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

<b>let </b><font color="Maroon">S</font> be  non <a href="struct_0.html#V3">empty</a> non <a href="msualg_1.html#V2">void</a>  <a href="msualg_1.html#L1">ManySortedSign</a> ;<br/><b>let </b><font color="Maroon">A</font> be  <a href="msualg_1.html#V5">non-empty</a>  <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">R</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K6">CongrLatt</a> <font color="Maroon">A</font></span>)</span>;<br/><b>let </b><font color="Maroon">F</font> be   <a href="closure2.html#NM1">SubsetFamily</a> of <span class="p1"><a href="pboole.html#K10">[|</a><span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font>,the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font></span><a href="pboole.html#K10">|]</a></span>;<br/>







<b>assume </b><a NAME="E1:40"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">R</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font>
 ;<br/>

<b>set </b><font color="Maroon">R0</font> =  <a href="mssubfam.html#K6">meet</a> <span class="p1"><a href="closure2.html#K5">|:</a><span class="default"><font color="Maroon">F</font></span><a href="closure2.html#K5">:|</a></span>;<br/>
<b>set </b><font color="Maroon">SF</font> = the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font>;<br/>
<b>set </b><font color="Maroon">I</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:40_1"/>
(  not <font color="Maroon">F</font> is <a href="xboole_0.html#V1">empty</a> or <font color="Maroon">F</font> is <a href="xboole_0.html#V1">empty</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_1"><b>suppose </b></a><a NAME="E1:40_1_1"/>
not <font color="Maroon">F</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">F1</font> = <font color="Maroon">F</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="closure2.html#NM1">SubsetFamily</a> of <span class="p1"><a href="pboole.html#K10">[|</a><span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font>,the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font></span><a href="pboole.html#K10">|]</a></span> ;<br/><a NAME="E3:40_1_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">F1</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K5">EqRelLatt</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_1_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:40_1_1_1"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F1</font>
 ;<br/>

<a NAME="E2:40_1_1_1"/><b>then </b>
<font color="Maroon">q</font> is   <a href="msualg_4.html#NM5">MSCongruence</a> of <font color="Maroon">A</font>
 
<b>by </b><i>, <a class="ref" href="msualg_5.html#D6">MSUALG_5:def 6</a></i>;<br/>
<b>hence </b><a NAME="E3:40_1_1_1"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K5">EqRelLatt</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font></span>)</span>
 <b>by </b><i><a class="ref" href="msualg_5.html#D5">MSUALG_5:def 5</a></i>;<br/>


</div><b>end;</b></div><a NAME="E4:40_1_1"/><b>then </b><i><font color="Green">E43</font></i>: 
 <a href="mssubfam.html#K6">meet</a> <span class="p1"><a href="closure2.html#K5">|:</a><span class="default"><font color="Maroon">F</font></span><a href="closure2.html#K5">:|</a></span> is  <a href="msualg_4.html#V2">MSEquivalence_Relation-like</a> <a href="msualg_4.html#NM3">ManySortedRelation</a> of the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="msualg_7.html#T9">MSUALG_7:9</a></i>;<br/><b>reconsider </b><font color="Maroon">R0</font> =  <a href="mssubfam.html#K6">meet</a> <span class="p1"><a href="closure2.html#K5">|:</a><span class="default"><font color="Maroon">F</font></span><a href="closure2.html#K5">:|</a></span> as   <a href="msualg_4.html#NM4">ManySortedRelation</a> of <font color="Maroon">A</font> <b>by </b><i>, <a class="ref" href="msualg_7.html#T9">MSUALG_7:9</a></i>;<br/><a NAME="E6:40_1_1"/>
<font color="Maroon">R0</font> is <a href="msualg_4.html#V3">MSEquivalence-like</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_1_2"><b>proof </b></a><div class="add">

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



<b>assume </b><a NAME="E1:40_1_1_2"/>
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> &amp; <font color="Maroon">R0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">R</font> )
 ;<br/>

<b>hence </b><a NAME="E2:40_1_1_2"/>
<font color="Maroon">R</font> is   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, <a class="ref" href="msualg_4.html#D3">MSUALG_4:def 3</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">R0</font> = <font color="Maroon">R0</font> as  <a href="msualg_4.html#V3">MSEquivalence-like</a> <a href="msualg_4.html#NM4">ManySortedRelation</a> of <font color="Maroon">A</font> ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">o</font> be   <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">a</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">o</font>,<font color="Maroon">A</font>, <font color="Maroon">b</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">o</font>,<font color="Maroon">A</font>;<br/><b>assume </b><a NAME="E1:40_1_1_3"/><i><font color="Green">E47</font></i>: 
for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">a</font> holds <br/><span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R0</font> <a href="msualg_4.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">n</font></span>)</span>
 ;<br/><b>set </b><font color="Maroon">r</font> =  <a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o</font>;<br/><b>consider </b><font color="Maroon">Q</font> being   <a href="setfam_1.html#NM1">Subset-Family</a> of <span class="p1">(<span class="default"><span class="p2"><a href="pboole.html#K10">[|</a><span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font>,the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font></span><a href="pboole.html#K10">|]</a></span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o</font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E3:40_1_1_3"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">Q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="closure2.html#K5">|:</a><span class="default"><font color="Maroon">F1</font></span><a href="closure2.html#K5">:|</a></span> <a href="mssubfam.html#K4">.</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o</font></span>)</span>
 <b>and </b><br/><a NAME="E4:40_1_1_3"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">R0</font> <a href="msualg_4.html#K2">.</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">Q</font>
 <b>by </b><i><a class="ref" href="mssubfam.html#D2">MSSUBFAM:def 2</a></i>;<br/><a NAME="E5:40_1_1_3"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">Q</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">s</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o</font></span>)</span></span>)</span> where <font color="Olive">s</font> is    <a href="closure2.html#M1">Element</a> of  <a href="closure2.html#K6">Bool</a> <span class="p1"><a href="pboole.html#K10">[|</a><span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font>,the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font></span><a href="pboole.html#K10">|]</a></span> : <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F1</font> </span>}</span> 
 <b>by </b><i>, <a class="ref" href="closure2.html#T15">CLOSURE2:15</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_1_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">Y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:40_1_1_3_1"/>
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font>
 ;<br/><b>then consider </b><font color="Maroon">s</font> being    <a href="closure2.html#M1">Element</a> of  <a href="closure2.html#K6">Bool</a> <span class="p1"><a href="pboole.html#K10">[|</a><span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font>,the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font></span><a href="pboole.html#K10">|]</a></span><b> such that </b><br/><a NAME="E3:40_1_1_3_1"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o</font></span>)</span>
 <b>and </b><br/><a NAME="E4:40_1_1_3_1"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F1</font>
 <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">s</font> = <font color="Maroon">s</font> as   <a href="msualg_4.html#NM5">MSCongruence</a> of <font color="Maroon">A</font> <b>by </b><i>, , <a class="ref" href="msualg_5.html#D6">MSUALG_5:def 6</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_1_3_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:40_1_1_3_1_1"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">a</font>
 ;<br/><b>set </b><font color="Maroon">t</font> = <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font>;<br/><b>consider </b><font color="Maroon">G</font> being   <a href="setfam_1.html#NM1">Subset-Family</a> of <span class="p1">(<span class="default"><span class="p2"><a href="pboole.html#K10">[|</a><span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font>,the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font></span><a href="pboole.html#K10">|]</a></span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E3:40_1_1_3_1_1"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="closure2.html#K5">|:</a><span class="default"><font color="Maroon">F1</font></span><a href="closure2.html#K5">:|</a></span> <a href="mssubfam.html#K4">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font></span>)</span>
 <b>and </b><br/><a NAME="E4:40_1_1_3_1_1"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">R0</font> <a href="msualg_4.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="mssubfam.html#D2">MSSUBFAM:def 2</a></i>;<br/><a NAME="E5:40_1_1_3_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">G</font>
 <b>by </b><i>, , </i>;<br/><a NAME="E6:40_1_1_3_1_1"/><b>then </b><i><font color="Green">E95</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K6">meet</a> <font color="Maroon">G</font>
 <b>by </b><i>, <a class="ref" href="setfam_1.html#D10">SETFAM_1:def 10</a></i>;<br/><a NAME="E7:40_1_1_3_1_1"/>
<font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">j</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font></span>)</span></span>)</span> where <font color="Olive">j</font> is    <a href="closure2.html#M1">Element</a> of  <a href="closure2.html#K6">Bool</a> <span class="p1"><a href="pboole.html#K10">[|</a><span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font>,the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font></span><a href="pboole.html#K10">|]</a></span> : <font color="Olive">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F1</font> </span>}</span> 
 <b>by </b><i>, <a class="ref" href="closure2.html#T15">CLOSURE2:15</a></i>;<br/><a NAME="E8:40_1_1_3_1_1"/><b>then </b>
<font color="Maroon">s</font> <a href="msualg_4.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E9:40_1_1_3_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">s</font> <a href="msualg_4.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">o</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E7:40_1_1_3_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<font color="Maroon">A</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<font color="Maroon">A</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>by </b><i>, <a class="ref" href="msualg_4.html#D6">MSUALG_4:def 6</a></i>;<br/></div><b>end;</b></div><a NAME="E7:40_1_1_3"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<font color="Maroon">A</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<font color="Maroon">A</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K6">meet</a> <font color="Maroon">Q</font>
 <b>by </b><i>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/><b>hence </b><a NAME="E8:40_1_1_3"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<font color="Maroon">A</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<font color="Maroon">A</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R0</font> <a href="msualg_4.html#K2">.</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="setfam_1.html#D10">SETFAM_1:def 10</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E9:40_1_1"/>
 <a href="mssubfam.html#K6">meet</a> <span class="p1"><a href="closure2.html#K5">|:</a><span class="default"><font color="Maroon">F</font></span><a href="closure2.html#K5">:|</a></span> is   <a href="msualg_4.html#NM5">MSCongruence</a> of <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="msualg_4.html#D6">MSUALG_4:def 6</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_2"><b>suppose </b></a><a NAME="E1:40_1_2"/>
<font color="Maroon">F</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><a NAME="E2:40_1_2"/><b>then </b>
<span class="p1"><a href="closure2.html#K5">|:</a><span class="default"><font color="Maroon">F</font></span><a href="closure2.html#K5">:|</a></span> <a href="pboole.html#R6">=</a>  <a href="pboole.html#K1">[0]</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
 <b>by </b><i><a class="ref" href="closure2.html#D4">CLOSURE2:def 4</a></i>;<br/><a NAME="E3:40_1_2"/><b>then </b>
 <a href="mssubfam.html#K6">meet</a> <span class="p1"><a href="closure2.html#K5">|:</a><span class="default"><font color="Maroon">F</font></span><a href="closure2.html#K5">:|</a></span> <a href="pboole.html#R6">=</a> <span class="p1"><a href="pboole.html#K10">[|</a><span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font>,the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font></span><a href="pboole.html#K10">|]</a></span>
 <b>by </b><i><a class="ref" href="mssubfam.html#T41">MSSUBFAM:41</a></i>;<br/><b>hence </b><a NAME="E4:40_1_2"/>
 <a href="mssubfam.html#K6">meet</a> <span class="p1"><a href="closure2.html#K5">|:</a><span class="default"><font color="Maroon">F</font></span><a href="closure2.html#K5">:|</a></span> is   <a href="msualg_4.html#NM5">MSCongruence</a> of <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="msualg_5.html#T20">MSUALG_5:20</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
