg_4.html#K15" title="MSUALG_4:func.15">MSNat_Hom</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon" title="c18">R</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c17">c</font>, <a href="msualg_4.html#K15" title="MSUALG_4:func.15">MSNat_Hom</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon" title="c18">R</font>]</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c20">F1</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c20">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msualg_4.html#K15" title="MSUALG_4:func.15">MSNat_Hom</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon" title="c18">R</font> &amp; <font color="Maroon" title="c20">F1</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon" title="c16">FF</font> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Maroon" title="c17">c</font> &amp;  ex <font color="Olive" title="b1">R</font> being  <a href="msualg_4.html#NM5" title="MSUALG_4:NM.5">MSCongruence</a> of <font color="Maroon">F<sub>2</sub></font>() st <br/>( <font color="Maroon" title="c17">c</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Olive" title="b1">R</font> &amp; <font color="Maroon" title="c20">F1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="msualg_4.html#K15" title="MSUALG_4:func.15">MSNat_Hom</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Olive" title="b1">R</font> ) )</span><br/>

<span class="kw">thus </span><a NAME="E5:3_1_7"/>
<font color="Maroon" title="c20">F1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="msualg_4.html#K15" title="MSUALG_4:func.15">MSNat_Hom</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon" title="c18">R</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c20">F1</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon" title="c16">FF</font> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Maroon" title="c17">c</font> &amp;  ex <font color="Olive" title="b1">R</font> being  <a href="msualg_4.html#NM5" title="MSUALG_4:NM.5">MSCongruence</a> of <font color="Maroon">F<sub>2</sub></font>() st <br/>( <font color="Maroon" title="c17">c</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Olive" title="b1">R</font> &amp; <font color="Maroon" title="c20">F1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="msualg_4.html#K15" title="MSUALG_4:func.15">MSNat_Hom</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Olive" title="b1">R</font> ) )</span><br/>

<a NAME="E6:3_1_7"/>
 <a href="msualg_4.html#K15" title="MSUALG_4:func.15">MSNat_Hom</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon" title="c18">R</font> <a href="msualg_3.html#R2" title="MSUALG_3:pred.2">is_epimorphism</a> <font color="Maroon">F<sub>2</sub></font>(), <a href="msualg_4.html#K13" title="MSUALG_4:func.13">QuotMSAlg</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon" title="c18">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_4.html#T3" title="MSUALG_4:th.3">MSUALG_4:3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:3_1_7"/>
<font color="Maroon" title="c20">F1</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon" title="c16">FF</font> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Maroon" title="c17">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:3_1_7"><span class="lab"><font color="Green" title="E18">A19</font></span></a>, <a class="ref" href="msualg_3.html#D10" title="MSUALG_3:def.10">MSUALG_3:def 10</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">R</font> being  <a href="msualg_4.html#NM5" title="MSUALG_4:NM.5">MSCongruence</a> of <font color="Maroon">F<sub>2</sub></font>() st <br/>( <font color="Maroon" title="c17">c</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Olive" title="b1">R</font> &amp; <font color="Maroon" title="c20">F1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="msualg_4.html#K15" title="MSUALG_4:func.15">MSNat_Hom</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Olive" title="b1">R</font> )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c18">R</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c17">c</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c18">R</font> &amp; <font color="Maroon" title="c20">F1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="msualg_4.html#K15" title="MSUALG_4:func.15">MSNat_Hom</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon" title="c18">R</font> )</span><br/>

<span class="kw">thus </span><a NAME="E8:3_1_7"/>
( <font color="Maroon" title="c17">c</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c18">R</font> &amp; <font color="Maroon" title="c20">F1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="msualg_4.html#K15" title="MSUALG_4:func.15">MSNat_Hom</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon" title="c18">R</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:3_1_7"><span class="lab"><font color="Green" title="E17">A18</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">consider </span><font color="Maroon" title="c17">FofI1</font> being   <a href="pboole.html#NM1" title="PBOOLE:NM.1">ManySortedSet</a> of <font color="Maroon" title="c8">CC</font><span class="kw"> such that </span><br/><a NAME="E33:3_1"/><span class="lab"><font color="Green" title="E17">A20</font></span>: 
for <font color="Olive" title="b1">c</font> being   <a href="closure2.html#M1" title="CLOSURE2:mode.1">Element</a> of <font color="Maroon" title="c8">CC</font> holds  <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">c</font>,<font color="Maroon" title="c17">FofI1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">c</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="pboole.html#S6" title="PBOOLE:sch.6">PBOOLE:sch 6</a>(<a class="txt" href="#E31:3_1"><span class="lab"><font color="Green" title="E16">A17</font></span></a>);<br/></span>
<a NAME="E34:3_1"/><span class="lab"><font color="Green" title="E18">A21</font></span>: 
for <font color="Olive" title="b1">c</font> being   <a href="closure2.html#M1" title="CLOSURE2:mode.1">Element</a> of <font color="Maroon" title="c8">CC</font>  ex <font color="Olive" title="b2">F1</font> being  <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon">F<sub>2</sub></font>(),<span class="p1">(<span class="default"><font color="Maroon" title="c16">FF</font> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Olive" title="b1">c</font></span>)</span> st <br/>( <font color="Olive" title="b2">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">FofI1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">c</font> &amp; <font color="Olive" title="b2">F1</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon" title="c16">FF</font> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Olive" title="b1">c</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_8"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c18">c</font> be    <a href="closure2.html#M1" title="CLOSURE2:mode.1">Element</a> of <font color="Maroon" title="c8">CC</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">F1</font> being  <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon">F<sub>2</sub></font>(),<span class="p1">(<span class="default"><font color="Maroon" title="c16">FF</font> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Maroon" title="c18">c</font></span>)</span> st <br/>( <font color="Olive" title="b1">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">FofI1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">c</font> &amp; <font color="Olive" title="b1">F1</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon" title="c16">FF</font> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Maroon" title="c18">c</font> )</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c19">F1</font> being   <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon">F<sub>2</sub></font>(),<span class="p1">(<span class="default"><font color="Maroon" title="c16">FF</font> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Maroon" title="c18">c</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E2:3_1_8"/><span class="lab"><font color="Green" title="E19">A22</font></span>: 
<font color="Maroon" title="c19">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">FofI1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">c</font>
 <span class="kw">and </span><br/><a NAME="E3:3_1_8"/><span class="lab"><font color="Green" title="E20">A23</font></span>: 
<font color="Maroon" title="c19">F1</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon" title="c16">FF</font> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Maroon" title="c18">c</font>
 <span class="kw">and </span><br/><a NAME="E4:3_1_8"/>
 ex <font color="Olive" title="b1">R</font> being  <a href="msualg_4.html#NM5" title="MSUALG_4:NM.5">MSCongruence</a> of <font color="Maroon">F<sub>2</sub></font>() st <br/>( <font color="Maroon" title="c18">c</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Olive" title="b1">R</font> &amp; <font color="Maroon" title="c19">F1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="msualg_4.html#K15" title="MSUALG_4:func.15">MSNat_Hom</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Olive" title="b1">R</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:3_1"><span class="lab"><font color="Green" title="E17">A20</font></span></a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c19">F1</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c19">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">FofI1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">c</font> &amp; <font color="Maroon" title="c19">F1</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon" title="c16">FF</font> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Maroon" title="c18">c</font> )</span><br/>

<span class="kw">thus </span><a NAME="E5:3_1_8"/>
( <font color="Maroon" title="c19">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">FofI1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">c</font> &amp; <font color="Maroon" title="c19">F1</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon" title="c16">FF</font> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Maroon" title="c18">c</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:3_1_8"><span class="lab"><font color="Green" title="E19">A22</font></span></a>, <a class="txt" href="#E3:3_1_8"><span class="lab"><font color="Green" title="E20">A23</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E35:3_1"/>
<font color="Maroon" title="c17">FofI1</font> is  <a href="funcop_1.html#V1" title="FUNCOP_1:attr.1">Function-yielding</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_9"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c18">c</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="funcop_1.html#D6" title="FUNCOP_1:def.6">FUNCOP_1:def 6</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c18">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <font color="Maroon" title="c17">FofI1</font> or <font color="Maroon" title="c17">FofI1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">c</font> is    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:3_1_9"/>
<font color="Maroon" title="c18">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c17">FofI1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c17">FofI1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">c</font> is    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> </span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c19">cc</font> = <font color="Maroon" title="c18">c</font> as    <a href="closure2.html#M1" title="CLOSURE2:mode.1">Element</a> of <font color="Maroon" title="c8">CC</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D4" title="PARTFUN1:def.4">PARTFUN1:def 4</a></span>;<br/>
<a NAME="E3:3_1_9"/>
 ex <font color="Olive" title="b1">F1</font> being  <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon">F<sub>2</sub></font>(),<span class="p1">(<span class="default"><font color="Maroon" title="c16">FF</font> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Maroon" title="c19">cc</font></span>)</span> st <br/>( <font color="Olive" title="b1">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">FofI1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">cc</font> &amp; <font color="Olive" title="b1">F1</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon" title="c16">FF</font> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Maroon" title="c19">cc</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:3_1"><span class="lab"><font color="Green" title="E18">A21</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:3_1_9"/>
<font color="Maroon" title="c17">FofI1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">c</font> is    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c18">FofI1</font> = <font color="Maroon" title="c17">FofI1</font> as   <a href="pboole.html#NM2" title="PBOOLE:NM.2">ManySortedFunction</a> of <font color="Maroon" title="c8">CC</font> ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c19">H</font> being   <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon">F<sub>2</sub></font>(),<span class="p1">(<span class="default"><a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <font color="Maroon" title="c16">FF</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E38:3_1"/><span class="lab"><font color="Green" title="E19">A24</font></span>: 
<font color="Maroon" title="c19">H</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon">F<sub>2</sub></font>(), <a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <font color="Maroon" title="c16">FF</font>
 <span class="kw">and </span><br/><a NAME="E39:3_1"/><span class="lab"><font color="Green" title="E20">A25</font></span>: 
for <font color="Olive" title="b1">c</font> being   <a href="closure2.html#M1" title="CLOSURE2:mode.1">Element</a> of <font color="Maroon" title="c8">CC</font> holds  <span class="p1">(<span class="default"><a href="pralg_3.html#K2" title="PRALG_3:func.2">proj</a> <font color="Maroon" title="c16">FF</font>,<font color="Olive" title="b1">c</font></span>)</span> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c19">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">FofI1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:3_1"><span class="lab"><font color="Green" title="E18">A21</font></span></a>, <a class="ref" href="pralg_3.html#T30" title="PRALG_3:th.30">PRALG_3:30</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_10"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c20">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c20">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() implies <span class="p1">(<span class="default"><a href="msualg_4.html#K17" title="MSUALG_4:func.17">MSCng</a> <font color="Maroon" title="c19">H</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">R0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">i</font> )</span><br/><span class="kw">assume </span><a NAME="E1:3_1_10"/>
<font color="Maroon" title="c20">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>()
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="msualg_4.html#K17" title="MSUALG_4:func.17">MSCng</a> <font color="Maroon" title="c19">H</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">R0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">i</font></span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c21">s</font> = <font color="Maroon" title="c20">i</font> as   <a href="msualg_1.html#NM1" title="MSUALG_1:NM.1">SortSymbol</a> of <font color="Maroon">F<sub>1</sub></font>() ;<br/><span class="kw">consider </span><font color="Maroon" title="c22">Q</font> being   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <span class="p1">(<span class="default"><span class="p2"><a href="pboole.html#K8" title="PBOOLE:func.8">[|</a><span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>(),the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>()</span><a href="pboole.html#K8" title="PBOOLE:func.8">|]</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">s</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E4:3_1_10"/><span class="lab"><font color="Green" title="E21">A26</font></span>: 
<font color="Maroon" title="c22">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="closure2.html#K5" title="CLOSURE2:func.5">|:</a><span class="default"><font color="Maroon" title="c8">CC</font></span><a href="closure2.html#K5" title="CLOSURE2:func.5">:|</a></span> <a href="mssubfam.html#K2" title="MSSUBFAM:func.2">.</a> <font color="Maroon" title="c21">s</font>
 <span class="kw">and </span><br/><a NAME="E5:3_1_10"/><span class="lab"><font color="Green" title="E22">A27</font></span>: 
<span class="p1">(<span class="default"><a href="mssubfam.html#K4" title="MSSUBFAM:func.4">meet</a> <span class="p2"><a href="closure2.html#K5" title="CLOSURE2:func.5">|:</a><span class="default"><font color="Maroon" title="c8">CC</font></span><a href="closure2.html#K5" title="CLOSURE2:func.5">:|</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c22">Q</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="mssubfam.html#D2" title="MSSUBFAM:def.2">MSSUBFAM:def 2</a></span>;<br/><a NAME="E6:3_1_10"/><span class="lab"><font color="Green" title="E23">A28</font></span>: 
<span class="p1"><a href="closure2.html#K5" title="CLOSURE2:func.5">|:</a><span class="default"><font color="Maroon" title="c8">CC</font></span><a href="closure2.html#K5" title="CLOSURE2:func.5">:|</a></span> <a href="mssubfam.html#K2" title="MSSUBFAM:func.2">.</a> <font color="Maroon" title="c21">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">s</font></span>)</span> where <font color="Olive" title="b1">t</font> is    <a href="closure2.html#M1" title="CLOSURE2:mode.1">Element</a> of  <a href="closure2.html#K6" title="CLOSURE2:func.6">Bool</a> <span class="p1"><a href="pboole.html#K8" title="PBOOLE:func.8">[|</a><span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>(),the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>()</span><a href="pboole.html#K8" title="PBOOLE:func.8">|]</a></span> : <font color="Olive" title="b1">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">CC</font> </span>}</span> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="closure2.html#T15" title="CLOSURE2:th.15">CLOSURE2:15</a></span>;<br/><a NAME="E7:3_1_10"/>
<span class="p1">(<span class="default"><a href="msualg_4.html#K17" title="MSUALG_4:func.17">MSCng</a> <font color="Maroon" title="c19">H</font></span>)</span> <a href="msualg_4.html#K2" title="MSUALG_4:func.2">.</a> <font color="Maroon" title="c21">s</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c12">R0</font> <a href="msualg_4.html#K2" title="MSUALG_4:func.2">.</a> <font color="Maroon" title="c21">s</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_10_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c23">a</font>, <font color="Maroon" title="c24">b</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="relat_1.html#D2" title="RELAT_1:def.2">RELAT_1:def 2</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( (  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c23">a</font>,<font color="Maroon" title="c24">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="msualg_4.html#K17" title="MSUALG_4:func.17">MSCng</a> <font color="Maroon" title="c19">H</font></span>)</span> <a href="msualg_4.html#K2" title="MSUALG_4:func.2">.</a> <font color="Maroon" title="c21">s</font> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c23">a</font>,<font color="Maroon" title="c24">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">R0</font> <a href="msualg_4.html#K2" title="MSUALG_4:func.2">.</a> <font color="Maroon" title="c21">s</font> ) &amp; (  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c23">a</font>,<font color="Maroon" title="c24">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">R0</font> <a href="msualg_4.html#K2" title="MSUALG_4:func.2">.</a> <font color="Maroon" title="c21">s</font> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c23">a</font>,<font color="Maroon" title="c24">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="msualg_4.html#K17" title="MSUALG_4:func.17">MSCng</a> <font color="Maroon" title="c19">H</font></span>)</span> <a href="msualg_4.html#K2" title="MSUALG_4:func.2">.</a> <font color="Maroon" title="c21">s</font> ) )</span><br/>

<div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c23">a</font>,<font color="Maroon" title="c24">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">R0</font> <a href="msualg_4.html#K2" title="MSUALG_4:func.2">.</a> <font color="Maroon" title="c21">s</font> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c23">a</font>,<font color="Maroon" title="c24">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="msualg_4.html#K17" title="MSUALG_4:func.17">MSCng</a> <font color="Maroon" title="c19">H</font></span>)</span> <a href="msualg_4.html#K2" title="MSUALG_4:func.2">.</a> <font color="Maroon" title="c21">s</font> )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:3_1_10_1_1"/><span class="lab"><font color="Green" title="E24">A29</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c23">a</font>,<font color="Maroon" title="c24">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="msualg_4.html#K17" title="MSUALG_4:func.17">MSCng</a> <font color="Maroon" title="c19">H</font></span>)</span> <a href="msualg_4.html#K2" title="MSUALG_4:func.2">.</a> <font color="Maroon" title="c21">s</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c23">a</font>,<font color="Maroon" title="c24">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">R0</font> <a href="msualg_4.html#K2" title="MSUALG_4:func.2">.</a> <font color="Maroon" title="c21">s</font></span><br/><a NAME="E2:3_1_10_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A30</font></span>: 
<font color="Maroon" title="c23">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">s</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/><a NAME="E3:3_1_10_1_1"/><span class="lab"><font color="Green" title="E26">A31</font></span>: 
<font color="Maroon" title="c24">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">s</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:3_1_10_1_1"><span class="lab"><font color="Green" title="E24">A29</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/><a NAME="E4:3_1_10_1_1"/><span class="lab"><font color="Green" title="E27">A32</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c23">a</font>,<font color="Maroon" title="c24">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msualg_4.html#K16" title="MSUALG_4:func.16">MSCng</a> <font color="Maroon" title="c19">H</font>,<font color="Maroon" title="c21">s</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:3_1"><span class="lab"><font color="Green" title="E19">A24</font></span></a>, <a class="txt" href="#E1:3_1_10_1_1"><span class="lab"><font color="Green" title="E24">A29</font></span></a>, <a class="ref" href="msualg_4.html#D20" title="MSUALG_4:def.20">MSUALG_4:def 20</a></span>;<br/><a NAME="E5:3_1_10_1_1"/><span class="lab"><font color="Green" title="E28">A33</font></span>: 
for <font color="Olive" title="b1">Y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c22">Q</font> holds <br/><span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c23">a</font>,<font color="Maroon" title="c24">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Y</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_10_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c25">Y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c25">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c22">Q</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c23">a</font>,<font color="Maroon" title="c24">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c25">Y</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:3_1_10_1_1_1"/>
<font color="Maroon" title="c25">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c22">Q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c23">a</font>,<font color="Maroon" title="c24">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c25">Y</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c26">t</font> being    <a href="closure2.html#M1" title="CLOSURE2:mode.1">Element</a> of  <a href="closure2.html#K6" title="CLOSURE2:func.6">Bool</a> <span class="p1"><a href="pboole.html#K8" title="PBOOLE:func.8">[|</a><span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>(),the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>()</span><a href="pboole.html#K8" title="PBOOLE:func.8">|]</a></span><span class="kw"> such that </span><br/><a NAME="E3:3_1_10_1_1_1"/><span class="lab"><font color="Green" title="E29">A34</font></span>: 
<font color="Maroon" title="c25">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">s</font>
 <span class="kw">and </span><br/><a NAME="E4:3_1_10_1_1_1"/><span class="lab"><font color="Green" title="E30">A35</font></span>: 
<font color="Maroon" title="c26">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">CC</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:3_1_10"><span class="lab"><font color="Green" title="E21">A26</font></span></a>, <a class="txt" href="#E6:3_1_10"><span class="lab"><font color="Green" title="E23">A28</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c27">t1</font> = <font color="Maroon" title="c26">t</font> as    <a href="closure2.html#M1" title="CLOSURE2:mode.1">Element</a> of <font color="Maroon" title="c8">CC</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:3_1_10_1_1_1"><span class="lab"><font color="Green" title="E30">A35</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c28">F1</font> being   <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon">F<sub>2</sub></font>(),<span class="p1">(<span class="default"><font color="Maroon" title="c16">FF</font> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Maroon" title="c27">t1</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E7:3_1_10_1_1_1"/><span class="lab"><font color="Green" title="E31">A36</font></span>: 
<font color="Maroon" title="c28">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">FofI1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">t1</font>
 <span class="kw">and </span><br/><a NAME="E8:3_1_10_1_1_1"/>
<font color="Maroon" title="c28">F1</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon" title="c16">FF</font> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Maroon" title="c27">t1</font>
 <span class="kw">and </span><br/><a NAME="E9:3_1_10_1_1_1"/><span class="lab"><font color="Green" title="E32">A37</font></span>: 
 ex <font color="Olive" title="b1">R</font> being  <a href="msualg_4.html#NM5" title="MSUALG_4:NM.5">MSCongruence</a> of <font color="Maroon">F<sub>2</sub></font>() st <br/>( <font color="Maroon" title="c27">t1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Olive" title="b1">R</font> &amp; <font color="Maroon" title="c28">F1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="msualg_4.html#K15" title="MSUALG_4:func.15">MSNat_Hom</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Olive" title="b1">R</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:3_1"><span class="lab"><font color="Green" title="E17">A20</font></span></a></span>;<br/>
<span class="p1">(<span class="default"><font color="Maroon" title="c28">F1</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c21">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">a</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pralg_3.html#K2" title="PRALG_3:func.2">proj</a> <font color="Maroon" title="c16">FF</font>,<font color="Maroon" title="c27">t1</font></span>)</span> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c19">H</font></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c21">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">a</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E39:3_1"><span class="lab"><font color="Green" title="E20">A25</font></span></a>, <a class="txt" href="#E7:3_1_10_1_1_1"><span class="lab"><font color="Green" title="E31">A36</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pralg_3.html#K2" title="PRALG_3:func.2">proj</a> <font color="Maroon" title="c16">FF</font>,<font color="Maroon" title="c27">t1</font></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c21">s</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">H</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c21">s</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">a</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_3.html#T2" title="MSUALG_3:th.2">MSUALG_3:2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_3.html#K2" title="PRALG_3:func.2">proj</a> <font color="Maroon" title="c16">FF</font>,<font color="Maroon" title="c27">t1</font></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c21">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">H</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c21">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">a</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:3_1_10_1_1"><span class="lab"><font color="Green" title="E25">A30</font></span></a>, <a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_3.html#K2" title="PRALG_3:func.2">proj</a> <font color="Maroon" title="c16">FF</font>,<font color="Maroon" title="c27">t1</font></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c21">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">H</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c21">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">b</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:3_1_10_1_1"><span class="lab"><font color="Green" title="E25">A30</font></span></a>, <a class="txt" href="#E3:3_1_10_1_1"><span class="lab"><font color="Green" title="E26">A31</font></span></a>, <a class="txt" href="#E4:3_1_10_1_1"><span class="lab"><font color="Green" title="E27">A32</font></span></a>, <a class="ref" href="msualg_4.html#D19" title="MSUALG_4:def.19">MSUALG_4:def 19</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pralg_3.html#K2" title="PRALG_3:func.2">proj</a> <font color="Maroon" title="c16">FF</font>,<font color="Maroon" title="c27">t1</font></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c21">s</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">H</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c21">s</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">b</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:3_1_10_1_1"><span class="lab"><font color="Green" title="E26">A31</font></span></a>, <a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pralg_3.html#K2" title="PRALG_3:func.2">proj</a> <font color="Maroon" title="c16">FF</font>,<font color="Maroon" title="c27">t1</font></span>)</span> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c19">H</font></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c21">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">b</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_3.html#T2" title="MSUALG_3:th.2">MSUALG_3:2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c28">F1</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c21">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">b</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E39:3_1"><span class="lab"><font color="Green" title="E20">A25</font></span></a>, <a class="txt" href="#E7:3_1_10_1_1_1"><span class="lab"><font color="Green" title="E31">A36</font></span></a></span>
;<br/>
<span class="kw">hence </span><a NAME="E11:3_1_10_1_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c23">a</font>,<font color="Maroon" title="c24">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c25">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:3_1_10_1_1"><span class="lab"><font color="Green" title="E25">A30</font></span></a>, <a class="txt" href="#E3:3_1_10_1_1"><span class="lab"><font color="Green" title="E26">A31</font></span></a>, <a class="txt" href="#E3:3_1_10_1_1_1"><span class="lab"><font color="Green" title="E29">A34</font></span></a>, <a class="txt" href="#E9:3_1_10_1_1_1"><span class="lab"><font color="Green" title="E32">A37</font></span></a>, <a class="ref" href="msualg_9.html#T34" title="MSUALG_9:th.34">MSUALG_9:34</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E6:3_1_10_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c23">a</font>,<font color="Maroon" title="c24">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">s</font></span>)</span>,<span class="p2">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">s</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:3_1_10_1_1"><span class="lab"><font color="Green" title="E24">A29</font></span></a></span>;<br/><a NAME="E7:3_1_10_1_1"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c23">a</font>,<font color="Maroon" title="c24">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="pboole.html#K8" title="PBOOLE:func.8">[|</a><span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>(),the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>()</span><a href="pboole.html#K8" title="PBOOLE:func.8">|]</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">s</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pboole.html#D21" title="PBOOLE:def.21">PBOOLE:def 21</a></span>;<br/><span class="kw">hence </span><a NAME="E8:3_1_10_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c23">a</font>,<font color="Maroon" title="c24">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">R0</font> <a href="msualg_4.html#K2" title="MSUALG_4:func.2">.</a> <font color="Maroon" title="c21">s</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:3_1_10"><span class="lab"><font color="Green" title="E22">A27</font></span></a>, <a class="txt" href="#E5:3_1_10_1_1"><span class="lab"><font color="Green" title="E28">A33</font></span></a>, <a class="ref" href="setfam_1.html#T58" title="SETFAM_1:th.58">SETFAM_1:58</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E2:3_1_10_1"/><span class="lab"><font color="Green" title="E24">A38</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c23">a</font>,<font color="Maroon" title="c24">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">R0</font> <a href="msualg_4.html#K2" title="MSUALG_4:func.2">.</a> <font color="Maroon" title="c21">s</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c23">a</font>,<font color="Maroon" title="c24">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="msualg_4.html#K17" title="MSUALG_4:func.17">MSCng</a> <font color="Maroon" title="c19">H</font></span>)</span> <a href="msualg_4.html#K2" title="MSUALG_4:func.2">.</a> <font color="Maroon" title="c21">s</font></span><br/>

<a NAME="E3:3_1_10_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A39</font></span>: 
<font color="Maroon" title="c23">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">s</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/>
<a NAME="E4:3_1_10_1"/><span class="lab"><font color="Green" title="E26">A40</font></span>: 
<font color="Maroon" title="c24">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">s</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:3_1_10_1"><span class="lab"><font color="Green" title="E24">A38</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/>
<a NAME="E5:3_1_10_1"/><span class="lab"><font color="Green" title="E27">A41</font></span>: 
for <font color="Olive" title="b1">c</font> being   <a href="closure2.html#M1" title="CLOSURE2:mode.1">Element</a> of <font color="Maroon" title="c8">CC</font> holds  <span class="p1">(<span class="default"><a href="card_3.html#K12" title="CARD_3:func.12">proj</a> <span class="p2">(<span class="default"><a href="pralg_2.html#K9" title="PRALG_2:func.9">Carrier</a> <font color="Maroon" title="c16">FF</font>,<font color="Maroon" title="c21">s</font></span>)</span>,<font color="Olive" title="b1">c</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">H</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c21">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="card_3.html#K12" title="CARD_3:func.12">proj</a> <span class="p2">(<span class="default"><a href="pralg_2.html#K9" title="PRALG_2:func.9">Carrier</a> <font color="Maroon" title="c16">FF</font>,<font color="Maroon" title="c21">s</font></span>)</span>,<font color="Olive" title="b1">c</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">H</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c21">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">b</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_10_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c25">c</font> be    <a href="closure2.html#M1" title="CLOSURE2:mode.1">Element</a> of <font color="Maroon" title="c8">CC</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="card_3.html#K12" title="CARD_3:func.12">proj</a> <span class="p2">(<span class="default"><a href="pralg_2.html#K9" title="PRALG_2:func.9">Carrier</a> <font color="Maroon" title="c16">FF</font>,<font color="Maroon" title="c21">s</font></span>)</span>,<font color="Maroon" title="c25">c</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">H</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c21">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="card_3.html#K12" title="CARD_3:func.12">proj</a> <span class="p2">(<span class="default"><a href="pralg_2.html#K9" title="PRALG_2:func.9">Carrier</a> <font color="Maroon" title="c16">FF</font>,<font color="Maroon" title="c21">s</font></span>)</span>,<font color="Maroon" title="c25">c</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">H</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c21">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">b</font></span>)</span></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c26">t1</font> = <font color="Maroon" title="c25">c</font> as   <a href="msualg_4.html#NM5" title="MSUALG_4:NM.5">MSCongruence</a> of <font color="Maroon">F<sub>2</sub></font>() <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:3_1"><span class="lab"><font color="Green" title="E11">A9</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c27">F1</font> being   <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon">F<sub>2</sub></font>(),<span class="p1">(<span class="default"><font color="Maroon" title="c16">FF</font> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Maroon" title="c25">c</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E3:3_1_10_1_2"/><span class="lab"><font color="Green" title="E28">A42</font></span>: 
<font color="Maroon" title="c27">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">FofI1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">c</font>
 <span class="kw">and </span><br/><a NAME="E4:3_1_10_1_2"/>
<font color="Maroon" title="c27">F1</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon" title="c16">FF</font> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Maroon" title="c25">c</font>
 <span class="kw">and </span><br/><a NAME="E5:3_1_10_1_2"/><span class="lab"><font color="Green" title="E29">A43</font></span>: 
 ex <font color="Olive" title="b1">R</font> being  <a href="msualg_4.html#NM5" title="MSUALG_4:NM.5">MSCongruence</a> of <font color="Maroon">F<sub>2</sub></font>() st <br/>( <font color="Maroon" title="c25">c</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Olive" title="b1">R</font> &amp; <font color="Maroon" title="c27">F1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="msualg_4.html#K15" title="MSUALG_4:func.15">MSNat_Hom</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Olive" title="b1">R</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:3_1"><span class="lab"><font color="Green" title="E17">A20</font></span></a></span>;<br/>
<a NAME="E6:3_1_10_1_2"/>
<font color="Maroon" title="c26">t1</font> is    <a href="closure2.html#M1" title="CLOSURE2:mode.1">Element</a> of  <a href="closure2.html#K6" title="CLOSURE2:func.6">Bool</a> <span class="p1"><a href="pboole.html#K8" title="PBOOLE:func.8">[|</a><span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>(),the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>()</span><a href="pboole.html#K8" title="PBOOLE:func.8">|]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="closure2.html#D1" title="CLOSURE2:def.1">CLOSURE2:def 1</a></span>;<br/>
<a NAME="E7:3_1_10_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c26">t1</font> <a href="msualg_4.html#K2" title="MSUALG_4:func.2">.</a> <font color="Maroon" title="c21">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="closure2.html#K5" title="CLOSURE2:func.5">|:</a><span class="default"><font color="Maroon" title="c8">CC</font></span><a href="closure2.html#K5" title="CLOSURE2:func.5">:|</a></span> <a href="mssubfam.html#K2" title="MSSUBFAM:func.2">.</a> <font color="Maroon" title="c21">s</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:3_1_10"><span class="lab"><font color="Green" title="E23">A28</font></span></a></span>;<br/>
<a NAME="E8:3_1_10_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A44</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c23">a</font>,<font color="Maroon" title="c24">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c26">t1</font> <a href="msualg_4.html#K2" title="MSUALG_4:func.2">.</a> <font color="Maroon" title="c21">s</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:3_1_10"><span class="lab"><font color="Green" title="E21">A26</font></span></a>, <a class="txt" href="#E5:3_1_10"><span class="lab"><font color="Green" title="E22">A27</font></span></a>, <a class="txt" href="#E2:3_1_10_1"><span class="lab"><font color="Green" title="E24">A38</font></span></a>, <a class="ref" href="setfam_1.html#T58" title="SETFAM_1:th.58">SETFAM_1:58</a></span>;<br/>
<span class="kw">thus </span><span class="p1">(<span class="default"><a href="card_3.html#K12" title="CARD_3:func.12">proj</a> <span class="p2">(<span class="default"><a href="pralg_2.html#K9" title="PRALG_2:func.9">Carrier</a> <font color="Maroon" title="c16">FF</font>,<font color="Maroon" title="c21">s</font></span>)</span>,<font color="Maroon" title="c25">c</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">H</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c21">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">a</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_3.html#K2" title="PRALG_3:func.2">proj</a> <font color="Maroon" title="c16">FF</font>,<font color="Maroon" title="c25">c</font></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c21">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">H</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c21">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">a</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="pralg_3.html#D3" title="PRALG_3:def.3">PRALG_3:def 3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pralg_3.html#K2" title="PRALG_3:func.2">proj</a> <font color="Maroon" title="c16">FF</font>,<font color="Maroon" title="c25">c</font></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c21">s</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">H</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c21">s</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">a</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:3_1_10_1"><span class="lab"><font color="Green" title="E25">A39</font></span></a>, <a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pralg_3.html#K2" title="PRALG_3:func.2">proj</a> <font color="Maroon" title="c16">FF</font>,<font color="Maroon" title="c25">c</font></span>)</span> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c19">H</font></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c21">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">a</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_3.html#T2" title="MSUALG_3:th.2">MSUALG_3:2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c27">F1</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c21">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">a</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E39:3_1"><span class="lab"><font color="Green" title="E20">A25</font></span></a>, <a class="txt" href="#E3:3_1_10_1_2"><span class="lab"><font color="Green" title="E28">A42</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c27">F1</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c21">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">b</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:3_1_10_1"><span class="lab"><font color="Green" title="E25">A39</font></span></a>, <a class="txt" href="#E4:3_1_10_1"><span class="lab"><font color="Green" title="E26">A40</font></span></a>, <a class="txt" href="#E8:3_1_10_1_2"><span class="lab"><font color="Green" title="E30">A44</font></span></a>, <a class="txt" href="#E5:3_1_10_1_2"><span class="lab"><font color="Green" title="E29">A43</font></span></a>, <a class="ref" href="msualg_9.html#T34" title="MSUALG_9:th.34">MSUALG_9:34</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pralg_3.html#K2" title="PRALG_3:func.2">proj</a> <font color="Maroon" title="c16">FF</font>,<font color="Maroon" title="c25">c</font></span>)</span> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c19">H</font></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c21">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">b</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E39:3_1"><span class="lab"><font color="Green" title="E20">A25</font></span></a>, <a class="txt" href="#E3:3_1_10_1_2"><span class="lab"><font color="Green" title="E28">A42</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pralg_3.html#K2" title="PRALG_3:func.2">proj</a> <font color="Maroon" title="c16">FF</font>,<font color="Maroon" title="c25">c</font></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c21">s</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">H</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c21">s</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">b</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_3.html#T2" title="MSUALG_3:th.2">MSUALG_3:2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_3.html#K2" title="PRALG_3:func.2">proj</a> <font color="Maroon" title="c16">FF</font>,<font color="Maroon" title="c25">c</font></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c21">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">H</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c21">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">b</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:3_1_10_1"><span class="lab"><font color="Green" title="E26">A40</font></span></a>, <a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="card_3.html#K12" title="CARD_3:func.12">proj</a> <span class="p2">(<span class="default"><a href="pralg_2.html#K9" title="PRALG_2:func.9">Carrier</a> <font color="Maroon" title="c16">FF</font>,<font color="Maroon" title="c21">s</font></span>)</span>,<font color="Maroon" title="c25">c</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">H</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c21">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">b</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="pralg_3.html#D3" title="PRALG_3:def.3">PRALG_3:def 3</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E6:3_1_10_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c19">H</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c21">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <font color="Maroon" title="c16">FF</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">s</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:3_1_10_1"><span class="lab"><font color="Green" title="E26">A40</font></span></a>, <a class="ref" href="funct_2.html#T7" title="FUNCT_2:th.7">FUNCT_2:7</a></span>;<br/>
<a NAME="E7:3_1_10_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E28">A45</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c19">H</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c21">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K9" title="PRALG_2:func.9">Carrier</a> <font color="Maroon" title="c16">FF</font>,<font color="Maroon" title="c21">s</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pralg_2.html#D17" title="PRALG_2:def.17">PRALG_2:def 17</a></span>;<br/>
<a NAME="E8:3_1_10_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c19">H</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c21">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <font color="Maroon" title="c16">FF</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">s</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:3_1_10_1"><span class="lab"><font color="Green" title="E25">A39</font></span></a>, <a class="ref" href="funct_2.html#T7" title="FUNCT_2:th.7">FUNCT_2:7</a></span>;<br/>
<a NAME="E9:3_1_10_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c19">H</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c21">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K9" title="PRALG_2:func.9">Carrier</a> <font color="Maroon" title="c16">FF</font>,<font color="Maroon" title="c21">s</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pralg_2.html#D17" title="PRALG_2:def.17">PRALG_2:def 17</a></span>;<br/>
<a NAME="E10:3_1_10_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c19">H</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c21">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">H</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c21">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:3_1_10_1"><span class="lab"><font color="Green" title="E28">A45</font></span></a>, <a class="txt" href="#E5:3_1_10_1"><span class="lab"><font color="Green" title="E27">A41</font></span></a>, <a class="ref" href="msualg_9.html#T15" title="MSUALG_9:th.15">MSUALG_9:15</a></span>;<br/>
<a NAME="E11:3_1_10_1"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c23">a</font>,<font color="Maroon" title="c24">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msualg_4.html#K16" title="MSUALG_4:func.16">MSCng</a> <font color="Maroon" title="c19">H</font>,<font color="Maroon" title="c21">s</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:3_1_10_1"><span class="lab"><font color="Green" title="E25">A39</font></span></a>, <a class="txt" href="#E4:3_1_10_1"><span class="lab"><font color="Green" title="E26">A40</font></span></a>, <a class="ref" href="msualg_4.html#D19" title="MSUALG_4:def.19">MSUALG_4:def 19</a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:3_1_10_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c23">a</font>,<font color="Maroon" title="c24">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="msualg_4.html#K17" title="MSUALG_4:func.17">MSCng</a> <font color="Maroon" title="c19">H</font></span>)</span> <a href="msualg_4.html#K2" title="MSUALG_4:func.2">.</a> <font color="Maroon" title="c21">s</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:3_1"><span class="lab"><font color="Green" title="E19">A24</font></span></a>, <a class="ref" href="msualg_4.html#D20" title="MSUALG_4:def.20">MSUALG_4:def 20</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E8:3_1_10"/>
<span class="p1">(<span class="default"><a href="msualg_4.html#K17" title="MSUALG_4:func.17">MSCng</a> <font color="Maroon" title="c19">H</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">R0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E41:3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E21">A46</font></span>: 
 <a href="msualg_4.html#K17" title="MSUALG_4:func.17">MSCng</a> <font color="Maroon" title="c19">H</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c12">R0</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pboole.html#T3" title="PBOOLE:th.3">PBOOLE:3</a></span>;<br/>
<a NAME="E42:3_1"/>
 <a href="msualg_4.html#K13" title="MSUALG_4:func.13">QuotMSAlg</a> <font color="Maroon">F<sub>2</sub></font>(),<span class="p1">(<span class="default"><a href="msualg_4.html#K17" title="MSUALG_4:func.17">MSCng</a> <font color="Maroon" title="c19">H</font></span>)</span>, <a href="msualg_3.html#K7" title="MSUALG_3:func.7">Image</a> <span class="p1">(<span class="default"><a href="msualg_4.html#K19" title="MSUALG_4:func.19">MSHomQuot</a> <font color="Maroon" title="c19">H</font></span>)</span> <a href="msualg_3.html#R6" title="MSUALG_3:pred.6">are_isomorphic</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:3_1"><span class="lab"><font color="Green" title="E19">A24</font></span></a>, <a class="ref" href="msualg_4.html#T4" title="MSUALG_4:th.4">MSUALG_4:4</a>, <a class="ref" href="msualg_9.html#T17" title="MSUALG_9:th.17">MSUALG_9:17</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c20">XX</font> being   <a href="msualg_1.html#V3" title="MSUALG_1:attr.3">strict</a>   <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">non-empty</a>   <a href="msualg_2.html#M1" title="MSUALG_2:mode.1">MSSubAlgebra</a> of  <a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <font color="Maroon" title="c16">FF</font><span class="kw"> such that </span><br/><a NAME="E44:3_1"/><span class="lab"><font color="Green" title="E22">A47</font></span>: 
<font color="Maroon" title="c13">A</font>,<font color="Maroon" title="c20">XX</font> <a href="msualg_3.html#R6" title="MSUALG_3:pred.6">are_isomorphic</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E41:3_1"><span class="lab"><font color="Green" title="E21">A46</font></span></a></span>;<br/>
<a NAME="E45:3_1"/>
for <font color="Olive" title="b1">cc</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">cc</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">CC</font> holds <br/> ex <font color="Olive" title="b2">A</font> being   <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon">F<sub>1</sub></font>() st <br/>( <font color="Olive" title="b2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">FF</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">cc</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b2">A</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_11"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c21">cc</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c21">cc</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">CC</font> implies  ex <font color="Olive" title="b1">A</font> being   <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon">F<sub>1</sub></font>() st <br/>( <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">FF</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">cc</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">A</font>] ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:3_1_11"/><span class="lab"><font color="Green" title="E23">A48</font></span>: 
<font color="Maroon" title="c21">cc</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">CC</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">A</font> being   <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon">F<sub>1</sub></font>() st <br/>( <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">FF</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">cc</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">A</font>] )</span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c22">c</font> = <font color="Maroon" title="c21">cc</font> as    <a href="closure2.html#M1" title="CLOSURE2:mode.1">Element</a> of <font color="Maroon" title="c8">CC</font> ;<br/>
<span class="kw">take </span><font color="Maroon" title="c23">V</font> = <font color="Maroon" title="c16">FF</font> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Maroon" title="c22">c</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c23">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">FF</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">cc</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c23">V</font>] )</span><br/>

<a NAME="E3:3_1_11"/><span class="lab"><font color="Green" title="E24">A49</font></span>: 
 ex <font color="Olive" title="b1">C</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K6" title="MSUALG_5:func.6">CongrLatt</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> st <br/>( <font color="Maroon" title="c21">cc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">C</font> &amp;  ex <font color="Olive" title="b2">R</font> being  <a href="msualg_4.html#NM5" title="MSUALG_4:NM.5">MSCongruence</a> of <font color="Maroon">F<sub>2</sub></font>() st <br/>( <font color="Olive" title="b2">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">C</font> &amp; <font color="Maroon">P<sub>1</sub></font>[ <a href="msualg_4.html#K13" title="MSUALG_4:func.13">QuotMSAlg</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Olive" title="b2">R</font>] ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:3_1_11"><span class="lab"><font color="Green" title="E23">A48</font></span></a></span>;<br/>
<a NAME="E4:3_1_11"/>
 ex <font color="Olive" title="b1">R</font> being  <a href="msualg_4.html#NM5" title="MSUALG_4:NM.5">MSCongruence</a> of <font color="Maroon">F<sub>2</sub></font>() st <br/>( <font color="Olive" title="b1">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">cc</font> &amp; <font color="Maroon" title="c16">FF</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">cc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msualg_4.html#K13" title="MSUALG_4:func.13">QuotMSAlg</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Olive" title="b1">R</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:3_1"><span class="lab"><font color="Green" title="E15">A16</font></span></a>, <a class="txt" href="#E1:3_1_11"><span class="lab"><font color="Green" title="E23">A48</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:3_1_11"/>
( <font color="Maroon" title="c23">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">FF</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">cc</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c23">V</font>] )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:3_1_11"><span class="lab"><font color="Green" title="E24">A49</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E46:3_1"/><span class="kw">then </span>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c20">XX</font>]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:3"><span class="lab"><font color="Green" title="E4">A2</font></span></a>, <a class="txt" href="#E3:3"><span class="lab"><font color="Green" title="E5">A3</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E47:3_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c13">A</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:3"><span class="lab"><font color="Green" title="E3">A1</font></span></a>, <a class="txt" href="#E44:3_1"><span class="lab"><font color="Green" title="E22">A47</font></span></a>, <a class="ref" href="msualg_3.html#T17" title="MSUALG_3:th.17">MSUALG_3:17</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c14">F</font> <a href="msualg_3.html#R2" title="MSUALG_3:pred.2">is_epimorphism</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon" title="c13">A</font> &amp; ( for <font color="Olive" title="b1">B</font> being  <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">non-empty</a>   <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon">F<sub>1</sub></font>()<br/> for <font color="Olive" title="b2">G</font> being  <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon">F<sub>2</sub></font>(),<font color="Olive" title="b1">B</font>  st <font color="Olive" title="b2">G</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Olive" title="b1">B</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">B</font>] holds <br/> ex <font color="Olive" title="b3">H</font> being  <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon" title="c13">A</font>,<font color="Olive" title="b1">B</font> st <br/>( <font color="Olive" title="b3">H</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon" title="c13">A</font>,<font color="Olive" title="b1">B</font> &amp; <font color="Olive" title="b3">H</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c14">F</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Olive" title="b2">G</font> &amp; ( for <font color="Olive" title="b4">K</font> being  <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon" title="c13">A</font>,<font color="Olive" title="b1">B</font>  st <font color="Olive" title="b4">K</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c14">F</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Olive" title="b2">G</font> holds <br/><font color="Olive" title="b3">H</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Olive" title="b4">K</font> ) ) ) )</span><br/>

<span class="kw">thus </span><a NAME="E48:3_1"/>
<font color="Maroon" title="c14">F</font> <a href="msualg_3.html#R2" title="MSUALG_3:pred.2">is_epimorphism</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon" title="c13">A</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_4.html#T3" title="MSUALG_4:th.3">MSUALG_4:3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">B</font> being  <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">non-empty</a>   <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon">F<sub>1</sub></font>()<br/> for <font color="Olive" title="b2">G</font> being  <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon">F<sub>2</sub></font>(),<font color="Olive" title="b1">B</font>  st <font color="Olive" title="b2">G</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Olive" title="b1">B</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">B</font>] holds <br/> ex <font color="Olive" title="b3">H</font> being  <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon" title="c13">A</font>,<font color="Olive" title="b1">B</font> st <br/>( <font color="Olive" title="b3">H</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon" title="c13">A</font>,<font color="Olive" title="b1">B</font> &amp; <font color="Olive" title="b3">H</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c14">F</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Olive" title="b2">G</font> &amp; ( for <font color="Olive" title="b4">K</font> being  <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon" title="c13">A</font>,<font color="Olive" title="b1">B</font>  st <font color="Olive" title="b4">K</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c14">F</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Olive" title="b2">G</font> holds <br/><font color="Olive" title="b3">H</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Olive" title="b4">K</font> ) )</span><br/>

<a NAME="E49:3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E23">A50</font></span>: 
<font color="Maroon" title="c14">F</font> is  <a href="msualg_3.html#V2" title="MSUALG_3:attr.2">"onto"</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_3.html#D10" title="MSUALG_3:def.10">MSUALG_3:def 10</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c21">B</font> be   <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">non-empty</a>   <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon">F<sub>1</sub></font>(); <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">G</font> being  <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon" title="c21">B</font>  st <font color="Olive" title="b1">G</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon" title="c21">B</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c21">B</font>] holds <br/> ex <font color="Olive" title="b2">H</font> being  <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon" title="c13">A</font>,<font color="Maroon" title="c21">B</font> st <br/>( <font color="Olive" title="b2">H</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon" title="c13">A</font>,<font color="Maroon" title="c21">B</font> &amp; <font color="Olive" title="b2">H</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c14">F</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Olive" title="b1">G</font> &amp; ( for <font color="Olive" title="b3">K</font> being  <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon" title="c13">A</font>,<font color="Maroon" title="c21">B</font>  st <font color="Olive" title="b3">K</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c14">F</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Olive" title="b1">G</font> holds <br/><font color="Olive" title="b2">H</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Olive" title="b3">K</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c22">G</font> be   <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon" title="c21">B</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c22">G</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon" title="c21">B</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c21">B</font>] implies  ex <font color="Olive" title="b1">H</font> being  <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon" title="c13">A</font>,<font color="Maroon" title="c21">B</font> st <br/>( <font color="Olive" title="b1">H</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon" title="c13">A</font>,<font color="Maroon" title="c21">B</font> &amp; <font color="Olive" title="b1">H</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c14">F</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c22">G</font> &amp; ( for <font color="Olive" title="b2">K</font> being  <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon" title="c13">A</font>,<font color="Maroon" title="c21">B</font>  st <font color="Olive" title="b2">K</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c14">F</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c22">G</font> holds <br/><font color="Olive" title="b1">H</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Olive" title="b2">K</font> ) ) )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E50:3_1"/><span class="lab"><font color="Green" title="E24">A51</font></span>: 
<font color="Maroon" title="c22">G</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon" title="c21">B</font>
 <span class="kw">and </span><br/><a NAME="E51:3_1"/><span class="lab"><font color="Green" title="E25">A52</font></span>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c21">B</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">H</font> being  <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon" title="c13">A</font>,<font color="Maroon" title="c21">B</font> st <br/>( <font color="Olive" title="b1">H</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon" title="c13">A</font>,<font color="Maroon" title="c21">B</font> &amp; <font color="Olive" title="b1">H</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c14">F</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c22">G</font> &amp; ( for <font color="Olive" title="b2">K</font> being  <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon" title="c13">A</font>,<font color="Maroon" title="c21">B</font>  st <font color="Olive" title="b2">K</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c14">F</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c22">G</font> holds <br/><font color="Olive" title="b1">H</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Olive" title="b2">K</font> ) )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c23">C</font> =  <a href="msualg_3.html#K7" title="MSUALG_3:func.7">Image</a> <span class="p1">(<span class="default"><a href="msualg_4.html#K19" title="MSUALG_4:func.19">MSHomQuot</a> <font color="Maroon" title="c22">G</font></span>)</span> as   <a href="msualg_1.html#V3" title="MSUALG_1:attr.3">strict</a>   <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">non-empty</a>   <a href="msualg_2.html#M1" title="MSUALG_2:mode.1">MSSubAlgebra</a> of <font color="Maroon" title="c21">B</font> ;<br/>
<a NAME="E53:3_1"/><span class="lab"><font color="Green" title="E26">A53</font></span>: 
 <a href="msualg_4.html#K13" title="MSUALG_4:func.13">QuotMSAlg</a> <font color="Maroon">F<sub>2</sub></font>(),<span class="p1">(<span class="default"><a href="msualg_4.html#K17" title="MSUALG_4:func.17">MSCng</a> <font color="Maroon" title="c22">G</font></span>)</span>,<font color="Maroon" title="c23">C</font> <a href="msualg_3.html#R6" title="MSUALG_3:pred.6">are_isomorphic</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E50:3_1"><span class="lab"><font color="Green" title="E24">A51</font></span></a>, <a class="ref" href="msualg_4.html#T4" title="MSUALG_4:th.4">MSUALG_4:4</a>, <a class="ref" href="msualg_9.html#T17" title="MSUALG_9:th.17">MSUALG_9:17</a></span>;<br/>
<a NAME="E54:3_1"/><span class="lab"><font color="Green" title="E27">A54</font></span>: 
 <a href="msualg_4.html#K17" title="MSUALG_4:func.17">MSCng</a> <font color="Maroon" title="c22">G</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K6" title="MSUALG_5:func.6">CongrLatt</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_5.html#D6" title="MSUALG_5:def.6">MSUALG_5:def 6</a></span>;<br/>
<a NAME="E55:3_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c23">C</font>]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:3"><span class="lab"><font color="Green" title="E4">A2</font></span></a>, <a class="txt" href="#E51:3_1"><span class="lab"><font color="Green" title="E25">A52</font></span></a></span>;<br/>
<a NAME="E56:3_1"/><span class="kw">then </span>
<font color="Maroon">P<sub>1</sub></font>[ <a href="msualg_4.html#K13" title="MSUALG_4:func.13">QuotMSAlg</a> <font color="Maroon">F<sub>2</sub></font>(),<span class="p1">(<span class="default"><a href="msualg_4.html#K17" title="MSUALG_4:func.17">MSCng</a> <font color="Maroon" title="c22">G</font></span>)</span>]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:3"><span class="lab"><font color="Green" title="E3">A1</font></span></a>, <a class="txt" href="#E53:3_1"><span class="lab"><font color="Green" title="E26">A53</font></span></a>, <a class="ref" href="msualg_3.html#T17" title="MSUALG_3:th.17">MSUALG_3:17</a></span>;<br/>
<a NAME="E57:3_1"/><span class="kw">then </span>
 <a href="msualg_4.html#K17" title="MSUALG_4:func.17">MSCng</a> <font color="Maroon" title="c22">G</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">CC</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E54:3_1"><span class="lab"><font color="Green" title="E27">A54</font></span></a></span>;<br/>
<a NAME="E58:3_1"/><span class="kw">then </span>
<font color="Maroon" title="c12">R0</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a>  <a href="msualg_4.html#K17" title="MSUALG_4:func.17">MSCng</a> <font color="Maroon" title="c22">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="closure2.html#T22" title="CLOSURE2:th.22">CLOSURE2:22</a>, <a class="ref" href="mssubfam.html#T43" title="MSSUBFAM:th.43">MSSUBFAM:43</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c24">H</font> being   <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon" title="c13">A</font>,<font color="Maroon" title="c21">B</font><span class="kw"> such that </span><br/><a NAME="E60:3_1"/><span class="lab"><font color="Green" title="E28">A55</font></span>: 
<font color="Maroon" title="c24">H</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon" title="c13">A</font>,<font color="Maroon" title="c21">B</font>
 <span class="kw">and </span><br/><a NAME="E61:3_1"/><span class="lab"><font color="Green" title="E29">A56</font></span>: 
<font color="Maroon" title="c22">G</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c24">H</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c14">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E50:3_1"><span class="lab"><font color="Green" title="E24">A51</font></span></a>, <a class="ref" href="msualg_9.html#T37" title="MSUALG_9:th.37">MSUALG_9:37</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c24">H</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c24">H</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon" title="c13">A</font>,<font color="Maroon" title="c21">B</font> &amp; <font color="Maroon" title="c24">H</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c14">F</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c22">G</font> &amp; ( for <font color="Olive" title="b1">K</font> being  <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon" title="c13">A</font>,<font color="Maroon" title="c21">B</font>  st <font color="Olive" title="b1">K</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c14">F</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c22">G</font> holds <br/><font color="Maroon" title="c24">H</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Olive" title="b1">K</font> ) )</span><br/>

<span class="kw">thus </span><a NAME="E62:3_1"/>
<font color="Maroon" title="c24">H</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon" title="c13">A</font>,<font color="Maroon" title="c21">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E60:3_1"><span class="lab"><font color="Green" title="E28">A55</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c24">H</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c14">F</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c22">G</font> &amp; ( for <font color="Olive" title="b1">K</font> being  <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon" title="c13">A</font>,<font color="Maroon" title="c21">B</font>  st <font color="Olive" title="b1">K</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c14">F</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c22">G</font> holds <br/><font color="Maroon" title="c24">H</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Olive" title="b1">K</font> ) )</span><br/>

<span class="kw">thus </span><a NAME="E63:3_1"/>
<font color="Maroon" title="c22">G</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c24">H</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c14">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E61:3_1"><span class="lab"><font color="Green" title="E29">A56</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">K</font> being  <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon" title="c13">A</font>,<font color="Maroon" title="c21">B</font>  st <font color="Olive" title="b1">K</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c14">F</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c22">G</font> holds <br/><font color="Maroon" title="c24">H</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Olive" title="b1">K</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c25">K</font> be   <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon" title="c13">A</font>,<font color="Maroon" title="c21">B</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c25">K</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c14">F</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c22">G</font> implies <font color="Maroon" title="c24">H</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c25">K</font> )</span><br/>

<span class="kw">assume </span><a NAME="E64:3_1"/>
<font color="Maroon" title="c25">K</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c14">F</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c22">G</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c24">H</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c25">K</font></span><br/>

<span class="kw">hence </span><a NAME="E65:3_1"/>
<font color="Maroon" title="c24">H</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c25">K</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E61:3_1"><span class="lab"><font color="Green" title="E29">A56</font></span></a>, <a class="txt" href="#E49:3_1"><span class="lab"><font color="Green" title="E23">A50</font></span></a>, <a class="ref" href="extens_1.html#T16" title="EXTENS_1:th.16">EXTENS_1:16</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
