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

<b>let </b><font color="Maroon" title="c1">I</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">M</font> be    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font>;<br/>



<b>set </b><font color="Maroon" title="c3">K</font> =  <a href="msualg_3.html#K2" title="MSUALG_3:func.2">id</a> <font color="Maroon" title="c2">M</font>;<br/>
<a NAME="E1:5"/>
 <a href="msualg_3.html#K2" title="MSUALG_3:func.2">id</a> <font color="Maroon" title="c2">M</font> is   <a href="msualg_5.html#NM1" title="MSUALG_5:NM.1">Equivalence_Relation</a> of <font color="Maroon" title="c2">M</font>
 
<b>by </b><i><a class="ref" href="msualg_7.html#T2" target="_self" title="MSUALG_7:th.2">Th2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c4">K</font> =  <a href="msualg_3.html#K2" title="MSUALG_3:func.2">id</a> <font color="Maroon" title="c2">M</font> as   <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#K5" title="MSUALG_5:func.5">EqRelLatt</a> <font color="Maroon" title="c2">M</font></span>)</span> <b>by </b><i><a class="ref" href="msualg_5.html#D5" title="MSUALG_5:def.5">MSUALG_5:def 5</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">a</font> be   <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#K5" title="MSUALG_5:func.5">EqRelLatt</a> <font color="Maroon" title="c2">M</font></span>)</span>;<br/><b>reconsider </b><font color="Maroon" title="c6">K'</font> = <font color="Maroon" title="c4">K</font>, <font color="Maroon" title="c7">a'</font> = <font color="Maroon" title="c5">a</font> as   <a href="msualg_5.html#NM1" title="MSUALG_5:NM.1">Equivalence_Relation</a> of <font color="Maroon" title="c2">M</font> <b>by </b><i><a class="ref" href="msualg_5.html#D5" title="MSUALG_5:def.5">MSUALG_5:def 5</a></i>;<br/><div><i><font color="Green" title="E3">A1</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:5_1_1"/><i><font color="Green" title="E4">A2</font></i>: 
<font color="Maroon" title="c8">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c9">i'</font> = <font color="Maroon" title="c8">i</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font> ;<br/><b>reconsider </b><font color="Maroon" title="c10">a2</font> = <font color="Maroon" title="c7">a'</font> <a href="msualg_4.html#K1" title="MSUALG_4:func.1">.</a> <font color="Maroon" title="c9">i'</font> as   <a href="eqrel_1.html#NM2" title="EQREL_1:NM.2">Equivalence_Relation</a> of <font color="Maroon" title="c2">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font> <b>by </b><i><a class="ref" href="msualg_4.html#D3" title="MSUALG_4:def.3">MSUALG_4:def 3</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c6">K'</font> <a href="pboole.html#K3" title="PBOOLE:func.3">/\</a> <font color="Maroon" title="c7">a'</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">K'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">a'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font></span>)</span>
<b>by </b><i><a class="txt" href="msualg_7.html#E1:5_1_1"><i><font color="Green" title="E4">A2</font></i></a>, <a class="ref" href="pboole.html#D8" title="PBOOLE:def.8">PBOOLE:def 8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font></span>)</span></span>)</span> <a href="eqrel_1.html#K2" title="EQREL_1:func.2">/\</a> <font color="Maroon" title="c10">a2</font>
<b>by </b><i><a class="ref" href="msualg_3.html#D1" title="MSUALG_3:def.1">MSUALG_3:def 1</a></i>
<br/>.= 
 <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font></span>)</span>
<b>by </b><i><a class="ref" href="eqrel_1.html#T17" title="EQREL_1:th.17">EQREL_1:17</a></i>
<br/>.= 
<font color="Maroon" title="c6">K'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font>
<b>by </b><i><a class="txt" href="msualg_7.html#E1:5_1_1"><i><font color="Green" title="E4">A2</font></i></a>, <a class="ref" href="msualg_3.html#D1" title="MSUALG_3:def.1">MSUALG_3:def 1</a></i>
;<br/></div><b>end;</b></div><b>thus </b><font color="Maroon" title="c4">K</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c5">a</font> = 
the <a href="lattices.html#U1" title="LATTICES:sel.1">L_meet</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K5" title="MSUALG_5:func.5">EqRelLatt</a> <font color="Maroon" title="c2">M</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c4">K</font>,<font color="Maroon" title="c5">a</font>
<b>by </b><i><a class="ref" href="lattices.html#D2" title="LATTICES:def.2">LATTICES:def 2</a></i>
<br/>.= 
<font color="Maroon" title="c6">K'</font> <a href="pboole.html#K3" title="PBOOLE:func.3">/\</a> <font color="Maroon" title="c7">a'</font>
<b>by </b><i><a class="ref" href="msualg_5.html#D5" title="MSUALG_5:def.5">MSUALG_5:def 5</a></i>
<br/>.= 
<font color="Maroon" title="c4">K</font>
<b>by </b><i><a class="txt" href="msualg_7.html#E2:5_1"><i><font color="Green" title="E3">A1</font></i></a>, <a class="ref" href="pboole.html#T3" title="PBOOLE:th.3">PBOOLE:3</a></i>
;<br/><b>hence </b><a NAME="E4:5_1"/>
<font color="Maroon" title="c5">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c4">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">K</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:5"/>
 <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <span class="p1">(<span class="default"><a href="msualg_5.html#K5" title="MSUALG_5:func.5">EqRelLatt</a> <font color="Maroon" title="c2">M</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msualg_3.html#K2" title="MSUALG_3:func.2">id</a> <font color="Maroon" title="c2">M</font>
 <b>by </b><i><a class="ref" href="lattices.html#D16" title="LATTICES:def.16">LATTICES:def 16</a></i>;<br/>


</div>
