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

<span class="kw">set </span><font color="Maroon" title="c1">EMF</font> =  the    <a href="pralg_2.html#M1" title="PRALG_2:mode.1">MSAlgebra-Family</a> of  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,<font color="Maroon">F<sub>1</sub></font>();<br/>
<span class="kw">set </span><font color="Maroon" title="c2">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#NM4" title="MSUALG_4:NM.4">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="c3">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="c4">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="c5">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="E2:3_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="c5">Xi</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</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#T5" onmouseover="rs('msualg_9/T5')" onmouseout="rh()">MSUALG_9:5</a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="3_1_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">i</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">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>() holds <br/> 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>  the    <a href="pralg_2.html#M1" title="PRALG_2:mode.1">MSAlgebra-Family</a> of  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,<font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">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="c5">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="Olive" title="b1">i</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c6">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">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>  the    <a href="pralg_2.html#M1" title="PRALG_2:mode.1">MSAlgebra-Family</a> of  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,<font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">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="c5">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="c6">i</font> )</span><br/><span class="kw">assume </span><a NAME="E1:3_1_1"/>
<font color="Maroon" title="c6">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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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>  the    <a href="pralg_2.html#M1" title="PRALG_2:mode.1">MSAlgebra-Family</a> of  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,<font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">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="c5">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="c6">i</font></span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">s</font> = <font color="Maroon" title="c6">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>  the    <a href="pralg_2.html#M1" title="PRALG_2:mode.1">MSAlgebra-Family</a> of  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,<font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">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> ( the    <a href="pralg_2.html#M1" title="PRALG_2:mode.1">MSAlgebra-Family</a> of  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon" title="c7">s</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="pralg_2.html#D10" onmouseover="rs('pralg_2/D10')" onmouseout="rh()">PRALG_2:def 10</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#T10" onmouseover="rs('card_3/T10')" onmouseout="rh()">CARD_3:10</a>, <a class="ref" href="pralg_2.html#D9" onmouseover="rs('pralg_2/D9')" onmouseout="rh()">PRALG_2:def 9</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="c7">s</font>

<br/>.= 
<span class="p1"><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">{</a><span class="default"><font color="Maroon" title="c5">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="c6">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:3_1"><span class="lab"><font color="Green" title="E6">A4</font></span></a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E4:3_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>  the    <a href="pralg_2.html#M1" title="PRALG_2:mode.1">MSAlgebra-Family</a> of  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,<font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</a> <span class="p1"><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">{</a><span class="default"><font color="Maroon" title="c5">Xi</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span>
 
;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c6">R</font> = <span class="p1"><a href="pboole.html#K6" title="PBOOLE:func.6">[|</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#K6" title="PBOOLE:func.6">|]</a></span> as   <a href="msualg_4.html#NM4" title="MSUALG_4:NM.4">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#T18" onmouseover="rs('msualg_5/T18')" onmouseout="rh()">MSUALG_5:18</a></span>;<br/>
<a NAME="E6:3_1"/>
<span class="p1"><a href="pboole.html#K6" title="PBOOLE:func.6">[|</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#K6" title="PBOOLE:func.6">|]</a></span> is   <a href="msualg_4.html#NM4" title="MSUALG_4:NM.4">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#T18" onmouseover="rs('msualg_5/T18')" onmouseout="rh()">MSUALG_5:18</a></span>;<br/>
<a NAME="E7:3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E8">A6</font></span>: 
<span class="p1"><a href="pboole.html#K6" title="PBOOLE:func.6">[|</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#K6" title="PBOOLE:func.6">|]</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" onmouseover="rs('msualg_5/D6')" onmouseout="rh()">MSUALG_5:def 6</a></span>;<br/>
<a NAME="E8:3_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="c6">R</font>)</span>)</span> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</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#T29" onmouseover="rs('msualg_9/T29')" onmouseout="rh()">MSUALG_9:29</a></span>;<br/>
<a NAME="E9:3_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="c6">R</font>), <a href="msafree2.html#K7" title="MSAFREE2:func.7">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#T26" onmouseover="rs('msualg_9/T26')" onmouseout="rh()">MSUALG_9:26</a></span>;<br/>
<a NAME="E10:3_1"/>
 for <font color="Olive" title="b1">i</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b1">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  holds <br/> ex <font color="Olive" title="b2">A</font> being    <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> over <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>  the    <a href="pralg_2.html#M1" title="PRALG_2:mode.1">MSAlgebra-Family</a> of  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,<font color="Maroon">F<sub>1</sub></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="E11:3_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>  the    <a href="pralg_2.html#M1" title="PRALG_2:mode.1">MSAlgebra-Family</a> of  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,<font color="Maroon">F<sub>1</sub></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="E12:3_1"/><span class="kw">then </span>
<font color="Maroon">P<sub>1</sub></font>[ <a href="msafree2.html#K7" title="MSAFREE2:func.7">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="#E4:3_1"><span class="lab"><font color="Green" title="E7">A5</font></span></a>, <a class="ref" href="msualg_9.html#T26" onmouseover="rs('msualg_9/T26')" onmouseout="rh()">MSUALG_9:26</a></span>;<br/>
<a NAME="E13:3_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="c6">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="#E9:3_1"><span class="lab"><font color="Green" title="E9">A7</font></span></a>, <a class="ref" href="msualg_3.html#T17" onmouseover="rs('msualg_3/T17')" onmouseout="rh()">MSUALG_3:17</a></span>;<br/>
<a NAME="E14:3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E10">A8</font></span>: 
<span class="p1"><a href="pboole.html#K6" title="PBOOLE:func.6">[|</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#K6" title="PBOOLE:func.6">|]</a></span> <a href="tarski.html#R2" title="TARSKI: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#NM4" title="MSUALG_4:NM.4">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="#E7:3_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">object</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ] <span class="kw">means </span> ex <font color="Olive" title="b1">R</font> being   <a href="msualg_4.html#NM4" title="MSUALG_4:NM.4">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#M2" title="HIDDEN:mode.2">set</a> ] <span class="kw">means </span> ex <font color="Olive" title="b1">R</font> being   <a href="msualg_4.html#NM4" title="MSUALG_4:NM.4">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" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="3_1_2"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">q</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b2">C</font> where <font color="Olive" title="b2">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="b3">R</font> being   <a href="msualg_4.html#NM4" title="MSUALG_4:NM.4">MSCongruence</a> of <font color="Maroon">F<sub>2</sub></font>() st <br/>( <font color="Olive" title="b3">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">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="b3">R</font>)] ) </span> } </span>  holds <br/><font color="Olive" title="b1">q</font> is   <a href="msualg_4.html#NM4" title="MSUALG_4:NM.4">MSCongruence</a> of <font color="Maroon">F<sub>2</sub></font>()</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c7">q</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">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#NM4" title="MSUALG_4:NM.4">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="c7">q</font> is   <a href="msualg_4.html#NM4" title="MSUALG_4:NM.4">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="c7">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#NM4" title="MSUALG_4:NM.4">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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">q</font> is   <a href="msualg_4.html#NM4" title="MSUALG_4:NM.4">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="c7">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#NM4" title="MSUALG_4:NM.4">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="c7">q</font> is   <a href="msualg_4.html#NM4" title="MSUALG_4:NM.4">MSCongruence</a> of <font color="Maroon">F<sub>2</sub></font>()
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E16:3_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#NM4" title="MSUALG_4:NM.4">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" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" 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="c7">q</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c7">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#NM4" title="MSUALG_4:NM.4">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="c7">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="c7">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#NM4" title="MSUALG_4:NM.4">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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">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="c7">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="#E15:3_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="c7">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" onmouseover="rs('msualg_5/D5')" onmouseout="rh()">MSUALG_5:def 5</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="c7">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#NM4" title="MSUALG_4:NM.4">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#K6" title="PBOOLE:func.6">[|</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#K6" title="PBOOLE:func.6">|]</a></span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:3_1"><span class="lab"><font color="Green" title="E10">A8</font></span></a>, <a class="ref" href="msualg_7.html#T5" onmouseover="rs('msualg_7/T5')" onmouseout="rh()">MSUALG_7:5</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">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="c7">CC</font></span><a href="closure2.html#K5" title="CLOSURE2:func.5">:|</a></span>;<br/>
<a NAME="E18:3_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="c7">CC</font></span><a href="closure2.html#K5" title="CLOSURE2:func.5">:|</a></span> is   <a href="msualg_4.html#V1" title="MSUALG_4:attr.1">MSEquivalence_Relation-like</a>  <a href="msualg_4.html#NM2" title="MSUALG_4:NM.2">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="#E16:3_1"><span class="lab"><font color="Green" title="E12">A10</font></span></a>, <a class="ref" href="msualg_7.html#T8" onmouseover="rs('msualg_7/T8')" onmouseout="rh()">MSUALG_7:8</a></span>;<br/>
<span class="kw">reconsider </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="c7">CC</font></span><a href="closure2.html#K5" title="CLOSURE2:func.5">:|</a></span> as   <a href="msualg_4.html#NM3" title="MSUALG_4:NM.3">ManySortedRelation</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="E12">A10</font></span></a>, <a class="ref" href="msualg_7.html#T8" onmouseover="rs('msualg_7/T8')" onmouseout="rh()">MSUALG_7:8</a></span>;<br/>
<a NAME="E20:3_1"/>
<font color="Maroon" title="c9">R0</font> is  <a href="msualg_4.html#V2" title="MSUALG_4:attr.2">MSEquivalence-like</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:3_1"><span class="lab"><font color="Green" title="E13">A11</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">R0</font> = <font color="Maroon" title="c9">R0</font> as   <a href="msualg_4.html#V2" title="MSUALG_4:attr.2">MSEquivalence-like</a>  <a href="msualg_4.html#NM3" title="MSUALG_4:NM.3">ManySortedRelation</a> of <font color="Maroon">F<sub>2</sub></font>() ;<br/>
<a NAME="E22:3_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" onmouseover="rs('domain_1/S7')" onmouseout="rh()">DOMAIN_1:sch 7</a>();<br/></span>
<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#NM4" title="MSUALG_4:NM.4">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#T28" onmouseover="rs('msualg_9/T28')" onmouseout="rh()">MSUALG_9:28</a></span>;<br/>
<span class="kw">take </span><font color="Maroon" title="c12">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="c11">R0</font>); <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="c12">A</font> st <br/>( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c12">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="c12">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> over <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="c12">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="c12">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#R8" title="PBOOLE:pred.8">=</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="c12">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#R8" title="PBOOLE:pred.8">=</a> <font color="Olive" title="b3">G</font> holds <br/><font color="Olive" title="b4">H</font> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</a> <font color="Olive" title="b5">K</font> ) ) ) )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c13">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="c11">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="c12">A</font> ;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c13">F</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c12">A</font>] &amp; <font color="Maroon" title="c13">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="c12">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> over <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="c12">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="c12">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="c13">F</font> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</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="c12">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="c13">F</font> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</a> <font color="Olive" title="b2">G</font> holds <br/><font color="Olive" title="b3">H</font> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</a> <font color="Olive" title="b4">K</font> ) ) ) )</span><br/>

<div><span class="lab"><font color="Green" title="E14">A12</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="3_1_4"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">c</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">CC</font> holds <br/> ex <font color="Olive" title="b2">w</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">c</font>,<font color="Olive" title="b2">w</font>]</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c14">c</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c14">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">CC</font> implies  ex <font color="Olive" title="b1">w</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c14">c</font>,<font color="Olive" title="b1">w</font>] )</span><br/><span class="kw">assume </span><a NAME="E1:3_1_4"/>
<font color="Maroon" title="c14">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">CC</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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">object</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c14">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="c15">cc</font> = <font color="Maroon" title="c14">c</font> as   <a href="msualg_4.html#NM4" title="MSUALG_4:NM.4">MSCongruence</a> of <font color="Maroon">F<sub>2</sub></font>() <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:3_1"><span class="lab"><font color="Green" title="E11">A9</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c16">w</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="c15">cc</font>) as    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  ;<br/><span class="kw">take </span><font color="Maroon" title="c17">w</font> = <font color="Maroon" title="c16">w</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c14">c</font>,<font color="Maroon" title="c17">w</font>]</span><br/><span class="kw">thus </span><a NAME="E4:3_1_4"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c14">c</font>,<font color="Maroon" title="c17">w</font>]
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="c14">FF</font> being   <a href="pboole.html#NM1" title="PBOOLE:NM.1">ManySortedSet</a> of <font color="Maroon" title="c7">CC</font><span class="kw"> such that </span><br/><a NAME="E27:3_1"/><span class="lab"><font color="Green" title="E15">A13</font></span>: 
 for <font color="Olive" title="b1">c</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">CC</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">c</font>,<font color="Maroon" title="c14">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" onmouseover="rs('pboole/S3')" onmouseout="rh()">PBOOLE:sch 3</a>(<a class="txt" href="#E25:3_1"><span class="lab"><font color="Green" title="E14">A12</font></span></a>);<br/></span>
<a NAME="E28:3_1"/>
<font color="Maroon" title="c14">FF</font> is    <a href="pralg_2.html#M1" title="PRALG_2:mode.1">MSAlgebra-Family</a> of <font color="Maroon" title="c7">CC</font>,<font color="Maroon">F<sub>1</sub></font>()
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="3_1_5"><span class="kw">proof </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">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="pralg_2.html#D5" onmouseover="rs('pralg_2/D5')" onmouseout="rh()">PRALG_2:def 5</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c15">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">CC</font> or <font color="Maroon" title="c14">FF</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">c</font> is    <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> over <font color="Maroon">F<sub>1</sub></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="c7">CC</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c14">FF</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">c</font> is    <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> over <font color="Maroon">F<sub>1</sub></font>()</span><br/>

<a NAME="E2:3_1_5"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">R</font> being   <a href="msualg_4.html#NM4" title="MSUALG_4:NM.4">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="c15">c</font> &amp; <font color="Maroon" title="c14">FF</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">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="#E27:3_1"><span class="lab"><font color="Green" title="E15">A13</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:3_1_5"/>
<font color="Maroon" title="c14">FF</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">c</font> is    <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> over <font color="Maroon">F<sub>1</sub></font>()
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="c15">FF</font> = <font color="Maroon" title="c14">FF</font> as    <a href="pralg_2.html#M1" title="PRALG_2:mode.1">MSAlgebra-Family</a> of <font color="Maroon" title="c7">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="c7">CC</font>,   <a href="hidden.html#M1" title="HIDDEN:mode.1">object</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="c15">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="c15">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#NM4" title="MSUALG_4:NM.4">MSCongruence</a> of <font color="Maroon">F<sub>2</sub></font>() st <br/>( $1 <a href="pboole.html#R8" title="PBOOLE:pred.8">=</a> <font color="Olive" title="b2">R</font> &amp; <font color="Olive" title="b1">F1</font> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</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="E30:3_1"/><span class="lab"><font color="Green" title="E16">A14</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="c7">CC</font>  ex <font color="Olive" title="b2">j</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</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" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" 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="closure2.html#M1" title="CLOSURE2:mode.1">Element</a> of <font color="Maroon" title="c7">CC</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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">object</a>  st <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c16">c</font>,<font color="Olive" title="b1">j</font>]</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c17">R</font> being   <a href="msualg_4.html#NM4" title="MSUALG_4:NM.4">MSCongruence</a> of <font color="Maroon">F<sub>2</sub></font>()<span class="kw"> such that </span><br/><a NAME="E2:3_1_6"/><span class="lab"><font color="Green" title="E17">A15</font></span>: 
<font color="Maroon" title="c17">R</font> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</a> <font color="Maroon" title="c16">c</font>
 <span class="kw">and </span><br/><a NAME="E3:3_1_6"/><span class="lab"><font color="Green" title="E18">A16</font></span>: 
<font color="Maroon" title="c15">FF</font> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</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="Maroon" title="c17">R</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:3_1"><span class="lab"><font color="Green" title="E15">A13</font></span></a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c18">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="c17">R</font>);<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c19">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="c17">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="c15">FF</font> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Maroon" title="c16">c</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:3_1_6"><span class="lab"><font color="Green" title="E18">A16</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="c17">R</font>)
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c16">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="c17">R</font>)]</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c19">F1</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c19">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="c17">R</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="c15">FF</font> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Maroon" title="c16">c</font> &amp;  ex <font color="Olive" title="b1">R</font> being   <a href="msualg_4.html#NM4" title="MSUALG_4:NM.4">MSCongruence</a> of <font color="Maroon">F<sub>2</sub></font>() st <br/>( <font color="Maroon" title="c16">c</font> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</a> <font color="Olive" title="b1">R</font> &amp; <font color="Maroon" title="c19">F1</font> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</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_6"/>
<font color="Maroon" title="c19">F1</font> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</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="c17">R</font>)
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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="c15">FF</font> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Maroon" title="c16">c</font> &amp;  ex <font color="Olive" title="b1">R</font> being   <a href="msualg_4.html#NM4" title="MSUALG_4:NM.4">MSCongruence</a> of <font color="Maroon">F<sub>2</sub></font>() st <br/>( <font color="Maroon" title="c16">c</font> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</a> <font color="Olive" title="b1">R</font> &amp; <font color="Maroon" title="c19">F1</font> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</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_6"/>
 <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="c17">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="c17">R</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_4.html#T3" onmouseover="rs('msualg_4/T3')" onmouseout="rh()">MSUALG_4:3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:3_1_6"/>
<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="c15">FF</font> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Maroon" title="c16">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:3_1_6"><span class="lab"><font color="Green" title="E18">A16</font></span></a>, <a class="ref" href="msualg_3.html#D8" onmouseover="rs('msualg_3/D8')" onmouseout="rh()">MSUALG_3:def 8</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">R</font> being   <a href="msualg_4.html#NM4" title="MSUALG_4:NM.4">MSCongruence</a> of <font color="Maroon">F<sub>2</sub></font>() st <br/>( <font color="Maroon" title="c16">c</font> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</a> <font color="Olive" title="b1">R</font> &amp; <font color="Maroon" title="c19">F1</font> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</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="c17">R</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c16">c</font> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</a> <font color="Maroon" title="c17">R</font> &amp; <font color="Maroon" title="c19">F1</font> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</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="c17">R</font>) )</span><br/>

<span class="kw">thus </span><a NAME="E8:3_1_6"/>
( <font color="Maroon" title="c16">c</font> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</a> <font color="Maroon" title="c17">R</font> &amp; <font color="Maroon" title="c19">F1</font> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</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="c17">R</font>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:3_1_6"><span class="lab"><font color="Green" title="E17">A15</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="c16">FofI1</font> being   <a href="pboole.html#NM1" title="PBOOLE:NM.1">ManySortedSet</a> of <font color="Maroon" title="c7">CC</font><span class="kw"> such that </span><br/><a NAME="E32:3_1"/><span class="lab"><font color="Green" title="E17">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="c7">CC</font> holds  <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">c</font>,<font color="Maroon" title="c16">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" onmouseover="rs('pboole/S6')" onmouseout="rh()">PBOOLE:sch 6</a>(<a class="txt" href="#E30:3_1"><span class="lab"><font color="Green" title="E16">A14</font></span></a>);<br/></span>
<a NAME="E33:3_1"/><span class="lab"><font color="Green" title="E18">A18</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="c7">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="c15">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="c16">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="c15">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" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" 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="c7">CC</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="c15">FF</font> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Maroon" title="c17">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="c16">FofI1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">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="c15">FF</font> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Maroon" title="c17">c</font> )</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c18">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="c15">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"> such that </span><br/><a NAME="E2:3_1_7"/><span class="lab"><font color="Green" title="E19">A19</font></span>: 
<font color="Maroon" title="c18">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">FofI1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</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="E20">A20</font></span>: 
<font color="Maroon" title="c18">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="c15">FF</font> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Maroon" title="c17">c</font>
 <span class="kw">and </span><br/><a NAME="E4:3_1_7"/>
 ex <font color="Olive" title="b1">R</font> being   <a href="msualg_4.html#NM4" title="MSUALG_4:NM.4">MSCongruence</a> of <font color="Maroon">F<sub>2</sub></font>() st <br/>( <font color="Maroon" title="c17">c</font> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</a> <font color="Olive" title="b1">R</font> &amp; <font color="Maroon" title="c18">F1</font> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</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="#E32:3_1"><span class="lab"><font color="Green" title="E17">A17</font></span></a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c18">F1</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c18">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">FofI1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">c</font> &amp; <font color="Maroon" title="c18">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="c15">FF</font> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Maroon" title="c17">c</font> )</span><br/>

<span class="kw">thus </span><a NAME="E5:3_1_7"/>
( <font color="Maroon" title="c18">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">FofI1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">c</font> &amp; <font color="Maroon" title="c18">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="c15">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="#E2:3_1_7"><span class="lab"><font color="Green" title="E19">A19</font></span></a>, <a class="txt" href="#E3:3_1_7"><span class="lab"><font color="Green" title="E20">A20</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E34:3_1"/>
<font color="Maroon" title="c16">FofI1</font> is  <a href="funcop_1.html#V1" title="FUNCOP_1:attr.1">Function-yielding</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" 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="c17">c</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="funcop_1.html#D6" onmouseover="rs('funcop_1/D6')" onmouseout="rh()">FUNCOP_1:def 6</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c17">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c16">FofI1</font> or <font color="Maroon" title="c16">FofI1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">c</font> is    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:3_1_8"/>
<font color="Maroon" title="c17">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c16">FofI1</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c16">FofI1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">c</font> is    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> </span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c18">cc</font> = <font color="Maroon" title="c17">c</font> as    <a href="closure2.html#M1" title="CLOSURE2:mode.1">Element</a> of <font color="Maroon" title="c7">CC</font> ;<br/>
<a NAME="E3:3_1_8"/>
 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="c15">FF</font> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Maroon" title="c18">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="c16">FofI1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">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="c15">FF</font> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Maroon" title="c18">cc</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:3_1"><span class="lab"><font color="Green" title="E18">A18</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:3_1_8"/>
<font color="Maroon" title="c16">FofI1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">c</font> is    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="c17">FofI1</font> = <font color="Maroon" title="c16">FofI1</font> as   <a href="pboole.html#NM2" title="PBOOLE:NM.2">ManySortedFunction</a> of <font color="Maroon" title="c7">CC</font> ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c18">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="c15">FF</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E37:3_1"/><span class="lab"><font color="Green" title="E19">A21</font></span>: 
<font color="Maroon" title="c18">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="c15">FF</font>
 <span class="kw">and </span><br/><a NAME="E38:3_1"/><span class="lab"><font color="Green" title="E20">A22</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="c7">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="c15">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="c18">H</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:3_1"><span class="lab"><font color="Green" title="E18">A18</font></span></a>, <a class="ref" href="pralg_3.html#T29" onmouseover="rs('pralg_3/T29')" onmouseout="rh()">PRALG_3:29</a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="3_1_9"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">i</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">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>() holds <br/><span class="p1">(<span class="default"><a href="msualg_4.html#K17" title="MSUALG_4:func.17">MSCng</a> <font color="Maroon" title="c18">H</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">R0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c19">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c19">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="c18">H</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">R0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">i</font> )</span><br/><span class="kw">assume </span><a NAME="E1:3_1_9"/>
<font color="Maroon" title="c19">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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="c18">H</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">R0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">i</font></span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c20">s</font> = <font color="Maroon" title="c19">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="c21">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#K6" title="PBOOLE:func.6">[|</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#K6" title="PBOOLE:func.6">|]</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">s</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E4:3_1_9"/><span class="lab"><font color="Green" title="E21">A23</font></span>: 
<font color="Maroon" title="c21">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="c7">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="c20">s</font>
 <span class="kw">and </span><br/><a NAME="E5:3_1_9"/><span class="lab"><font color="Green" title="E22">A24</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="c7">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="c20">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="c21">Q</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="mssubfam.html#D1" onmouseover="rs('mssubfam/D1')" onmouseout="rh()">MSSUBFAM:def 1</a></span>;<br/><a NAME="E6:3_1_9"/><span class="lab"><font color="Green" title="E23">A25</font></span>: 
<span class="p1"><a href="closure2.html#K5" title="CLOSURE2:func.5">|:</a><span class="default"><font color="Maroon" title="c7">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="c20">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="c20">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#K6" title="PBOOLE:func.6">[|</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#K6" title="PBOOLE:func.6">|]</a></span> : <font color="Olive" title="b1">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c7">CC</font> </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="closure2.html#T14" onmouseover="rs('closure2/T14')" onmouseout="rh()">CLOSURE2:14</a></span>;<br/><a NAME="E7:3_1_9"/>
<span class="p1">(<span class="default"><a href="msualg_4.html#K17" title="MSUALG_4:func.17">MSCng</a> <font color="Maroon" title="c18">H</font></span>)</span> <a href="msualg_4.html#K2" title="MSUALG_4:func.2">.</a> <font color="Maroon" title="c20">s</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c11">R0</font> <a href="msualg_4.html#K2" title="MSUALG_4:func.2">.</a> <font color="Maroon" title="c20">s</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="3_1_9_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c22">a</font>, <font color="Maroon" title="c23">b</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="relat_1.html#D2" onmouseover="rs('relat_1/D2')" onmouseout="rh()">RELAT_1:def 2</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="c22">a</font>,<font color="Maroon" title="c23">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="c18">H</font></span>)</span> <a href="msualg_4.html#K2" title="MSUALG_4:func.2">.</a> <font color="Maroon" title="c20">s</font> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c22">a</font>,<font color="Maroon" title="c23">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="c11">R0</font> <a href="msualg_4.html#K2" title="MSUALG_4:func.2">.</a> <font color="Maroon" title="c20">s</font> ) &amp; (  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c22">a</font>,<font color="Maroon" title="c23">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="c11">R0</font> <a href="msualg_4.html#K2" title="MSUALG_4:func.2">.</a> <font color="Maroon" title="c20">s</font> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c22">a</font>,<font color="Maroon" title="c23">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="c18">H</font></span>)</span> <a href="msualg_4.html#K2" title="MSUALG_4:func.2">.</a> <font color="Maroon" title="c20">s</font> ) )</span><br/>

<div><span class="kw">hereby </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="c22">a</font>,<font color="Maroon" title="c23">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="c11">R0</font> <a href="msualg_4.html#K2" title="MSUALG_4:func.2">.</a> <font color="Maroon" title="c20">s</font> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c22">a</font>,<font color="Maroon" title="c23">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="c18">H</font></span>)</span> <a href="msualg_4.html#K2" title="MSUALG_4:func.2">.</a> <font color="Maroon" title="c20">s</font> )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:3_1_9_1_1"/><span class="lab"><font color="Green" title="E24">A26</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c22">a</font>,<font color="Maroon" title="c23">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="c18">H</font></span>)</span> <a href="msualg_4.html#K2" title="MSUALG_4:func.2">.</a> <font color="Maroon" title="c20">s</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="c22">a</font>,<font color="Maroon" title="c23">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="c11">R0</font> <a href="msualg_4.html#K2" title="MSUALG_4:func.2">.</a> <font color="Maroon" title="c20">s</font></span><br/><a NAME="E2:3_1_9_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A27</font></span>: 
<font color="Maroon" title="c22">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="c20">s</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T87" onmouseover="rs('zfmisc_1/T87')" onmouseout="rh()">ZFMISC_1:87</a></span>;<br/><a NAME="E3:3_1_9_1_1"/><span class="lab"><font color="Green" title="E26">A28</font></span>: 
<font color="Maroon" title="c23">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="c20">s</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:3_1_9_1_1"><span class="lab"><font color="Green" title="E24">A26</font></span></a>, <a class="ref" href="zfmisc_1.html#T87" onmouseover="rs('zfmisc_1/T87')" onmouseout="rh()">ZFMISC_1:87</a></span>;<br/><a NAME="E4:3_1_9_1_1"/><span class="lab"><font color="Green" title="E27">A29</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c22">a</font>,<font color="Maroon" title="c23">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="c18">H</font>,<font color="Maroon" title="c20">s</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:3_1"><span class="lab"><font color="Green" title="E19">A21</font></span></a>, <a class="txt" href="#E1:3_1_9_1_1"><span class="lab"><font color="Green" title="E24">A26</font></span></a>, <a class="ref" href="msualg_4.html#D18" onmouseover="rs('msualg_4/D18')" onmouseout="rh()">MSUALG_4:def 18</a></span>;<br/><a NAME="E5:3_1_9_1_1"/><span class="lab"><font color="Green" title="E28">A30</font></span>: 
 for <font color="Olive" title="b1">Y</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b1">Y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c21">Q</font> holds <br/><span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c22">a</font>,<font color="Maroon" title="c23">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" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="3_1_9_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c24">Y</font> be    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c24">Y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c21">Q</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c22">a</font>,<font color="Maroon" title="c23">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="c24">Y</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:3_1_9_1_1_1"/>
<font color="Maroon" title="c24">Y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c21">Q</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="c22">a</font>,<font color="Maroon" title="c23">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="c24">Y</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c25">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#K6" title="PBOOLE:func.6">[|</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#K6" title="PBOOLE:func.6">|]</a></span><span class="kw"> such that </span><br/><a NAME="E3:3_1_9_1_1_1"/><span class="lab"><font color="Green" title="E29">A31</font></span>: 
<font color="Maroon" title="c24">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">s</font>
 <span class="kw">and </span><br/><a NAME="E4:3_1_9_1_1_1"/><span class="lab"><font color="Green" title="E30">A32</font></span>: 
<font color="Maroon" title="c25">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c7">CC</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:3_1_9"><span class="lab"><font color="Green" title="E21">A23</font></span></a>, <a class="txt" href="#E6:3_1_9"><span class="lab"><font color="Green" title="E23">A25</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c26">t1</font> = <font color="Maroon" title="c25">t</font> as    <a href="closure2.html#M1" title="CLOSURE2:mode.1">Element</a> of <font color="Maroon" title="c7">CC</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:3_1_9_1_1_1"><span class="lab"><font color="Green" title="E30">A32</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="c15">FF</font> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Maroon" title="c26">t1</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E7:3_1_9_1_1_1"/><span class="lab"><font color="Green" title="E31">A33</font></span>: 
<font color="Maroon" title="c27">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="c26">t1</font>
 <span class="kw">and </span><br/><a NAME="E8:3_1_9_1_1_1"/>
<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="c15">FF</font> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Maroon" title="c26">t1</font>
 <span class="kw">and </span><br/><a NAME="E9:3_1_9_1_1_1"/><span class="lab"><font color="Green" title="E32">A34</font></span>: 
 ex <font color="Olive" title="b1">R</font> being   <a href="msualg_4.html#NM4" title="MSUALG_4:NM.4">MSCongruence</a> of <font color="Maroon">F<sub>2</sub></font>() st <br/>( <font color="Maroon" title="c26">t1</font> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</a> <font color="Olive" title="b1">R</font> &amp; <font color="Maroon" title="c27">F1</font> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</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="#E32:3_1"><span class="lab"><font color="Green" title="E17">A17</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="c20">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">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="c15">FF</font>,<font color="Maroon" title="c26">t1</font>)</span>)</span> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c18">H</font></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c20">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">a</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:3_1"><span class="lab"><font color="Green" title="E20">A22</font></span></a>, <a class="txt" href="#E7:3_1_9_1_1_1"><span class="lab"><font color="Green" title="E31">A33</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="c15">FF</font>,<font color="Maroon" title="c26">t1</font>)</span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c20">s</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">H</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c20">s</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">a</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_3.html#T2" onmouseover="rs('msualg_3/T2')" onmouseout="rh()">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="c15">FF</font>,<font color="Maroon" title="c26">t1</font>)</span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c20">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="c18">H</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c20">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">a</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:3_1_9_1_1"><span class="lab"><font color="Green" title="E25">A27</font></span></a>, <a class="ref" href="funct_2.html#T15" onmouseover="rs('funct_2/T15')" onmouseout="rh()">FUNCT_2:15</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="c15">FF</font>,<font color="Maroon" title="c26">t1</font>)</span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c20">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="c18">H</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c20">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">b</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:3_1_9_1_1"><span class="lab"><font color="Green" title="E25">A27</font></span></a>, <a class="txt" href="#E3:3_1_9_1_1"><span class="lab"><font color="Green" title="E26">A28</font></span></a>, <a class="txt" href="#E4:3_1_9_1_1"><span class="lab"><font color="Green" title="E27">A29</font></span></a>, <a class="ref" href="msualg_4.html#D17" onmouseover="rs('msualg_4/D17')" onmouseout="rh()">MSUALG_4:def 17</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="c15">FF</font>,<font color="Maroon" title="c26">t1</font>)</span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c20">s</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">H</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c20">s</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">b</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:3_1_9_1_1"><span class="lab"><font color="Green" title="E26">A28</font></span></a>, <a class="ref" href="funct_2.html#T15" onmouseover="rs('funct_2/T15')" onmouseout="rh()">FUNCT_2:15</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="c15">FF</font>,<font color="Maroon" title="c26">t1</font>)</span>)</span> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c18">H</font></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c20">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">b</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_3.html#T2" onmouseover="rs('msualg_3/T2')" onmouseout="rh()">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="c20">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">b</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:3_1"><span class="lab"><font color="Green" title="E20">A22</font></span></a>, <a class="txt" href="#E7:3_1_9_1_1_1"><span class="lab"><font color="Green" title="E31">A33</font></span></a></span>
;<br/>
<span class="kw">hence </span><a NAME="E11:3_1_9_1_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c22">a</font>,<font color="Maroon" title="c23">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="c24">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:3_1_9_1_1"><span class="lab"><font color="Green" title="E25">A27</font></span></a>, <a class="txt" href="#E3:3_1_9_1_1"><span class="lab"><font color="Green" title="E26">A28</font></span></a>, <a class="txt" href="#E3:3_1_9_1_1_1"><span class="lab"><font color="Green" title="E29">A31</font></span></a>, <a class="txt" href="#E9:3_1_9_1_1_1"><span class="lab"><font color="Green" title="E32">A34</font></span></a>, <a class="ref" href="msualg_9.html#T33" onmouseover="rs('msualg_9/T33')" onmouseout="rh()">MSUALG_9:33</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E6:3_1_9_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c22">a</font>,<font color="Maroon" title="c23">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="c20">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="c20">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_9_1_1"><span class="lab"><font color="Green" title="E24">A26</font></span></a></span>;<br/><a NAME="E7:3_1_9_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="c22">a</font>,<font color="Maroon" title="c23">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#K6" title="PBOOLE:func.6">[|</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#K6" title="PBOOLE:func.6">|]</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">s</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pboole.html#D16" onmouseover="rs('pboole/D16')" onmouseout="rh()">PBOOLE:def 16</a></span>;<br/><span class="kw">hence </span><a NAME="E8:3_1_9_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c22">a</font>,<font color="Maroon" title="c23">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="c11">R0</font> <a href="msualg_4.html#K2" title="MSUALG_4:func.2">.</a> <font color="Maroon" title="c20">s</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:3_1_9"><span class="lab"><font color="Green" title="E22">A24</font></span></a>, <a class="txt" href="#E5:3_1_9_1_1"><span class="lab"><font color="Green" title="E28">A30</font></span></a>, <a class="ref" href="setfam_1.html#T43" onmouseover="rs('setfam_1/T43')" onmouseout="rh()">SETFAM_1:43</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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_9_1"/><span class="lab"><font color="Green" title="E24">A35</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c22">a</font>,<font color="Maroon" title="c23">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="c11">R0</font> <a href="msualg_4.html#K2" title="MSUALG_4:func.2">.</a> <font color="Maroon" title="c20">s</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="c22">a</font>,<font color="Maroon" title="c23">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="c18">H</font></span>)</span> <a href="msualg_4.html#K2" title="MSUALG_4:func.2">.</a> <font color="Maroon" title="c20">s</font></span><br/>

<a NAME="E3:3_1_9_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A36</font></span>: 
<font color="Maroon" title="c22">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="c20">s</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T87" onmouseover="rs('zfmisc_1/T87')" onmouseout="rh()">ZFMISC_1:87</a></span>;<br/>
<a NAME="E4:3_1_9_1"/><span class="lab"><font color="Green" title="E26">A37</font></span>: 
<font color="Maroon" title="c23">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="c20">s</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:3_1_9_1"><span class="lab"><font color="Green" title="E24">A35</font></span></a>, <a class="ref" href="zfmisc_1.html#T87" onmouseover="rs('zfmisc_1/T87')" onmouseout="rh()">ZFMISC_1:87</a></span>;<br/>
<a NAME="E5:3_1_9_1"/><span class="lab"><font color="Green" title="E27">A38</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="c7">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="c15">FF</font>,<font color="Maroon" title="c20">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="c18">H</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c20">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">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="c15">FF</font>,<font color="Maroon" title="c20">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="c18">H</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c20">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">b</font></span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="3_1_9_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c24">c</font> be    <a href="closure2.html#M1" title="CLOSURE2:mode.1">Element</a> of <font color="Maroon" title="c7">CC</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="c15">FF</font>,<font color="Maroon" title="c20">s</font>)</span>)</span>,<font color="Maroon" title="c24">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="c18">H</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c20">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">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="c15">FF</font>,<font color="Maroon" title="c20">s</font>)</span>)</span>,<font color="Maroon" title="c24">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="c18">H</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c20">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">b</font></span>)</span></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c25">t1</font> = <font color="Maroon" title="c24">c</font> as   <a href="msualg_4.html#NM4" title="MSUALG_4:NM.4">MSCongruence</a> of <font color="Maroon">F<sub>2</sub></font>() <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:3_1"><span class="lab"><font color="Green" title="E11">A9</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c26">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="c15">FF</font> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Maroon" title="c24">c</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E3:3_1_9_1_2"/><span class="lab"><font color="Green" title="E28">A39</font></span>: 
<font color="Maroon" title="c26">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="c24">c</font>
 <span class="kw">and </span><br/><a NAME="E4:3_1_9_1_2"/>
<font color="Maroon" title="c26">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="c15">FF</font> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Maroon" title="c24">c</font>
 <span class="kw">and </span><br/><a NAME="E5:3_1_9_1_2"/><span class="lab"><font color="Green" title="E29">A40</font></span>: 
 ex <font color="Olive" title="b1">R</font> being   <a href="msualg_4.html#NM4" title="MSUALG_4:NM.4">MSCongruence</a> of <font color="Maroon">F<sub>2</sub></font>() st <br/>( <font color="Maroon" title="c24">c</font> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</a> <font color="Olive" title="b1">R</font> &amp; <font color="Maroon" title="c26">F1</font> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</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="#E32:3_1"><span class="lab"><font color="Green" title="E17">A17</font></span></a></span>;<br/>
<a NAME="E6:3_1_9_1_2"/>
<font color="Maroon" title="c25">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#K6" title="PBOOLE:func.6">[|</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#K6" title="PBOOLE:func.6">|]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="closure2.html#D1" onmouseover="rs('closure2/D1')" onmouseout="rh()">CLOSURE2:def 1</a></span>;<br/>
<a NAME="E7:3_1_9_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c25">t1</font> <a href="msualg_4.html#K2" title="MSUALG_4:func.2">.</a> <font color="Maroon" title="c20">s</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="closure2.html#K5" title="CLOSURE2:func.5">|:</a><span class="default"><font color="Maroon" title="c7">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="c20">s</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:3_1_9"><span class="lab"><font color="Green" title="E23">A25</font></span></a></span>;<br/>
<a NAME="E8:3_1_9_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A41</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c22">a</font>,<font color="Maroon" title="c23">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">t1</font> <a href="msualg_4.html#K2" title="MSUALG_4:func.2">.</a> <font color="Maroon" title="c20">s</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:3_1_9"><span class="lab"><font color="Green" title="E21">A23</font></span></a>, <a class="txt" href="#E5:3_1_9"><span class="lab"><font color="Green" title="E22">A24</font></span></a>, <a class="txt" href="#E2:3_1_9_1"><span class="lab"><font color="Green" title="E24">A35</font></span></a>, <a class="ref" href="setfam_1.html#T43" onmouseover="rs('setfam_1/T43')" onmouseout="rh()">SETFAM_1:43</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="c15">FF</font>,<font color="Maroon" title="c20">s</font>)</span>)</span>,<font color="Maroon" title="c24">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="c18">H</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c20">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">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="c15">FF</font>,<font color="Maroon" title="c24">c</font>)</span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c20">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="c18">H</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c20">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">a</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="pralg_3.html#D2" onmouseover="rs('pralg_3/D2')" onmouseout="rh()">PRALG_3:def 2</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="c15">FF</font>,<font color="Maroon" title="c24">c</font>)</span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c20">s</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">H</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c20">s</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">a</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:3_1_9_1"><span class="lab"><font color="Green" title="E25">A36</font></span></a>, <a class="ref" href="funct_2.html#T15" onmouseover="rs('funct_2/T15')" onmouseout="rh()">FUNCT_2:15</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="c15">FF</font>,<font color="Maroon" title="c24">c</font>)</span>)</span> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c18">H</font></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c20">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">a</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_3.html#T2" onmouseover="rs('msualg_3/T2')" onmouseout="rh()">MSUALG_3:2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c26">F1</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c20">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">a</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:3_1"><span class="lab"><font color="Green" title="E20">A22</font></span></a>, <a class="txt" href="#E3:3_1_9_1_2"><span class="lab"><font color="Green" title="E28">A39</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c26">F1</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c20">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">b</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:3_1_9_1"><span class="lab"><font color="Green" title="E25">A36</font></span></a>, <a class="txt" href="#E4:3_1_9_1"><span class="lab"><font color="Green" title="E26">A37</font></span></a>, <a class="txt" href="#E8:3_1_9_1_2"><span class="lab"><font color="Green" title="E30">A41</font></span></a>, <a class="txt" href="#E5:3_1_9_1_2"><span class="lab"><font color="Green" title="E29">A40</font></span></a>, <a class="ref" href="msualg_9.html#T33" onmouseover="rs('msualg_9/T33')" onmouseout="rh()">MSUALG_9:33</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="c15">FF</font>,<font color="Maroon" title="c24">c</font>)</span>)</span> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c18">H</font></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c20">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">b</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:3_1"><span class="lab"><font color="Green" title="E20">A22</font></span></a>, <a class="txt" href="#E3:3_1_9_1_2"><span class="lab"><font color="Green" title="E28">A39</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="c15">FF</font>,<font color="Maroon" title="c24">c</font>)</span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c20">s</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">H</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c20">s</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">b</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_3.html#T2" onmouseover="rs('msualg_3/T2')" onmouseout="rh()">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="c15">FF</font>,<font color="Maroon" title="c24">c</font>)</span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c20">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="c18">H</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c20">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">b</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:3_1_9_1"><span class="lab"><font color="Green" title="E26">A37</font></span></a>, <a class="ref" href="funct_2.html#T15" onmouseover="rs('funct_2/T15')" onmouseout="rh()">FUNCT_2:15</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="c15">FF</font>,<font color="Maroon" title="c20">s</font>)</span>)</span>,<font color="Maroon" title="c24">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="c18">H</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c20">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">b</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="pralg_3.html#D2" onmouseover="rs('pralg_3/D2')" onmouseout="rh()">PRALG_3:def 2</a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E6:3_1_9_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c18">H</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c20">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">b</font> <a href="tarski.html#R2" title="TARSKI: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="c15">FF</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">s</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:3_1_9_1"><span class="lab"><font color="Green" title="E26">A37</font></span></a>, <a class="ref" href="funct_2.html#T5" onmouseover="rs('funct_2/T5')" onmouseout="rh()">FUNCT_2:5</a></span>;<br/>
<a NAME="E7:3_1_9_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E28">A42</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c18">H</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c20">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">b</font> <a href="tarski.html#R2" title="TARSKI: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="c15">FF</font>,<font color="Maroon" title="c20">s</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pralg_2.html#D10" onmouseover="rs('pralg_2/D10')" onmouseout="rh()">PRALG_2:def 10</a></span>;<br/>
<a NAME="E8:3_1_9_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c18">H</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c20">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">a</font> <a href="tarski.html#R2" title="TARSKI: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="c15">FF</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">s</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:3_1_9_1"><span class="lab"><font color="Green" title="E25">A36</font></span></a>, <a class="ref" href="funct_2.html#T5" onmouseover="rs('funct_2/T5')" onmouseout="rh()">FUNCT_2:5</a></span>;<br/>
<a NAME="E9:3_1_9_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c18">H</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c20">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">a</font> <a href="tarski.html#R2" title="TARSKI: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="c15">FF</font>,<font color="Maroon" title="c20">s</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pralg_2.html#D10" onmouseover="rs('pralg_2/D10')" onmouseout="rh()">PRALG_2:def 10</a></span>;<br/>
<a NAME="E10:3_1_9_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c18">H</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c20">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">H</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c20">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:3_1_9_1"><span class="lab"><font color="Green" title="E28">A42</font></span></a>, <a class="txt" href="#E5:3_1_9_1"><span class="lab"><font color="Green" title="E27">A38</font></span></a>, <a class="ref" href="msualg_9.html#T14" onmouseover="rs('msualg_9/T14')" onmouseout="rh()">MSUALG_9:14</a></span>;<br/>
<a NAME="E11:3_1_9_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="c22">a</font>,<font color="Maroon" title="c23">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="c18">H</font>,<font color="Maroon" title="c20">s</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:3_1_9_1"><span class="lab"><font color="Green" title="E25">A36</font></span></a>, <a class="txt" href="#E4:3_1_9_1"><span class="lab"><font color="Green" title="E26">A37</font></span></a>, <a class="ref" href="msualg_4.html#D17" onmouseover="rs('msualg_4/D17')" onmouseout="rh()">MSUALG_4:def 17</a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:3_1_9_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c22">a</font>,<font color="Maroon" title="c23">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="c18">H</font></span>)</span> <a href="msualg_4.html#K2" title="MSUALG_4:func.2">.</a> <font color="Maroon" title="c20">s</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:3_1"><span class="lab"><font color="Green" title="E19">A21</font></span></a>, <a class="ref" href="msualg_4.html#D18" onmouseover="rs('msualg_4/D18')" onmouseout="rh()">MSUALG_4:def 18</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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_9"/>
<span class="p1">(<span class="default"><a href="msualg_4.html#K17" title="MSUALG_4:func.17">MSCng</a> <font color="Maroon" title="c18">H</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">R0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">i</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E40:3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E21">A43</font></span>: 
 <a href="msualg_4.html#K17" title="MSUALG_4:func.17">MSCng</a> <font color="Maroon" title="c18">H</font> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</a> <font color="Maroon" title="c11">R0</font>
 
;<br/>
<a NAME="E41:3_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="c18">H</font></span>)</span>), <a href="msualg_3.html#K6" title="MSUALG_3:func.6">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="c18">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="#E37:3_1"><span class="lab"><font color="Green" title="E19">A21</font></span></a>, <a class="ref" href="msualg_4.html#T4" onmouseover="rs('msualg_4/T4')" onmouseout="rh()">MSUALG_4:4</a>, <a class="ref" href="msualg_9.html#T16" onmouseover="rs('msualg_9/T16')" onmouseout="rh()">MSUALG_9:16</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c19">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="c15">FF</font><span class="kw"> such that </span><br/><a NAME="E43:3_1"/><span class="lab"><font color="Green" title="E22">A44</font></span>: 
<font color="Maroon" title="c12">A</font>,<font color="Maroon" title="c19">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="#E40:3_1"><span class="lab"><font color="Green" title="E21">A43</font></span></a></span>;<br/>
<a NAME="E44:3_1"/>
 for <font color="Olive" title="b1">cc</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b1">cc</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c7">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> over <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="c15">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" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="3_1_10"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c20">cc</font> be    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c20">cc</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c7">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> over <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="c15">FF</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">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_10"/><span class="lab"><font color="Green" title="E23">A45</font></span>: 
<font color="Maroon" title="c20">cc</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c7">CC</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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> over <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="c15">FF</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">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="c21">c</font> = <font color="Maroon" title="c20">cc</font> as    <a href="closure2.html#M1" title="CLOSURE2:mode.1">Element</a> of <font color="Maroon" title="c7">CC</font> ;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c15">FF</font> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Maroon" title="c21">c</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c15">FF</font> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Maroon" title="c21">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">FF</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">cc</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c15">FF</font> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Maroon" title="c21">c</font>] )</span><br/>

<a NAME="E3:3_1_10"/><span class="lab"><font color="Green" title="E24">A46</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="c20">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#NM4" title="MSUALG_4:NM.4">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_10"><span class="lab"><font color="Green" title="E23">A45</font></span></a></span>;<br/>
<a NAME="E4:3_1_10"/>
 ex <font color="Olive" title="b1">R</font> being   <a href="msualg_4.html#NM4" title="MSUALG_4:NM.4">MSCongruence</a> of <font color="Maroon">F<sub>2</sub></font>() st <br/>( <font color="Olive" title="b1">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">cc</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="c20">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="#E27:3_1"><span class="lab"><font color="Green" title="E15">A13</font></span></a>, <a class="txt" href="#E1:3_1_10"><span class="lab"><font color="Green" title="E23">A45</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:3_1_10"/>
( <font color="Maroon" title="c15">FF</font> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Maroon" title="c21">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">FF</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">cc</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c15">FF</font> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Maroon" title="c21">c</font>] )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:3_1_10"><span class="lab"><font color="Green" title="E24">A46</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E45:3_1"/><span class="kw">then </span>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c19">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="E46:3_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c12">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="#E43:3_1"><span class="lab"><font color="Green" title="E22">A44</font></span></a>, <a class="ref" href="msualg_3.html#T17" onmouseover="rs('msualg_3/T17')" onmouseout="rh()">MSUALG_3:17</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c13">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="c12">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> over <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="c12">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="c12">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="c13">F</font> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</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="c12">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="c13">F</font> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</a> <font color="Olive" title="b2">G</font> holds <br/><font color="Olive" title="b3">H</font> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</a> <font color="Olive" title="b4">K</font> ) ) ) )</span><br/>

<span class="kw">thus </span><a NAME="E47:3_1"/>
<font color="Maroon" title="c13">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="c12">A</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_4.html#T3" onmouseover="rs('msualg_4/T3')" onmouseout="rh()">MSUALG_4:3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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> over <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="c12">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="c12">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="c13">F</font> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</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="c12">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="c13">F</font> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</a> <font color="Olive" title="b2">G</font> holds <br/><font color="Olive" title="b3">H</font> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</a> <font color="Olive" title="b4">K</font> ) )</span><br/>

<a NAME="E48:3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E23">A47</font></span>: 
<font color="Maroon" title="c13">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#D8" onmouseover="rs('msualg_3/D8')" onmouseout="rh()">MSUALG_3:def 8</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c20">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> over <font color="Maroon">F<sub>1</sub></font>(); <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="c20">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="c20">B</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c20">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="c12">A</font>,<font color="Maroon" title="c20">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="c12">A</font>,<font color="Maroon" title="c20">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="c13">F</font> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</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="c12">A</font>,<font color="Maroon" title="c20">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="c13">F</font> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</a> <font color="Olive" title="b1">G</font> holds <br/><font color="Olive" title="b2">H</font> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</a> <font color="Olive" title="b3">K</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c21">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="c20">B</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c21">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="c20">B</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c20">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="c12">A</font>,<font color="Maroon" title="c20">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="c12">A</font>,<font color="Maroon" title="c20">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="c13">F</font> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</a> <font color="Maroon" title="c21">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="c12">A</font>,<font color="Maroon" title="c20">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="c13">F</font> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</a> <font color="Maroon" title="c21">G</font> holds <br/><font color="Olive" title="b1">H</font> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</a> <font color="Olive" title="b2">K</font> ) ) )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E49:3_1"/><span class="lab"><font color="Green" title="E24">A48</font></span>: 
<font color="Maroon" title="c21">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="c20">B</font>
 <span class="kw">and </span><br/><a NAME="E50:3_1"/><span class="lab"><font color="Green" title="E25">A49</font></span>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c20">B</font>]
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="c12">A</font>,<font color="Maroon" title="c20">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="c12">A</font>,<font color="Maroon" title="c20">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="c13">F</font> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</a> <font color="Maroon" title="c21">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="c12">A</font>,<font color="Maroon" title="c20">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="c13">F</font> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</a> <font color="Maroon" title="c21">G</font> holds <br/><font color="Olive" title="b1">H</font> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</a> <font color="Olive" title="b2">K</font> ) )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c22">C</font> =  <a href="msualg_3.html#K6" title="MSUALG_3:func.6">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="c21">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="c20">B</font> ;<br/>
<a NAME="E52:3_1"/><span class="lab"><font color="Green" title="E26">A50</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="c21">G</font></span>)</span>),<font color="Maroon" title="c22">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="#E49:3_1"><span class="lab"><font color="Green" title="E24">A48</font></span></a>, <a class="ref" href="msualg_4.html#T4" onmouseover="rs('msualg_4/T4')" onmouseout="rh()">MSUALG_4:4</a>, <a class="ref" href="msualg_9.html#T16" onmouseover="rs('msualg_9/T16')" onmouseout="rh()">MSUALG_9:16</a></span>;<br/>
<a NAME="E53:3_1"/><span class="lab"><font color="Green" title="E27">A51</font></span>: 
 <a href="msualg_4.html#K17" title="MSUALG_4:func.17">MSCng</a> <font color="Maroon" title="c21">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" onmouseover="rs('msualg_5/D6')" onmouseout="rh()">MSUALG_5:def 6</a></span>;<br/>
<a NAME="E54:3_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c22">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="#E50:3_1"><span class="lab"><font color="Green" title="E25">A49</font></span></a></span>;<br/>
<a NAME="E55:3_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="c21">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="#E52:3_1"><span class="lab"><font color="Green" title="E26">A50</font></span></a>, <a class="ref" href="msualg_3.html#T17" onmouseover="rs('msualg_3/T17')" onmouseout="rh()">MSUALG_3:17</a></span>;<br/>
<a NAME="E56:3_1"/><span class="kw">then </span>
 <a href="msualg_4.html#K17" title="MSUALG_4:func.17">MSCng</a> <font color="Maroon" title="c21">G</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c7">CC</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E53:3_1"><span class="lab"><font color="Green" title="E27">A51</font></span></a></span>;<br/>
<a NAME="E57:3_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">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="c21">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="closure2.html#T21" onmouseover="rs('closure2/T21')" onmouseout="rh()">CLOSURE2:21</a>, <a class="ref" href="mssubfam.html#T43" onmouseover="rs('mssubfam/T43')" onmouseout="rh()">MSSUBFAM:43</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c23">H</font> being   <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon" title="c12">A</font>,<font color="Maroon" title="c20">B</font><span class="kw"> such that </span><br/><a NAME="E59:3_1"/><span class="lab"><font color="Green" title="E28">A52</font></span>: 
<font color="Maroon" title="c23">H</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon" title="c12">A</font>,<font color="Maroon" title="c20">B</font>
 <span class="kw">and </span><br/><a NAME="E60:3_1"/><span class="lab"><font color="Green" title="E29">A53</font></span>: 
<font color="Maroon" title="c21">G</font> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</a> <font color="Maroon" title="c23">H</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c13">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E49:3_1"><span class="lab"><font color="Green" title="E24">A48</font></span></a>, <a class="ref" href="msualg_9.html#T36" onmouseover="rs('msualg_9/T36')" onmouseout="rh()">MSUALG_9:36</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c23">H</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c23">H</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon" title="c12">A</font>,<font color="Maroon" title="c20">B</font> &amp; <font color="Maroon" title="c23">H</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c13">F</font> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</a> <font color="Maroon" title="c21">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="c12">A</font>,<font color="Maroon" title="c20">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="c13">F</font> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</a> <font color="Maroon" title="c21">G</font> holds <br/><font color="Maroon" title="c23">H</font> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</a> <font color="Olive" title="b1">K</font> ) )</span><br/>

<span class="kw">thus </span><a NAME="E61:3_1"/>
<font color="Maroon" title="c23">H</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon" title="c12">A</font>,<font color="Maroon" title="c20">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E59:3_1"><span class="lab"><font color="Green" title="E28">A52</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c23">H</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c13">F</font> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</a> <font color="Maroon" title="c21">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="c12">A</font>,<font color="Maroon" title="c20">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="c13">F</font> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</a> <font color="Maroon" title="c21">G</font> holds <br/><font color="Maroon" title="c23">H</font> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</a> <font color="Olive" title="b1">K</font> ) )</span><br/>

<span class="kw">thus </span><a NAME="E62:3_1"/>
<font color="Maroon" title="c21">G</font> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</a> <font color="Maroon" title="c23">H</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c13">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E60:3_1"><span class="lab"><font color="Green" title="E29">A53</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="c12">A</font>,<font color="Maroon" title="c20">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="c13">F</font> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</a> <font color="Maroon" title="c21">G</font> holds <br/><font color="Maroon" title="c23">H</font> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</a> <font color="Olive" title="b1">K</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c24">K</font> be   <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon" title="c12">A</font>,<font color="Maroon" title="c20">B</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c24">K</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c13">F</font> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</a> <font color="Maroon" title="c21">G</font> implies <font color="Maroon" title="c23">H</font> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</a> <font color="Maroon" title="c24">K</font> )</span><br/>

<span class="kw">assume </span><a NAME="E63:3_1"/>
<font color="Maroon" title="c24">K</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c13">F</font> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</a> <font color="Maroon" title="c21">G</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c23">H</font> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</a> <font color="Maroon" title="c24">K</font></span><br/>

<span class="kw">hence </span><a NAME="E64:3_1"/>
<font color="Maroon" title="c23">H</font> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</a> <font color="Maroon" title="c24">K</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E60:3_1"><span class="lab"><font color="Green" title="E29">A53</font></span></a>, <a class="txt" href="#E48:3_1"><span class="lab"><font color="Green" title="E23">A47</font></span></a>, <a class="ref" href="extens_1.html#T12" onmouseover="rs('extens_1/T12')" onmouseout="rh()">EXTENS_1:12</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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