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<b>let </b><font color="Maroon">L</font> be  <a href="waybel_5.html#V1" target="_self">completely-distributive</a> <a href="waybel_0.html#NM8">LATTICE</a>;<br/><b>let </b><font color="Maroon">X</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font>;<br/><b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/>





<b>set </b><font color="Maroon">A</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <font color="Olive">y</font></span>)</span> where <font color="Olive">y</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> </span>}</span> ;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:53_1"/>
( <font color="Maroon">X</font> is <a href="xboole_0.html#V1">empty</a> or  not <font color="Maroon">X</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="53_1_1"><b>suppose </b></a><a NAME="E1:53_1_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">X</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:53_1_1_1"/><i><font color="Green">E36</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <font color="Olive">y</font></span>)</span> where <font color="Olive">y</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> </span>}</span>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><b>consider </b><font color="Maroon">z</font> being    <a href="subset_1.html#M1">Element</a> of <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <font color="Olive">y</font></span>)</span> where <font color="Olive">y</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> </span>}</span> ;<br/><a NAME="E3:53_1_1_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <font color="Olive">y</font></span>)</span> where <font color="Olive">y</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="waybel_5.html#E5:53_1_1_1"><i><font color="Green">E38</font></i></a></i>;<br/><b>then consider </b><font color="Maroon">y</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E5:53_1_1_1"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">y</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 ;<br/><b>thus </b><a NAME="E6:53_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="waybel_5.html#E2"><i><font color="Green">Lemma37</font></i></a>, </i>;<br/></div><b>end;</b></div><a NAME="E3:53_1_1"/><b>then </b>
(  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font> &amp;  <a href="yellow_0.html#K1">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <font color="Olive">y</font></span>)</span> where <font color="Olive">y</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> </span>}</span> ,<font color="Maroon">L</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font> )
 <b>by </b><i><a class="txt" href="waybel_5.html#E2"><i><font color="Green">Lemma37</font></i></a>, <a class="ref" href="yellow_0.html#D11">YELLOW_0:def 11</a></i>;<br/><b>hence </b><a NAME="E4:53_1_1"/>
<font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">X</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K1">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <font color="Olive">y</font></span>)</span> where <font color="Olive">y</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> </span>}</span> ,<font color="Maroon">L</font>
 <b>by </b><i><a class="ref" href="waybel_1.html#T4">WAYBEL_1:4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_2"><b>suppose </b></a><a NAME="E1:53_1_2"/>
not <font color="Maroon">X</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:53_1_2"/>
<font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">X</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K1">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <font color="Olive">y</font></span>)</span> where <font color="Olive">y</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> </span>}</span> ,<font color="Maroon">L</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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