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<b>let </b><font color="Maroon">I</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">M</font> be    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">I</font>;<br/>



<b>set </b><font color="Maroon">K</font> = <span class="p1"><a href="pboole.html#K10">[|</a><span class="default"><font color="Maroon">M</font>,<font color="Maroon">M</font></span><a href="pboole.html#K10">|]</a></span>;<br/>
<a NAME="E1:6"/>
<span class="p1"><a href="pboole.html#K10">[|</a><span class="default"><font color="Maroon">M</font>,<font color="Maroon">M</font></span><a href="pboole.html#K10">|]</a></span> is   <a href="msualg_5.html#NM1">Equivalence_Relation</a> of <font color="Maroon">M</font>
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">K</font> = <span class="p1"><a href="pboole.html#K10">[|</a><span class="default"><font color="Maroon">M</font>,<font color="Maroon">M</font></span><a href="pboole.html#K10">|]</a></span> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K5">EqRelLatt</a> <font color="Maroon">M</font></span>)</span> <b>by </b><i><a class="ref" href="msualg_5.html#D5">MSUALG_5:def 5</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K5">EqRelLatt</a> <font color="Maroon">M</font></span>)</span>;<br/><b>reconsider </b><font color="Maroon">K'</font> = <font color="Maroon">K</font>, <font color="Maroon">a'</font> = <font color="Maroon">a</font> as   <a href="msualg_5.html#NM1">Equivalence_Relation</a> of <font color="Maroon">M</font> <b>by </b><i><a class="ref" href="msualg_5.html#D5">MSUALG_5:def 5</a></i>;<br/><div><i><font color="Green">E34</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:6_1_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 ;<br/><b>then reconsider </b><font color="Maroon">i'</font> = <font color="Maroon">i</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> ;<br/><b>consider </b><font color="Maroon">K1</font> being   <a href="msualg_4.html#NM3">ManySortedRelation</a> of <font color="Maroon">M</font><b> such that </b><br/><a NAME="E4:6_1_1"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">K1</font> <a href="pboole.html#R6">=</a> <font color="Maroon">K'</font> <a href="pboole.html#K2">\/</a> <font color="Maroon">a'</font> &amp; <font color="Maroon">K'</font> <a href="msualg_5.html#K4">"\/"</a> <font color="Maroon">a'</font> <a href="pboole.html#R6">=</a>  <a href="msualg_5.html#K3">EqCl</a> <font color="Maroon">K1</font> )
 <b>by </b><i><a class="ref" href="msualg_5.html#D4">MSUALG_5:def 4</a></i>;<br/><b>reconsider </b><font color="Maroon">K2</font> = <font color="Maroon">K'</font> <a href="msualg_4.html#K1">.</a> <font color="Maroon">i'</font>, <font color="Maroon">a2</font> = <font color="Maroon">a'</font> <a href="msualg_4.html#K1">.</a> <font color="Maroon">i'</font> as   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">M</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/><span class="p1">(<span class="default"><font color="Maroon">K'</font> <a href="pboole.html#K2">\/</a> <font color="Maroon">a'</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> = 
<span class="p1">(<span class="default"><font color="Maroon">K'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><font color="Maroon">a'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i>, <a class="ref" href="pboole.html#D7">PBOOLE:def 7</a></i>
<br/>.= 
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><font color="Maroon">a'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i>, <a class="ref" href="pboole.html#D21">PBOOLE:def 21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="eqrel_1.html#K1">nabla</a> <span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="eqrel_1.html#K3">\/</a> <font color="Maroon">a2</font>
<b>by </b><i><a class="ref" href="eqrel_1.html#D1">EQREL_1:def 1</a></i>
<br/>.= 
 <a href="eqrel_1.html#K1">nabla</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i><a class="ref" href="msualg_5.html#T4">MSUALG_5:4</a></i>
<br/>.= 
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
<b>by </b><i><a class="ref" href="eqrel_1.html#D1">EQREL_1:def 1</a></i>
<br/>.= 
<font color="Maroon">K'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i>, <a class="ref" href="pboole.html#D21">PBOOLE:def 21</a></i>
;<br/><b>hence </b><span class="p1">(<span class="default"><font color="Maroon">K'</font> <a href="msualg_5.html#K4">"\/"</a> <font color="Maroon">a'</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> = 
 <a href="msualg_5.html#K1">EqCl</a> <font color="Maroon">K2</font>
<b>by </b><i>, <a class="ref" href="msualg_5.html#D3">MSUALG_5:def 3</a></i>
<br/>.= 
<font color="Maroon">K'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i><a class="ref" href="msualg_5.html#T3">MSUALG_5:3</a></i>
;<br/><br/></div><b>end;</b></div><b>thus </b><font color="Maroon">K</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">a</font> = 
the <a href="lattices.html#U2">L_join</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K5">EqRelLatt</a> <font color="Maroon">M</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">K</font>,<font color="Maroon">a</font>
<b>by </b><i><a class="ref" href="lattices.html#D1">LATTICES:def 1</a></i>
<br/>.= 
<font color="Maroon">K'</font> <a href="msualg_5.html#K4">"\/"</a> <font color="Maroon">a'</font>
<b>by </b><i><a class="ref" href="msualg_5.html#D5">MSUALG_5:def 5</a></i>
<br/>.= 
<font color="Maroon">K</font>
<b>by </b><i>, <a class="ref" href="pboole.html#T3">PBOOLE:3</a></i>
;<br/><b>hence </b><a NAME="E4:6_1"/>
<font color="Maroon">a</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">K</font> <a href="hidden.html#R1">=</a> <font color="Maroon">K</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:6"/>
 <a href="lattices.html#K6">Top</a> <span class="p1">(<span class="default"><a href="msualg_5.html#K5">EqRelLatt</a> <font color="Maroon">M</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="pboole.html#K10">[|</a><span class="default"><font color="Maroon">M</font>,<font color="Maroon">M</font></span><a href="pboole.html#K10">|]</a></span>
 <b>by </b><i><a class="ref" href="lattices.html#D17">LATTICES:def 17</a></i>;<br/>


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