<?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>set </b><font color="Maroon">M</font> = the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font>;<br/>
<b>set </b><font color="Maroon">E</font> =  <a href="msualg_5.html#K5">EqRelLatt</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font>;<br/>
<b>set </b><font color="Maroon">C</font> =  <a href="msualg_5.html#K6">CongrLatt</a> <font color="Maroon">A</font>;<br/>
<b>let </b><font color="Maroon">B</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/>

<a NAME="E1:14"/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K6">CongrLatt</a> <font color="Maroon">A</font></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K5">EqRelLatt</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font></span>)</span>
 
<b>by </b><i><a class="ref" href="nat_lat.html#D16">NAT_LAT:def 16</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">B1</font> = <font color="Maroon">B</font> as   <a href="struct_0.html#NM2">Subset</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="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">B1</font> = <font color="Maroon">B1</font> as   <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> <b>by </b><i><a class="ref" href="msualg_7.html#T6">MSUALG_7:6</a></i>;<br/>
<b>set </b><font color="Maroon">d'</font> =  <a href="lattice3.html#K16">"/\"</a> <font color="Maroon">B</font>,<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>;<br/>
<b>reconsider </b><font color="Maroon">d</font> =  <a href="lattice3.html#K16">"/\"</a> <font color="Maroon">B</font>,<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> as   <a href="msualg_5.html#NM1">Equivalence_Relation</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_5.html#D5">MSUALG_5:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">d</font> = <font color="Maroon">d</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> <b>by </b><i><a class="ref" href="msualg_4.html#D5">MSUALG_4:def 5</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:14_1"/>
( <font color="Maroon">B</font> is <a href="xboole_0.html#V1">empty</a> or  not <font color="Maroon">B</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="14_1_1"><b>suppose </b></a><a NAME="E1:14_1_1"/>
<font color="Maroon">B</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">B'</font> = <font color="Maroon">B</font> as  <a href="xboole_0.html#V1">empty</a> <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/><a NAME="E3:14_1_1"/>
for <font color="Olive">q</font> being  <a href="struct_0.html#NM1">Element</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>  st <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B'</font> holds <br/> <a href="lattices.html#K6">Top</a> <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> <a href="lattices.html#R3">[=</a> <font color="Olive">q</font>
 ;<br/><a NAME="E4:14_1_1"/><b>then </b><i><font color="Green">E41</font></i>: 
 <a href="lattices.html#K6">Top</a> <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> <a href="lattice3.html#R3">is_less_than</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="lattice3.html#D16">LATTICE3:def 16</a></i>;<br/><a NAME="E5:14_1_1"/>
for <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</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>  st <font color="Olive">b</font> <a href="lattice3.html#R3">is_less_than</a> <font color="Maroon">B</font> holds <br/><font color="Olive">b</font> <a href="lattices.html#R3">[=</a>  <a href="lattices.html#K6">Top</a> <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="lattices.html#T45">LATTICES:45</a></i>;<br/><b>then </b> <a href="lattice3.html#K16">"/\"</a> <font color="Maroon">B</font>,<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> = 
 <a href="lattices.html#K6">Top</a> <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="lattice3.html#T34">LATTICE3:34</a></i>
<br/>.= 
<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="msualg_7.html#T5">MSUALG_7:5</a></i>
;<br/><b>hence </b><a NAME="E7:14_1_1"/>
 <a href="lattice3.html#K16">"/\"</a> <font color="Maroon">B</font>,<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> 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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1_2"><b>suppose </b></a><a NAME="E1:14_1_2"/><i><font color="Green">E42</font></i>: 
not <font color="Maroon">B</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><a NAME="E2:14_1_2"/>
for <font color="Olive">o</font> being  <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S</font><br/> for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="msualg_1.html#K3">Args</a> <font color="Olive">o</font>,<font color="Maroon">A</font>  st ( 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="relat_1.html#K1">dom</a> <font color="Olive">x</font> holds <br/><span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">x</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive">y</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">d</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="Olive">o</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">n</font></span>)</span> ) holds <br/><span class="p1"><a href="domain_1.html#K1">[</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="Olive">o</font>,<font color="Maroon">A</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">x</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="Olive">o</font>,<font color="Maroon">A</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">y</font></span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">d</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="Olive">o</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_2_1"><b>proof </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">x</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">y</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="E1:14_1_2_1"/><i><font color="Green">E43</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="relat_1.html#K1">dom</a> <font color="Maroon">x</font> holds <br/><span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">y</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">d</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>reconsider </b><font color="Maroon">m</font> =  <a href="mssubfam.html#K6">meet</a> <span class="p1"><a href="closure2.html#K5">|:</a><span class="default"><font color="Maroon">B1</font></span><a href="closure2.html#K5">:|</a></span> as   <a href="msualg_5.html#NM1">Equivalence_Relation</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/>
<div><i><font color="Green">E62</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_2_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">q</font> be   <a href="msualg_4.html#NM5">MSCongruence</a> of <font color="Maroon">A</font>;<br/><b>assume </b><a NAME="E1:14_1_2_1_1"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_2_1_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="E1:14_1_2_1_1_1"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">x</font>
 ;<br/><a NAME="E2:14_1_2_1_1_1"/>
<font color="Maroon">m</font> <a href="pboole.html#R2">c=</a> <font color="Maroon">q</font>
 <b>by </b><i>, <a class="ref" href="msualg_7.html#T8">MSUALG_7:8</a></i>;<br/><a NAME="E3:14_1_2_1_1_1"/><b>then </b><i><font color="Green">E90</font></i>: 
<font color="Maroon">m</font> <a href="msualg_4.html#K1">.</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="tarski.html#R1">c=</a> <font color="Maroon">q</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="pboole.html#D5">PBOOLE:def 5</a></i>;<br/><a NAME="E4:14_1_2_1_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">y</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">d</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>, </i>;<br/><a NAME="E5:14_1_2_1_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">y</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">m</font> <a href="msualg_4.html#K1">.</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="msualg_7.html#T11">MSUALG_7:11</a></i>;<br/><b>hence </b><a NAME="E6:14_1_2_1_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">y</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">q</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/>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:14_1_2_1_1"/>
<span class="p1"><a href="domain_1.html#K1">[</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">x</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">y</font></span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">q</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="msualg_4.html#D6">MSUALG_4:def 6</a></i>;<br/></div><b>end;</b></div>
<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="E5:14_1_2_1"/><i><font color="Green">E92</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">B1</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> &amp; <font color="Maroon">m</font> <a href="msualg_4.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> <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/>
<b>reconsider </b><font color="Maroon">B1'</font> = <font color="Maroon">B1</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> <b>by </b><i/>;<br/>
<a NAME="E7:14_1_2_1"/>
<span class="p1"><a href="closure2.html#K5">|:</a><span class="default"><font color="Maroon">B1'</font></span><a href="closure2.html#K5">:|</a></span> is <a href="relat_1.html#V2">non-empty</a>
 
;<br/>
<a NAME="E8:14_1_2_1"/><b>then </b><i><font color="Green">E94</font></i>: 
<font color="Maroon">Q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i>, <a class="ref" href="pboole.html#D16">PBOOLE:def 16</a></i>;<br/>
<a NAME="E9:14_1_2_1"/><i><font color="Green">E95</font></i>: 
<span class="p1"><a href="closure2.html#K5">|:</a><span class="default"><font color="Maroon">B1</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> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">x1</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">x1</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">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</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="14_1_2_1_2"><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="E1:14_1_2_1_2"/>
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="closure2.html#K5">|:</a><span class="default"><font color="Maroon">B1</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>
 ;<br/><b>then consider </b><font color="Maroon">z</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="E3:14_1_2_1_2"/><i><font color="Green">E97</font></i>: 
( <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</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> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> )
 <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">z'</font> = <font color="Maroon">z</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/><a NAME="E5:14_1_2_1_2"/>
<span class="p1"><a href="domain_1.html#K1">[</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">x</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">y</font></span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">z'</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>, </i>;<br/><b>hence </b><a NAME="E6:14_1_2_1_2"/>
<span class="p1"><a href="domain_1.html#K1">[</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">x</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">y</font></span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<a NAME="E11:14_1_2_1"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1">[</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">x</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">y</font></span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K6">meet</a> <span class="p1">(<span class="default"><span class="p2"><a href="closure2.html#K5">|:</a><span class="default"><font color="Maroon">B1</font></span><a href="closure2.html#K5">:|</a></span> <a href="mssubfam.html#K4">.</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>by </b><i>, , <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>
<a NAME="E12:14_1_2_1"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1">[</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">x</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">y</font></span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">m</font> <a href="msualg_4.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>by </b><i>, , <a class="ref" href="setfam_1.html#D10">SETFAM_1:def 10</a></i>;<br/>
<b>hence </b><a NAME="E13:14_1_2_1"/>
<span class="p1"><a href="domain_1.html#K1">[</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">x</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">y</font></span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">d</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="msualg_7.html#T11">MSUALG_7:11</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:14_1_2"/>
 <a href="lattice3.html#K16">"/\"</a> <font color="Maroon">B</font>,<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> 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><b>end;</b></div>

</div>
